Facts on File DICTIONARY of MATHEMATICS

The Facts On File DICTIONARY of MATHEMATICS

Fourth Edition

Edited by John Daintith Richard Rennie The Facts On File Dictionary of Mathematics Fourth Edition

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This book is printed on acid-free paper. PREFACE

This dictionary is one of a series designed for use in schools. It is intended for students of mathematics, but we hope that it will also be helpful to other science students and to anyone interested in science. Facts On File also publishes dictionaries in a variety of disciplines, including biology, chemistry, forensic science, marine science, physics, space and astronomy, and weather and climate.

The Facts On File Dictionary of Mathematics was first published in 1980 and the third edition was published in 1999. This fourth edition of the dictionary has been extensively revised and extended. The dictionary now contains over 2,000 headwords covering the terminology of modern mathematics. A totally new feature of this edition is the inclusion of over 800 pronunciations for terms that are not in everyday use. A number of appendixes have been included at the end of the book containing useful information, including symbols and notation, symbols for physical quantities, areas and volumes, expansions, derivatives, integrals, trigonometric formulae, a table of powers and roots, and a Greek alphabet. There is also a list of Web sites and a bibliography. A guide to using the dictionary has also been added to this latest version of the book.

We would like to thank all the people who have cooperated in producing this book. A list of contributors is given on the acknowledgments page. We are also grateful to the many people who have given additional help and advice.

v ACKNOWLEDGMENTS

Contributors

Norman Cunliffe B.Sc. Eric Deeson M.Sc., F.C.P., F.R.A.S. Claire Farmer B.Sc. Jane Farrill Southern B.Sc., M.Sc. Carol Gibson B.Sc. Valerie Illingworth B.Sc., M.Phil. Alan Isaacs B.Sc., Ph.D. Sarah Mitchell B.A. Roger Adams B.Sc. Roger Picken B.Sc. Janet Triggs B.Sc.

Pronunciations

William Gould B.A.

vi CONTENTS

Preface v

Acknowledgments vi

Guide to Using the Dictionary viii

Pronunciation Key x

Entries A to Z 1

Appendixes I. Symbols and Notation 245 II. Symbols for Physical 248 Quantities III. Areas and Volumes 249 IV. Expansions 250 V. Derivatives 251 VI. Integrals 252 VII. Trigonometric Formulae 253 VIII. Conversion Factors 254 IX. Powers and Roots 257 X. The Greek Alphabet 260 XI. Web Sites 261

Bibliography 262

vii GUIDE TO USING THE DICTIONARY

The main features of dictionary entries are as follows.

Headwords The main term being defined is in bold type:

absolute Denoting a number or measurement that does not depend on a standard reference value.

Plurals Irregular plurals are given in brackets after the headword.

abscissa (pl. abscissas or abscissae) The horizontal or x-coordinate in a two-dimensional rectangular Cartesian coordinate system.

Variants Sometimes a word has a synonym or alternative spelling. This is placed in brackets after the headword, and is also in bold type:

angular frequency (pulsatance) Symbol: ω The number of complete rotations per unit time.

Here, ‘pulsatance’ is another word for angular frequency. Generally, the entry for the synonym consists of a simple cross-reference:

pulsatance See angular frequency.

Abbreviations Abbreviations for terms are treated in the same way as variants:

cosecant (cosec; csc) A trigonometric function of an angle equal to the reciprocal of its sine ....

The entry for the synonym consists of a simple cross-reference:

cosec See cosecant.

Multiple definitions Some terms have two or more distinct senses. These are numbered in bold type

base 1. In geometry, the lower side of a triangle, or other plane figure, or the lower face of a pyramid or other solid. 2. In a number system, the number of different symbols used, including zero.

viii Cross-references These are references within an entry to other entries that may give additional useful information. Cross-references are indicated in two ways. When the word appears in the definition, it is printed in small capitals:

Abelian group /ă-beel-ee-ăn / (commutative group) A type of GROUP in which the elements can also be related to each other in pairs by a commutative operation.

In this case the cross-reference is to the entry for ‘group’.

Alternatively, a cross-reference may be indicated by ‘See’, ‘See also’, or ‘Compare’, usually at the end of an entry:

angle of depression The angle between the horizontal and a line from an observer to an object situated below the eye level of the observer. See also angle.

Hidden entries Sometimes it is convenient to define one term within the entry for another term:

arc A part of a continuous curve. If the circumference of a circle is divided into two unequal parts, the smaller is known as the minor arc and…

Here, ‘minor arc’ is a hidden entry under arc, and is indicated by italic type. The entry for ‘minor arc’ consists of a simple cross-reference:

minor arc See arc.

Pronunciations Where appropriate pronunciations are indicated immediately after the headword, enclosed in forward slashes:

abacus /ab-ă-kŭs/ A calculating device consisting of rows of beads strung on wire and mounted in a frame.

Note that simple words in everyday language are not given pronunciations. Also headwords that are two-word phrases do not have pronunciations if the component words are pronounced elsewhere in the dictionary.

ix Pronunciation Key

Bold type indicates a stressed syllable. In pronunciations, a consonant is sometimes dou- bled to prevent accidental mispronunciation of a syllable resembling a familiar word; for example, /ass-id/ (acid), rather than /as-id/ and /ul-tră- sonn-iks/ (ultrasonics), rather than /ul-tră-son-iks/. An apostrophe is used: (a) between two consonants forming a syllable, as in /den-t’l/ (dental), and (b) between two letters when the syllable might otherwise be mis- pronounced through resembling a familiar word, as in /th’e-ră-pee/ (therapy) and /tal’k/ (talc). The symbols used are:

/a/ as in back /bak/, active /ak-tiv/ /ng/ as in sing /sing/ /ă/ as in abduct /ăb-dukt/, gamma /gam-ă/ /nk/ as in rank /rank/, bronchus /bronk-ŭs/ /ah/ as in palm /pahm/, father /fah-ther/, /o/ as in pot /pot/ /air/ as in care /kair/, aerospace /air-ŏ- /ô/ as in dog /dôg/ spays/ /ŏ/ as in buttock /but-ŏk/ /ar/ as in tar /tar/, starfish /star-fish/, heart /oh/ as in home /hohm/, post /pohst/ /hart/ /oi/ as in boil /boil/ /aw/ as in jaw /jaw/, gall /gawl/, taut /tawt/ /oo/ as in food /food/, croup /kroop/, fluke /ay/ as in mania /may-niă/ ,grey /gray/ /flook/ /b/ as in bed /bed/ /oor/ as in pruritus /proor-ÿ-tis/ /ch/ as in chin /chin/ /or/ as in organ /or-găn/, wart /wort/ /d/ as in day /day/ /ow/ as in powder /pow-der/, pouch /e/ as in red /red/ /powch/ /ĕ/ as in bowel /bow-ĕl/ /p/ as in pill /pil/ /ee/ as in see /see/, haem /heem/, caffeine /r/ as in rib /rib/ /kaf-een/, baby /bay-bee/ /s/ as in skin /skin/, cell /sel/ /eer/ as in fear /feer/, serum /seer-ŭm/ /sh/ as in shock /shok/, action /ak-shŏn/ /er/ as in dermal /der-măl/, labour /lay-ber/ /t/ as in tone /tohn/ /ew/ as in dew /dew/, nucleus /new-klee-ŭs/ /th/ as in thin /thin/, stealth /stelth/ /ewr/ as in epidural /ep-i-dewr-ăl/ /th/ as in then /then/, bathe /bayth/ /f/ as in fat /fat/, phobia /foh-biă/, rough /u/ as in pulp /pulp/, blood /blud/ /ruf/ /ŭ/ as in typhus /tÿ-fŭs/ /g/ as in gag /gag/ /û/ as in pull /pûl/, hook /hûk/ /h/ as in hip /hip/ /v/ as in vein /vayn/ /i/ as in fit /fit/, reduction /ri-duk-shăn/ /w/ as in wind /wind/ /j/ as in jaw /jaw/, gene /jeen/, ridge /rij/ /y/ as in yeast /yeest/ /k/ as in kidney /kid-nee/, chlorine /klor- /ÿ/ as in bite /bÿt/, high /hÿ/, hyperfine /hÿ- een/, crisis /krÿ-sis/ per-fÿn/ /ks/ as in toxic /toks-ik/ /yoo/ as in unit /yoo-nit/, formula /form- /kw/ as in quadrate /kwod-rayt/ yoo-lă/ /l/ as in liver /liv-er/, seal /seel/ /yoor/ as in pure /pyoor/, ureter /yoor-ee- /m/ as in milk /milk/ ter/ /n/ as in nit /nit/ /ÿr/ as in fire /fÿr/

x A

Abacus: the number 3258 is shown on the right-hand side.

abacus /ab-ă-kŭs/ A calculating device abscissa /ab-siss-ă/ (pl. abscissas or ab- consisting of rows of beads strung on wire scissae) The horizontal or x-coordinate in a and mounted in a frame. An abacus with two-dimensional rectangular Cartesian co- nine beads in each row can be used for ordinate system. See Cartesian coordi- counting in ordinary arithmetic. The low- nates. est wire counts the digits, 1, 2, … 9, the next tens, 10, 20, … 90, the next hundreds, absolute Denoting a number or measure- 100, 200, … 900, and so on. The number ment that does not depend on a standard 342, for example, would be counted out reference value. For example, absolute by, starting with all the beads on the right, density is measured in kilograms per cubic pushing two beads to the left hand side of meter but relative density is the ratio of the bottom row, four to the left of the sec- density to that of a standard density (i.e. ond row, and three to the left of the third the density of a reference substance under row. Abaci, of various types, are still used standard conditions). Compare relative. for calculating in some countries; experts with them can perform calculations very absolute convergence The convergence rapidly. of the sum of the absolute values of terms in a series of positive and negative terms. Abelian group /ă-beel-ee-ăn / (commuta- For example, the series: tive group) A type of GROUP in which the 1 – (1/2)2 + (1/3)3 – (1/4)4 + … elements can also be related to each other is absolutely convergent because in pairs by a commutative operation. For 1 + (1/2)2 + (1/3)3 + (1/4)4 + … example, if the operation is multiplication is also convergent. A series that is conver- and the elements are rational numbers, gent but has a divergent series of absolute then the set is an Abelian group because for values is conditionally convergent. For ex- any two elements a and b, a × b = b × a, and ample, all three numbers, a, b, and a × b are el- 1 – 1/2 + 1/3 – 1/4 + … ements in the set. All cyclic groups are is conditionally convergent because Abelian groups. See also cyclic group. 1 + 1/2 + 1/3 + 1/4 + …

1 absolute error is divergent. See also convergent series. acceleration due to gravity See acceleration of free fall. absolute error The difference between the measured value of a quantity and its acceleration of free fall (acceleration due true value. Compare relative error. See also to gravity) Symbol: g The constant accel- error. eration of a mass falling freely (without friction) in the Earth’s gravitational field. absolute maximum See maximum point. The acceleration is toward the surface of the Earth. g is a measure of gravitational absolute minimum See minimum point. field strength – the force on unit mass. The force on a mass m is its weight W, where W absolute value The MODULUS of a real = mg. number or of a complex number. For ex- The value of g varies with distance from ample, the absolute value of –2.3, written the Earth’s surface. Near the surface it is |–2.3|, is 2.3. The absolute value of a com- just under 10 meters per second per second –2 plex number is also the modulus; for ex- (9.806 65 m s is the standard value). It ample the absolute value of 2 + 3i is √(22 + varies with latitude, in part because the 32). Earth is not perfectly spherical (it is flat- tened near the poles). abstract algebra See algebra; algebraic structure. acceptance region When considering a hypothesis, the sample space is divided into abstract number A number regarded two regions – the acceptance region and simply as a number, without reference to the rejection region (or critical region). The any material objects or specific examples. acceptance region is the one in which the sample must lie if the hypothesis is to be ac- For example, the number ‘three’ when it cepted. does not refer to three objects, quantities, etc., but simply to the abstract concept of access time The time needed for the read- ‘three’. ing out of, or writing into, the memory of a computer, i.e. the time it takes for the acceleration Symbol: a The rate of memory to transfer data from or to the change of speed or velocity with respect to CPU (see central processor). time. The SI unit is the meter per second –2 per second (m s ). A body moving in a accumulation point (cluster point) For a straight line with increasing speed has a given set S, a point that can be approached positive acceleration. A body moving in a arbitrarily closely by members of that set. curved path with uniform (constant) speed Another way of saying this is that an accu- also has an acceleration, since the velocity mulation point is the limit of a sequence of (a vector depending on direction) is chang- points in the set. An accumulation point of ing. In the case of motion in a circle the ac- a set need not necessarily be a member of 2 celeration is v /r directed toward the center the set itself, although it can be. For exam- of the circle (radius r). ple, any rational number is an accumula- For constant acceleration: tion point of the set of rationals. But 0 is an a = (v2 – v1)/t accumulation point of the set {1,½,¼,⅛,… v1 is the speed or velocity when timing } although it is not itself a member of the starts; v2 is the speed or velocity after time set. t. (This is one of the equations of motion.) This equation gives the mean acceleration accuracy The number of significant fig- over the time interval t. If the acceleration ures in a number representing a measure- is not constant ment or value of a quantity. If a length is a = dv/dt, or d2x/dt2 written as 2.314 meters, then it is normally See also Newton’s laws of motion. assumed that all of the four figures are

2 adjacent meaningful, and that the length has been tions in each MATRIX are added arithmeti- measured to the nearest millimeter. It is in- cally. correct to write a number to a precision of, for example, four significant figures when addition formulae Equations that ex- the accuracy of the value is only three sig- press trigonometric functions of the sum or nificant figures, unless the error in the esti- difference of two angles in terms of sepa- mate is indicated. For example, 2.310 ± rate functions of the angles. 0.005 meters is equivalent to 2.31 meters. sin(x + y) = sinx cosy + cosx siny sin(x – y) = sinx cosy – cosx siny acre A unit of area equal to 4840 square cos(x + y) = cosx cosy – sinx siny yards. It is equivalent to 0.404 68 hectare. cos(x – y) = cosx cosy + sinx siny tan(x + y) = (tanx + tany)/(1 – tanx action An out-dated term for force. See tany) reaction. tan(x – y) = (tanx – tany)/(1 + tanx tany) action at a distance An effect in which They are used to simplify trigonometric ex- one body affects another through space pressions, for example, in solving an equa- with no apparent contact or transfer be- tion. From the addition formulae the tween them. See field. following formulae can be derived: The double-angle formulae: actuary An expert in statistics who calcu- sin(2x) = 2 sinx cosx lates insurance risks and relates them to the cos(2x) = cos2x – sin2x premiums to be charged. tan(2x) = 2tanx/(1 – tan2x) The half-angle formulae: acute Denoting an angle that is less than a sin(x/2) = ±√[(1 – cosx)/2] right angle; i.e. an angle less than 90° (or cos(x/2) = ±√[(1 + cosx)/2] π/2 radian). Compare obtuse; reflex. tan(x/2) = sinx/(1 + cosx) = (1 – addend /ad-end, ă-dend/ One of the num- cosx)/sinx The product formulae: bers added together in a sum. See also au- ½ gend. sinx cosy = [sin(x + y) + sin(x – y)] cosx siny = ½[sin(x + y) – sin(x – y)] ½ adder The circuitry in a computer that cosx cosy = [cos(x + y) + cos(x – y)] ½ adds digital signals (i.e. the ADDEND, AU- sinx siny = [cos(x – y) – cos(x + y)] GEND and carry digit) to produce a sum and a carry digit. addition of fractions See fractions. addition Symbol: + The operation of find- addition of matrices See matrix. ing the SUM of two or more quantities. In arithmetic, the addition of numbers is com- address See store. mutative (4 + 5 = 5 + 4), associative (2 + (3 + 4) = (2 + 3) + 4), and the identity element ad infinitum /ad in-fă-nÿ-tŭn/ To infinity; is zero (5 + 0 = 5). The inverse operation to an infinite number of times. Often abbrevi- addition is subtraction. In vector addition, ated to ad inf. the direction of the two vectors affects the sum. Two vectors are added by placing adjacent 1. Denoting one of the sides them head-to-tail to form two sides of a tri- forming a given angle in a triangle. In a angle. The length and direction of the third right-angled triangle it is the side between side is the VECTOR SUM. Matrix addition the given angle and the right angle. In can only be carried out between matrices trigonometry, the ratios of this adjacent with the same number of rows and side to the other side lengths are used to de- columns, and the sum has the same dimen- fine the cosine and tangent functions of the sions. The elements in corresponding posi- angle.

3 adjoint

2. Denoting two sides of a polygon that algebra The branch of mathematics in share a common vertex. which symbols are used to represent num- 3. Denoting two angles that share a com- bers or variables in arithmetical opera- mon vertex and a common side. tions. For example, the relationship: 4. Denoting two faces of a polyhedron that 3 × (4 + 2) = (3 × 4) + (3 × 2) share a common edge. belongs to arithmetic. It applies only to this particular set of numbers. On the other adjoint (of a matrix) See cofactor. hand the equation: x(y + z) = xy + xz admissible hypothesis Any hypothesis is an expression in algebra. It is true for any that could possibly be true; in other three numbers denoted by x, y, and z. The words, an hypothesis that has not been above equation is a statement of the dis- ruled out. tributive law of arithmetic; similar statements can be written for the associative aether /ee-th’er/ See ether. and commutative laws. Much of elementary algebra consists of affine geometry /ă-fÿn/ The study of methods of manipulating equations to put properties left invariant under the group of them in a more convenient form. For ex- affine transformations. See affine transfor- ample, the equation: mation; geometry. x + 3y = 15 can be changed by subtracting 3y from affine transformation A transformation both sides of the equation, giving: of the form x + 3y – 3y = 15 – 3y ′ x = 15 – 3y x = a1x + b1y + c1, y′ = a x + b y + c The effect is that of moving a term (+3y) 2 2 2 from one side of the equation to the other where a b – a b ≠ 0. An affine transfor- 1 2 2 1 and changing the sign. Similarly a multipli- mation maps parallel lines into parallel cation on one side of the equation becomes lines, finite points into finite points, leaves a division when the term is moved to the the line at infinity fixed, and preserves the other side; for example: ratio of the distances separating three xy = 5 points that lie on a straight line. An affine becomes: transformation can always be factored into x = 5/y the product of the following important ‘Ordinary’ algebra is a generalization of special cases: ′ ′ arithmetic. Other forms of higher algebra 1. translations: x = x + a, y = y + b also exist, concerned with mathematical ′ θ θ ′ θ 2. rotations: x = xcos + ysin , y = –xsin entities other than numbers. For example, θ + ycos matrix algebra is concerned with the rela- ′ ′ 3. stretchings or shrinkings: x = tx, y = ty tions between matrices; vector algebra ′ 4. reflections in the x-axis or y-axis: x = x, with vectors; Boolean algebra is applicable ′ ′ ′ y = –y or x = –x, y = y to logical propositions and to sets; etc. An ′ ′ 5. elongations or compressions: x = x, y = algebra consists of a number of mathemat- ′ ′ ty or x = tx, y = y ical entities (e.g. matrices or sets) and operations (e.g. addition or set inclusion) with aleph /ah-lef, ay-/ The first letter of the formal rules for the relationships between Hebrew alphabet, used to denote transfi- the mathematical entities. Such a system is ' nite cardinal numbers. 0, the smallest called an algebraic structure. transfinite cardinal number, is the number ' of elements in the set of integers. 1, is the algebra, Boolean See Boolean algebra. ' number of subsets of any set with 0, mem- ' bers. In general n+1 is defined in the same algebraic structure A structure imposed way as the number of subsets of a set with on elements of a set by certain operations ' n members. that act on or combine the elements. The 4 ambiguous combination of the set and the operations tangent to a CIRCLE and a chord drawn satisfy particular axioms that define the from the point of contact of the tangent is structure. Examples of algebraic structures equal to any angle subtended by the chord are FIELDS, GROUPS, and RINGS. The study of in the alternate segment of the circle, where algebraic structure is sometimes called ab- the alternative segment is on the side of the stract algebra. chord opposite (alternate to) the angle. algorithm /al-gŏ-rith-’m/ A mechanical alternating series A series in which the procedure for performing a given calcula- terms are alternately positive and negative, tion or solving a problem in a series of for example: n steps. One example is the common method Sn = –1 + 1/2 – 1/3 + 1/4 … + (–1) /n of long division in steps. Another is the Eu- Such a series is convergent if the absolute clidean algorithm for finding the highest value of each term is less than the preced- common factor of two positive integers. ing one. The example above is a convergent series. allometry /ă-lom-ĕ-tree/ A relation be- An alternating series can be constructed tween two variables that can be expressed from the sum of two series, one with posi- by the equation tive terms and one with negative terms. In y = axb this case, if both are convergent separately where x and y are the variables, a is a con- then the alternating series is also conver- stant and b is a growth coefficient. it is gent, even if the absolute value of each used to describe the systematic growth of term is not always smaller than the one be- an organism, in which case y is the mass of fore it. For instance, the series: n a particular part of the organism and x is S1 = 1/2 + 1/4 + 1/8 + … + 1/2 its total mass. and S2 = –1/2 – 1/3 – 1/4 – 1/5 – …(–1)/(n + 1) alphanumeric Describing any of the are both convergent, and so their sum: characters (or their codes) that stand for Sn = S1 + S2 the letters of the alphabet or numerals, es- = 1/2 – 1/2 + 1/4 – 1/3 + 1/8 – 1/4 + pecially in computer science. Punctuation … marks and mathematical symbols are not is also convergent. regarded as alphanumeric characters. alternation See disjunction. alternate angles A pair of equal angles formed by two parallel lines and a third altitude The perpendicular distance from line crossing both. For example, the two the base of a figure (e.g. a triangle, pyra- acute angles in the letter Z are alternate an- mid, or cone) to the vertex opposite. gles. Compare corresponding angles. ambiguous Having more than one possi- alternate-segment theorem A result in ble meaning, value, or solution. For exam- geometry stating that the angle between a ple, an ambiguous case occurs in finding

x •

• x

Alternate angles: alternate angles formed by a line cutting two parallel lines.

5 ampere the sides and angles of a triangle when two record. Alternatively it can be used to con- sides and an angle other then the included trol the process that produces the data en- angle are known. One solution is an acute- tering the computer. angled triangle and the other is an obtuse- Analog computers operate in real time angled triangle. and are used, for example, in the automatic control of certain industrial processes and ampere /am-pair/ Symbol: A The SI base in a variety of scientific experiments. They unit of electric current, defined as the con- can perform much more complicated stant current that, maintained in two mathematics than digital computers but straight parallel infinite conductors of neg- are less accurate and are less flexible in the ligible circular cross-section placed one kind of things they can do. See also hybrid meter apart in vacuum, would produce a computer. force between the conductors of 2 × 10–7 newton per meter. The ampere is named analog/digital converter A device that for the French physicist André Marie Am- converts analog signals (the output from père (1775–1836). an ANALOG COMPUTER) into digital signals, so that they can be dealt with by a digital amplitude The maximum value of a vary- computer. See computer. ing quantity from its mean or base value. In the case of a simple harmonic motion – a analog electronics A branch of electron- wave or vibration – it is half the maximum ics in which inputs and outputs can have a peak-to-peak value. range of voltages rather than fixed values. A frequently used circuit in analog elec- analog computer A type of COMPUTER in tronics is the operational amplifier, so which numerical information (generally called because it can perform mathematical called data) is represented in the form of a operations such as differentiation and inte- quantity, usually a voltage, that can vary gration. continuously. This varying quantity is an analog of the actual data, i.e. it varies in the analogy A general similarity between two same manner as the data, but is easier to problems or methods. Analogy is used to manipulate in the mathematical operations indicate the results of one problem from performed by the analog computer. The the known results of the other. data is obtained from some process, experiment, etc.; it could be the changing tem- analysis The branch of mathematics con- perature or pressure in a system or the cerned with the limit process and the con- varying speed of flow of a liquid. There cept of convergence. It includes the theory may be several sets of data, each repre- of differentiation, integration, infinite se- sented by a varying voltage. ries, and analytic functions. Traditionally, The data is converted into its voltage it includes the study of functions of real analog or analogs and calculations and and complex variables arising from differ- other sorts of mathematical operations, es- ential and integral calculus. pecially the solution of differential equations, can then be performed on the analytic A function of a real or complex voltage(s) (and hence on the data they rep- variable is analytic (or holomorphic) at a resent). This is done by the user selecting a point if there is a neighborhood N of this group of electronic devices in the computer point such that the function is differen- to which the voltage(s) are to be applied. tiable at every point of N. An alternative These devices rapidly add voltages, and (and equivalent) definition is that a func- multiply them, integrate them, etc., as re- tion is analytic at a point if it can be repre- quired. The resulting voltage is propor- sented in a neighborhood of this point by tional to the result of the operation. It can its Taylor series about it. A function is said be fed to a recording device to produce a to be analytic in a region if it is analytic at graph or some other form of permanent every point of that region.

6 anharmonic oscillator analytical geometry (coordinate geome- astronomer Anders Jonas Ångstrom try) The use of coordinate systems and al- (1814–74). gebraic methods in geometry. In a plane Cartesian coordinate system a point is rep- angular acceleration Symbol: α The ro- resented by a set of numbers and a curve is tational acceleration of an object about an an equation for a set of points. The geo- axis; i.e. the rate of change of angular ve- metric properties of curves and figures can locity with time: thus be investigated by algebra. Analytical α = dω/dt geometry also enables geometrical inter- or pretations to be given to equations. α = d2θ/dt2 where ω is angular velocity and θ is angu- anchor ring See torus. lar displacement. Angular acceleration is analogous to linear acceleration. See rota- and See conjunction. tional motion.

AND gate See logic gate. angular displacement Symbol: θ The rotational displacement of an object about an angle (plane angle) The spatial relation- axis. If the object (or a point on it) moves ship between two straight lines. If two lines from point P1 to point P2 in a plane per- θ are parallel, the angle between them is pendicular to the axis, is the angle P1OP2, zero. Angles are measured in degrees or, al- where O is the point at which the perpen- ternatively, in radians. A complete revolu- dicular plane meets the axis. See also rotation is 360 degrees (360° or 2π radians). A tional motion. straight line forms an angle of 180° (π radians) and a right angle is 90° (π/2 radi- angular frequency (pulsatance) Symbol: ans). ω The number of complete rotations per The angle between a line and a plane is unit time. Angular frequency is often used the angle between the line and its orthogo- to describe vibrations. Thus, a simple har- nal projection on the plane. monic motion of frequency f can be repre- The angle between two planes is the sented by a point moving in a circular path angle between lines drawn perpendicular at constant speed. The foot of a perpendic- to the common edge from a point – one line ular from the point to a diameter of the cir- in each plane. The angle between two in- cle moves with simple harmonic motion. tersecting curves is the angle between their The angular frequency of this motion is tangents at the point of intersection. equal to 2πf, where f is the frequency of the See also solid angle. simple harmonic motion. The unit of angular frequency, like frequency, is the hertz. angle of depression The angle between the horizontal and a line from an observer angular momentum Symbol: L The to an object situated below the eye level of product of the moment of inertia of a body the observer. See also angle. and its angular velocity. Angular momentum is analogous to linear momentum, mo- angle of elevation The angle between the ment of inertia being the equivalent of horizontal and a line from an observer to mass for ROTATIONAL MOTION. an object situated above the eye level of the observer. See also angle of depression. angular velocity Symbol: ω The rate of change of angular displacement with time: ångstrom /ang-strŏm/ Symbol: Å A unit ω = dθ/dt. See also rotational motion. of length defined as 10–10 meter. The ångstrom is sometimes used for expressing anharmonic oscillator /an-har-mon-ik/ wavelengths of light or ultraviolet radia- A system whose vibration, while still peri- tion or for the sizes of molecules. The unit odic, cannot be described in terms of sim- is named for the Swedish physicist and ple harmonic motions (i.e. sinusoidal

7 annuity

straight angle

obtuse angle

reflex angle

acute angle

right angle

reflex angle

Angle: types of angle

8 Apollonius’ circle

equal to f(x) for all values of x in the domain of f. The function g(x) is said to be the antiderivative of f(x). The indefinite in- R tegral ∫f(x)dx does not specify all the anti- derivatives of f(x) since an arbitrary r constant c can be added to any antiderivative. Thus, if both g1(x) and g2(x) are anti- derivatives of a continuous function f(x) then g1(x) and g2(x) can only differ by a constant.

antilogarithm /an-tee-lôg-ă-rith-logarithms, the antilogarithm of x is 10x. In natural loga- x Annulus (shown shaded) rithms, the antilogarithm of x is e .

antinode /an-tee-nohd/ A point of maxi- motions). In such cases, the period of oscil- mum vibration in a stationary wave pat- lation is not independent of the amplitude. tern. Compare node. See also stationary wave. annuity A pension in which an insurance company pays the annuitant fixed regular antinomy /an-tin-ŏ-mee/ See paradox. sums of money in return for sums of money paid to it either in installments or as a lump antiparallel /an-tee-pa-ră-lel/ Parallel but sum. An annuity certain is paid for a fixed acting in opposite directions, said of vec- number of years as opposed to an annuity tors. that is payable only while the annuitant is alive. antisymmetric matrix /an-tee-să-met-rik/ (skew-symmetric matrix) A square matrix annulus /an-yŭ-lŭs/ (pl. annuli or annu- A that satisfies the relation AT = –A, where luses) The region between two concentric AT is the TRANSPOSE of A. The definition of π 2 2 circles. The area of an annulus is (R – r ), an antisymmetric matrix means that all en- where R is the radius of the larger circle tries aij of the matrix have to satisfy aij = and r is the radius of the smaller. –aji for all the i and j in the matrix. This, in turn, means that all diagonal entries in the In logic, the first part of a antecedent matrix must have a value of zero, i.e. a = conditional statement; a proposition or ii 0 for all i in the matrix. statement that is said to imply another. For example, in the statement ‘if it is raining apex The point at the top of a solid, such then the streets are wet’, ‘it is raining’ is the antecedent. Compare consequent. See also as a pyramid, or of a plane figure, such as implication. a triangle. anticlockwise (counterclockwise) Rotat- Apollonius’ circle /ap-ŏ-loh-nee-ŭs/ A ing in the opposite sense to the hands of a circle defined as the locus of all points P clock. See clockwise. that satisfy the relation AP/BP = c, where A and B are points in a plane and c is a con- antiderivative /an-tee-dĕ-riv-ă-tiv/ A stant. In the case of c = 1 a straight line is function g(x) that is related to a real func- obtained. This case can be considered to be tion f(x) by the fact that the derivative of a particular case of a circle or can be ex- g(x) with respect to x, denoted by g′(x), is plicitly left out as a special case.

9 Apollonius’ theorem

P clidean geometry in surveying, architecture, navigation, or science is applied geometry. The term ‘applied mathematics’ is used especially for mechanics – the study AB of forces and motion. Compare pure mathematics.

approximate Describing a value of some Apollonius’ circle quantity that is not exact but close enough to the correct value for some specific purpose, as within certain boundaries of error. It is also used as a verb meaning ‘to find the The circle is named for the Greek mathe- value of a quantity within certain bounds matician Apollonius of Perga (c. 261 BC–c. of accuracy, but not exactly’. For example, 190 BC). one can approximate an irrational number, such as π, by finding its decimal expansion Apollonius’ theorem /ap-ŏ-loh-nee-ŭs/ to a certain number of places. The equation that relates the length of a median in a triangle to the lengths of its approximate integration Any of various sides. If a is the length of one side and b is techniques for finding an approximate the length of another, and the third side is value for a definite integral. There are divided into two equal lengths c by a me- many integrals that cannot be evaluated dian of length m, then: exactly and approximation techniques are a2 + b2 = 2m2 + 2c2 used to estimate values for such integrals. Both analytical and numerical approxi- apothem /ap-ŏ-th’em/ (short radius) A mate integration techniques exist. In some line segment from the center of a regular analytical approximate integration tech- polygon perpendicular to the center of a niques the value of the integral is expressed side. as an ASYMPTOTIC SERIES. Two examples of numerical approximate integration are applications program A computer pro- SIMPSON’S RULE and the TRAPEZIUM RULE. gram designed to be used for a specific pur- See also numerical integration. pose (such as stock control or word processing). approximately equal to Symbol ≅ A symbol used to relate two quantities in applied mathematics The study of the which one of the quantities is a good ap- mathematical techniques that are used to proximation to the other quantity but is solve problems. Strictly speaking it is the not exactly equal to it. An example of the application of mathematics to any ‘real’ use of this symbol is π≅22/7. system. For instance, pure geometry is the study of entities – lines, points, angles, etc. approximation /ă-proks-ă-may-shŏn/ A – based on certain axioms. The use of Eu- calculation of a quantity that gives values

b a m

c c

Apollonius’ theorem: a2 + b2 = 2m2 + 2c2

10 arc tangent

arc cotangent (arc cot) An inverse cotan- major gent. See inverse trigonometric functions. arc arc coth An inverse hyperbolic cotangent. See inverse hyperbolic functions.

minor arc Archimedean solid /ar-kă-mee-dee-ăn, - m…brevel-dee-ăn/ See semi-regular polyhedron. Archimedean solids are named for the Greek mathematician Archimedes (287 BC–212 BC).

Archimedean spiral A particular type of Arc: major and minor arcs of a circle. SPIRAL that is described in POLAR COORDI- NATES by the equation r = aθ, where a is a positive constant. It can be considered to represent the locus of a point moving along that are not, in general, exact but are close a radius vector with a uniform velocity to the exact values. There are many inte- while the radius vector itself is moving grals and differential equations in mathe- about a pole with a constant angular ve- matics and its physical applications that locity. A spiral of this type asymptotically cannot be solved exactly and require the approaches a circle of radius a. use of approximation techniques. See Newton’s method; numerical integration; Archimedes’ principle /ar-kă-mee-deez/ Simpson’s rule; trapezium rule. The upward force of an object totally or partly submerged in a fluid is equal to the apse /aps/ Any point on the orbit at which weight of fluid displaced by the object. The the motion of the orbiting body is at right upward force, often called the upthrust, re- angles to the central radius vector of the sults from the fact that the pressure in a orbit. The distance from an apse to the cen- fluid (liquid or gas) increases with depth. If tre of motion (the apsidal distance) equals the object displaces a volume V of fluid of the maximum or minimum value of the RA- density ρ, then: DIUS VECTOR. upthrust u = Vρg where g is the acceleration of free fall. If the arc A part of a continuous curve. If the cir- upthrust on the object equals the object’s cumference of a circle is divided into two weight, the object will float. unequal parts, the smaller is known as the minor arc and the larger is known as the (arc sec) An inverse secant. See major arc. arc secant inverse trigonometric functions. arc cosecant (arc cosec; arc csc) An inverse cosecant. See inverse trigonometric arc sech An inverse hyperbolic secant. See functions. inverse hyperbolic functions. arc cosech An inverse hyperbolic cose- arc sine (arc sin) An inverse sine. See in- cant. See inverse hyperbolic functions. verse trigonometric functions. arc cosh An inverse hyperbolic cosine. See arc sinh An inverse hyperbolic sine. See inverse hyperbolic functions. inverse hyperbolic functions. arc cosine (arc cos) An inverse cosine. See arc tangent (arc tan) An inverse tangent. inverse trigonometric functions. See inverse trigonometric functions.

11 arc tanh

A curved area can be found by dividing it into rectangles and adding the areas of the rectangles. The more rectangles, the better the approximation. arc tanh An inverse hyperbolic tangent. solve problems containing numerical infor- See inverse hyperbolic functions. mation. It also involves an understanding of the structure of the number system and are /air/ A metric unit of area equal to 100 the facility to change numbers from one square meters. It is equivalent to 119.60 sq form to another; for example, the changing yd. See also hectare. of fractions to decimals, and vice versa. area Symbol: A The extent of a plane fig- arithmetic and logic unit (ALU) See ure or a surface, measured in units of central processor. length squared. The SI unit of area is the square meter (m2). The area of a rectangle arithmetic mean See mean. is the product of its length and breadth. The area of a triangle is the product of the arithmetic sequence (arithmetic progres- altitude and half the base. Closed figures sion) A SEQUENCE in which the difference bounded by straight lines have areas that between each term and the one after it is can be determined by subdividing them constant, for example, {9, 11, 13, 15, …}. into triangles. Areas for other figures can The difference between successive terms is be found by using integral calculus. called the common difference. The general formula for the nth term of an arithmetic Argand diagram /ar-gănd/ See complex sequence is: number. nn = a + (n – 1)d where a is the first term of the sequence argument (amplitude) 1. In a complex and d is the common difference. Compare number written in the form r(cosθ + i sinθ), geometric sequence. See also arithmetic se- the angle θ is the argument. It is therefore ries. the angle that the vector representing the complex number makes with the horizon- arithmetic series A SERIES in which the tal axis in an Argand diagram. See also difference between each term and the one complex number; modulus. after it is constant, for example, 3 + 7 + 11 2. In LOGIC, a sequence of propositions or + 15 + …. The general formula for an arith- statements, starting with a set of premisses metic series is: (initial assumptions) and ending with a Sn = a + (a + d) + (a + 2d) + … conclusion. + [a + (n – 1)d] = ∑[a + (n – 1)d] arithmetic The study of the skills neces- In the example, the first term, a, is 3, the sary to manipulate numbers in order to common difference, d, is 4, and so the

12 asymptotic series nth term, a + (n – 1)d, is 3 + (n – 1)4. The associative Denoting an operation that is sum to n terms of an arithmetic series is independent of grouping. An operation • is n[2a + (n – 1)d]/2 or n(a + l)/2 where l is associative if the last (nth) term. Compare geometric a•(b•c) = (a•b)•c series. for all values of a, b, and c. In ordinary arithmetic, addition and multiplication are arm One of the lines forming a given associative operations. This is sometimes angle. referred to as the associative law of addition and the associative law of multiplica- array An ordered arrangement of num- tion. Subtraction and division are not bers or other items of information, such as associative. See also commutative; distrib- those in a list or table. In computing, an utive. array has its own name, or identifier, and each member of the array is identified by a astroid /ass-troid/ A star-shaped curved θ subscript used with the identifier. An array defined in terms of the parameter by: x = 3θ 3θ can be examined by a program and a par- a cos , y = a sin , where a is a constant. ticular item of information extracted by using this identifier and subscript. astronomical unit (au, AU) The mean distance between the Sun and the Earth, artificial intelligence The branch of used as a unit of distance in astronomy for computer science that is concerned with measurements within the solar system. It is programs for carrying out tasks which re- defined as 149 597 870 km. quire intelligence when they are done by /ay-să-met-ră-kăl/ Denot- humans. Many of these tasks involve a lot asymmetrical ing any figure that cannot be divided into more computation than is immediately ap- two parts that are mirror images of each parent because much of the computation is other. The letter R, for example, is asym- unconscious in humans, making it hard to metrical, as is any solid object that has a simulate. Programs now exist that play left-handed or right-handed characteristic. chess and other games at a high level, take Compare symmetrical. decisions based on available evidence, prove theorems in certain branches of asymptote /ass-im-toht/ A straight line mathematics, recognize connected speech towards which a curve approaches but using a limited vocabulary, and use televi- never meets. A hyperbola, for example, has sion cameras to recognize objects. Al- two asymptotes. In two-dimensional though these examples sound impressive, Cartesian coordinates, the curve with the the programs have limited ability, no cre- equation y = 1/x has the lines x = 0 and y = ativity, and each can only carry out a lim- 0 as asymptotes, since y becomes infinitely ited range of tasks. There is still a lot more small, but never reaches zero, as x in- research to be done before the ultimate creases, and vice versa. goal of artificial intelligence is achieved, which is to understand intelligence well asymptotic series /ass-im-tot-ik/ A series 2 enough to make computers more intelli- of the form a0 + a1/x + a2/x + ... an/xn rep- gent than people. In fact there is consider- resenting a function f(x) is an asymptotic able controversy about the whole subject, series if |f(x) – Sn(x)| tends to zero as |x|→∞ with many people postulating that the for a fixed n, where Sn(x) is the sum of the human thought process is different in kind first n terms of the series. Asymptotic series to the computational operation of com- can be defined for either real or complex puter processes. variables. They are usually divergent although some are convergent. Asymptotic assembler See program. series are used extensively in mathematical analysis and its physical applications. assembly language See program. For example, many integrals that cannot

13 atmosphere

4 —

3 —

2 —

1 —

-4 -3 -2 -1 0

— — — — — — — — x — — — — — 1 2 3 4 5 6 7 8

— -1

— -2

— -3

— -4 Asymptote: the x-axis and the y-axis are asymptotes to this curve. be calculated precisely can be calculated atto- Symbol: a A prefix denoting 10–18. approximately in terms of asymptotic For example, 1 attometer (am) = 10–18 series. meter (m). atmosphere A unit of pressure equal attractor The point or set of points in to 760 mmHg. It is equivalent to phase space to which a system moves with 101 325 newtons per square meters time. The attractor of a system may be a (101 325 N m–2). single point (in which case the system reaches a fixed state that is independent of atmospheric pressure See pressure of the time), or it may be a closed curve known as atmosphere. a limit cycle. This is the type of behavior

Attractor: an example of a strange attractor. The variable shown on the left displays chaotic behavior. The plot in phase space on the right shows a non-intersecting curve.

14 azimuth found in oscillating systems. In some sys- different real roots, m1 and m2, then the tems, the attractor is a curve that is not complementary function is given by y = A closed and does not repeat itself. This, exp (m1x) + B exp (m2x), where A and B known as a strange attractor, is character- are constants. If the auxiliary equation has istic of chaotic systems. See chaos theory. one root m that occurs twice then the complementary function is given by y = (A + AU (au) See astronomical unit. Bx) exp (mx). augend /aw-jend, aw-jend/ A number to average See mean. which another, the ADDEND, is added in a sum. axial plane /aks-ee-ăl/ A fixed reference plane in a three-dimensional coordinate automorphism /aw-tŏ-mor-fiz-ăm/ In system. For example, in rectangular Carte- very general terms, a transformation on el- sian coordinates, the planes defined by x = ements belonging to a set that has some 0, y = 0, and z = 0 are axial planes. The x- structure, in which the structural relations coordinate of a point is its perpendicular between the elements are unchanged by the distance from the plane x = 0, and the y- transformations. The concept of automor- and z-coordinates are the perpendicular phism applies to structure in algebra and distances from the y = 0 and z = 0 planes re- geometry. An example of an automor- spectively. See also coordinates. phism is a transformation among the points of space that takes every figure into axial vector (pseudovector) A quantity a similar figure. The set of automorphisms that acts like a vector but changes sign if on a set of elements is a GROUP called the the coordinate system is changed from a automorphism group. In physics the set of right-handed system to a left-handed sys- transformations between frames of refer- tem. An example of an axial vector is a vec- ence forms the physical automorphism tor formed by the (cross) vector product of group. two vectors. See also polar vector. auxiliary equation An equation that is axiom /aks-ee-ŏm/ (postulate) In a math- used in the solution of an inhomogeneous ematical or logical system, an initial propo- second-order linear DIFFERENTIAL EQUATION sition or statement that is accepted as true of the form ad2y/dx2 + bdy/dx + cy = f(x) without proof and from which further where a, b, and c are constants and f(x) is statements, or theorems, can be derived. In a function of x. The solution is found in a mathematical proof, the axioms are often terms of the solution to the simpler homo- well-known formulae. geneous equation ad2y/dx2 + bdy/dx + cy = 0. A result in the theory of differential axiom of choice See choice; axiom of. equations shows that the general solution of the first equation can be written in the axis (pl. axes) 1. A line about which a fig- form y = g(x) + y1(x), where g(x) is a func- ure is symmetrical. tion, known as the complementary func- 2. One of the fixed reference lines used in a tion, which is the general solution of the graph or a coordinate system. See coordi- second equation, and y1(x) is a particular nates. solution of the first equation, known as the 3. A line about which a curve or body ro- particular integral. The complementary tates or revolves. function can be found by taking y = exp 4. The line of intersection of two or more (mx) to be a solution of the second equa- coaxial planes. tion. Taking this solution gives rise to the auxiliary equation am2 + bm + c = 0. The azimuth /az-ă-mŭth/ The angle θ meas- form the complementary function takes de- ured in a horizontal plane from the x-axis pends on the nature of the roots of the aux- in spherical polar coordinates. It is the iliary equation. If the equation has two same as the longitude of a point.

15 azimuth

z z

z = 0 y y

x = 0 0 0 x x

y = 0 In three-dimensional rectangular Cartesion coordinates, the x- and y- axes lie in the axial plane z = 0, y the y- and z-axes in the axial plane x = 0, and the x- and z-axes in the axial plane y = 0.

0 x

Axial planes

16 B

backing store See disk; magnetic tape; white flowers grow from a packet of mixed store. seeds. See also histogram. ballistic pendulum A device for measur- barn Symbol: b A unit of area defined as ing the momentum (or velocity) of a pro- 10–28 square meter. The barn is sometimes jectile (e.g. a bullet). It consists of a heavy used to express the effective cross-sections pendulum, which is struck by the projec- of atoms or nuclei in the scattering or ab- tile. The momentum can be calculated by sorption of particles. measuring the displacement of the pendulum and using the law of constant momen- barrel A US unit of capacity used to meas- tum. If the mass of the projectile is known ure solids or liquids. It is equal to 7056 its velocity can be found. cubic inches (0.115 6 m3). In the petroleum industry, 1 barrel = 42 US gallons, 35 im- ballistics The study of the motion of ob- perial gallons, or 159.11 liters. jects that are propelled by an external force (i.e. the motion of projectiles). barrel printer See line printer. bar A unit of pressure defined as 105 pas- barycenter /ba-ră-sen-ter/ See center of cals. The millibar (mb) is more common; it mass. Used in particular for the center of is used for measuring atmospheric pressure mass of a system of separate objects con- in meteorology. sidered as a whole. bar chart A GRAPH consisting of bars barycentric coordinates /ba-ră-sen-trik/ whose lengths are proportional to quanti- Coordinates that relate the center of mass ties in a set of data. It can be used when one of separate objects to the center of mass of axis cannot have a numerical scale; e.g. to the system of several objects as a whole. show how many pink, red, yellow, and Consider three objects with masses m1, m2,

frequency 15 –

10 – This bar chart shows the results when a group of 40 school students were asked how many books they each had read in the previous week. 13 had read none, 13 had read one, 8 5 – had read two, 5 had read three, 1 had read four, and no-one had read five or more.

0 1 2 3 4 number of books read in a week

Bar chart

17 base and m3 with m1 + m2 + m3 = 1, and their Similarly, in three dimensions a vector can centers of mass at the points p1 = (x1,y1,z1), be written as the sum of multiples of three p2 = (x2,y2,z2), p3 = (x3,y3,z3). Then the cen- basis vectors. ter of mass of the three objects together is the point batch processing A method of operation, p = m1p1 + m2p2 + m3p3 = used especially in large computer systems, (m1x1+m2x2+m3x3, m1y1+ m2y2+m3y3, in which a number of programs are col- m1z1+m2z2+m3z3) lected together and input to the computer (m1,m2,m3) are said to be the barycentric as a single unit. The programs forming a coordinates of the point p with respect to batch can either be submitted at a central the points p1, p2, and p3. site or at a remote job entry site; there can be any number of remote job entry sites, base 1. In geometry, the lower side of a which can be situated at considerable dis- triangle, or other plane figure, or the lower tance from the computer. The programs face of a pyramid or other solid. The alti- are then executed as time becomes avail- tude is measured from the base and at right able in the system. Compare time sharing. angles to it. 2. In a number system, the number of dif- Bayes’ theorem /bayz/ A formula ex- ferent symbols used, including zero. For pressing the probability of an intersection example, in the decimal number system, of two or more sets as a product of the in- the base is ten. Ten units, ten tens, etc., are dividual probabilities for each. It is used to grouped together and represented by the calculate the probability that a particular figure 1 in the next position. In binary event Bi has occurred when it is known numbers, the base is two and the symbols that at least one of the set {B1, B2, … Bn} used are 0 and 1. has occurred and that another event A has 3. In logarithms, the number that is raised also occurred. This conditional probability to the power equal to the value of the log- is written as P(Bi|A). B1, … Bn form a par- arithm. In common logarithms the base is tition of the sample space s such that ∪ ∪ ∪ ∩ 10; for example, the logarithm to the base B1 B2 … Bn = s and Bi Bj = 0, for all i 10 of 100 is 2: and j. If the probabilities of B1, B2, … Bn log10100 = 2 and all of the conditional probabilities 2 100 = 10 P(A|Bj) are known, then P(Bi|A) is given by P(Bi)P(A|Bj) base unit A unit that is defined in terms of The theorem is named for the English reproducible physical phenomena or pro- mathematician Thomas Bayes (1702–61). totypes, rather than of other units. The second, for example, is a base unit (of time) in beam compass See compasses. the SI, being defined in terms of the frequency of radiation associated with a par- bearing The horizontal angle between a ticular atomic transition. Conventionally, line and a fixed reference direction, usually seven units are chosen as base units in the measured clockwise from the direction of SI. See also SI units. north. Bearings are expressed in degrees (e.g. 125°); bearings of less than 100° are BASIC See program. written with an initial 0 (e.g. 030°). They have important applications in radar, basis vectors In two dimensions, two sonar, and surveying. nonparallel VECTORS, scalar multiples of which are added to form any other vector bel See decibel. in the same plane. For example, the unit vectors i and j in the directions of the x- Bernoulli trial An experiment in which and y-axes of a Cartesian coordinate sys- there are two possible independent out- tem are basis vectors. The position vector comes, for example, tossing a coin. The OP of the point P(2,3) is equal to 2i + 3j. experiment is named for the Swiss mathe-

18 biconditional

E E C C

C C

E E

E E C C

C C

E E

Bezier curve: construction of a curve by successive iterations.

matician Jakob (or Jacques) Bernoulli points. The midpoints of these lines can (1654–1705), who made an important then be used to get an even better approxi- contribution to the discipline of probabil- mation, and so on. The Bezier curve is the ity theory. limit of this recursive process. Bezier curves can be defined mathematically by cubic Bessel functions /bess-ăl/ A set of func- polynomials. A similar type of curve ob- tions, denoted by the letter J, that are solu- tained by using control points is known as tions to equations involving waves, which a B-spline. The curve is named for the are expressed in cylindrical polar coordi- French engineer and mathematician Pierre nates. The solutions form an infinite series Bezier (1910–99). and are listed in tables. The set of functions is named for the German astronomer bias A property of a statistical sample that Friedrich Wilhelm Bessel (1784–1846), makes it unrepresentative of the whole who used them in his work on planetary population. For example, if medical data is movements. See also partial differential based on a survey of patients in a hospital, equation. then the sample is a biased estimate of the general population, since healthy people beva- Symbol: B A prefix used in the USA will be excluded. to denote 109. It is equivalent to the SI prefix giga-. biconditional /bÿ-kŏn-dish-ŏ-năl/ Sym- bol: ↔ or ≡ In logic, the relationship if and Bezier curve /bez-ee-ay/ A type of curve only if (often abbreviated to iff) that holds used in computer graphics. A simple Bezier between a pair of propositions or state- curve is defined by four points. Two of ments P and Q only when they are both these are the end points of the curve. The other two are control points and lie off the P Q P ≡ Q curve. A way of producing the curve is to T T T first join the four points by three straight T F F lines. If the midpoints of these three lines F T F are taken, together with the original two F F T end points, a better approximation is to draw five straight lines joining these Biconditional

19 bilateral symmetry true or both false. It is also the relationship binary relation between two real numbers. of logical equivalence; the truth of P is both In general, the existence of a relation be- a necessary and a sufficient CONDITION for tween a and b can be indicated by aRb. the truth of Q (and vice versa). The truth- table definition for the biconditional is binomial /bÿ-moh-mee-ăl/ An algebraic shown. See also truth table. expression with two variables in it. For example, 2x + y and 4a + b = 0 are binomials. bilateral symmetry A type of symmetry Compare trinomial. in which a shape is SYMMETRICAL about a plane, i.e. each half of the shape is a mirror binomial coefficient The factor multi- image of the other. plying the variables in a term of a BINOMIAL EXPANSION. For example, in (x + y)2 = x2 + billion A number equal to 109 (i.e. one 2xy + y2 the binomial coefficients are 1, 2, thousand million). This has always been and 1 respectively. the definition in the USA. In the UK a billion was formerly 1012 (i.e. one million mil- binomial distribution The distribution lion). In the 1960s, the British Treasury of the number of successes in an experi- started using the term billion in the Ameri- ment in which there are two possible out- can sense of 109, and this usage has now comes, i.e. success and failure. The supplanted the original ‘million million’. probability of k successes is b(k,n,p) = n!/k!(n – k)! × pn × qn–k bimodal /bÿ-moh-d’l/ Describing distribu- where p is the probability of success and q tions of numerical data that have two (= 1 – p) the probability of failure on each peaks (modes) in frequency. trial. These probabilities are given by the terms in the binomial theorem expansion binary /bÿ-nă-ree, -nair-ee/ Denoting or of (p + q)n. The distribution has a mean np based on two. A binary number is made up and variance npq. If n is large and p small using only two different digits, 0 and 1, in- it can be approximated by a Poisson distri- stead of ten in the decimal system. Each bution with mean np. If n is large and p is digit represents a unit, twos, fours, eights, not near 0 or 1, it can be approximated by sixteens, etc., instead of units, tens, hun- a normal distribution with mean np and dreds, etc. For example, 2 is written as 10, variance npq 3 is 11, 16 (= 24) is 10 000. Computers calculate using binary numbers. The digits 1 binomial expansion A rule for the ex- and 0 correspond to on/off conditions in an pansion of an expression of the form (x + electronic switching circuit or to presence y)n. x and y can be any real numbers and n or absence of magnetization on a disk or is an integer. The general formula, some- tape. Compare decimal; duodecimal; hexa- times called the binomial theorem, is decimal; octal. (x + y)n = xn + nxn–1y + [n(n – 1)/2!]xn–2y2 + … + yn binary logarithm A LOGARITHM to the This can be written in the form: n n–1 n–2 2 base two. The binary logarithm of 2 (writ- x + nC1x y + nC2x y + … n–r r ten log22) is 1. + nCrx y + … The coefficients nC1, nC2, etc., are called bi- binary operation A mathematical pro- nomial coefficients. In general, the rth co- cedure that combines two numbers, quan- efficient, nCr, is given by tities, etc., to give a third. For example, n!/(n – r)!r! See also combination. multiplication of two numbers in arithmetic is a binary operation. binomial theorem See binomial expansion. binary relation A relation between two elements a and b of a set S. For example, birectangular /bÿ-rek-tang-gyŭ-ler/ Hav- the symbol > for ‘greater than’ indicates a ing two right angles. See spherical triangle.

20 boundary condition bisector /bÿ-sek-ter, bÿ-sek-/ A straight not be in the set; e.g. the set {1,½,¼,…} is line or a plane that divides a line, a plane, bounded and infinite and it does have an or an angle into two equal parts. accumulation point, namely 0, but that point is not in the set. The theorem is bistable circuit /bÿ-stay-băl/ An elec- named for the Czech philosopher, mathe- tronic circuit that has two stable states. matician and theologian Bernard Bolzano The circuit will remain in one state until (1781–1848) and the German mathemati- the application of a suitable pulse, which cian Karl Weierstrass (1815–97). will cause it to assume the other state. Bistable circuits are used extensively in Boolean algebra /boo-lee-ăn/ A system of computer equipment for storing data and mathematical logic that uses symbols and for counting. They usually have two input set theory to represent logic operations in a terminals to which pulses can be applied. A mathematical form. It was the first system pulse on one input causes the circuit to of logic using algebraic methods for com- change state; it will remain in that state bining symbols in proofs and deductions. until a pulse on the other input causes it to Various systems have been developed and assume the alternative state. These circuits are used in probability theory and comput- are often called flip-flops. ing. Boolean algebra is named for the British mathematician George Boole bit Abbreviation of binary digit, i.e. either (1815–64). of the digits 0 or 1 used in binary notation. Bits are the basic units of information in bound 1. In a set of numbers, a value be- computing as they can represent the states yond which there are no members of the of a two-valued system. For example, the set. A lower bound is less than or equal to passage of an electric pulse along a wire every number in the set. An upper bound is could be represented by a 1 while a 0 greater than or equal to every number in would mean that no pulse had passed. the set. The least upper bound (or supre- Again, the two states of magnetization of mum) is the lowest of its upper bounds and spots on a magnetic tape or disk can be the greatest lower bound (or infimum) is represented by either a 1 or 0. See also bi- the largest of its lower bounds. For exam- nary; byte; word. ple, the set {0.6, 0.66, 0.666, …} has a least upper bound of 2/3. bitmap /bit-map/ See computer graphics. 2. A bound of a function is a bound of the set of values that the function can take for bivariate /bÿ-vair-ee-ayt/ Containing two the range of values of the variable. For ex- variable quantities. A plane vector, for example, if the variable x can be any real ample, is bivariate because it has both mag- number, then the function f(x) = x2 has a nitude and direction. lower bound of 0. A bivariate random variable (X,Y) has 3. In formal logic a variable is said to be the joint probability P(x,y); i.e. the proba- bound if it is within the scope of a quanti- bility that X and Y have the values x and y fier. For example, in the sentence Fy → respectively, is equal to P(x) × P(y), when X (∃x)Fx, x is a bound variable, whereas y is and Y are independent. not. A variable which is not bound is said to be free. Board of Trade unit (BTU) A unit of energy equivalent to the kilowatt-hour (3.6 × boundary condition In a DIFFERENTIAL 106 joules). It was formerly used in the UK EQUATION, the value of the variables at a for the sale of electricity. certain point, or information about their relationship at a point, that enables the ar- Bolzano–Weierstrass theorem /bohl- bitrary constants in the solution to be de- tsah-noh vÿ-er-shtrahss/ The theorem that termined. For example, the equation any bounded infinite set has an accumula- d2y/dx2 – 4dy/dx + 3y = 0 tion point. The accumulation point need has a general solution

21 boundary line

y = Ae–x + Be–3x maximum points, minimum points, or where A and B are arbitrary constants. If cusps. the boundary conditions are y = 1 at x = 0 2. A departure from the normal sequential and dy/dx = 3 at x = 0, the first can be sub- execution of instructions in a computer stituted to obtain B = 1 – A. Differentiating program. Control is thus transferred to an- the general solution for y gives other part of the program rather than pass- dy/dx = –Ae–x – 3Be–3x ing in strict sequence from one instruction and substituting the second boundary con- to the next. The branch will be either un- dition then gives conditional, i.e. it will always occur, or it 3 = –A – 3B = 2A – 3 will be conditional, i.e. the transfer of con- That is, A = 3 and B = –2. trol will depend on the result of some arithmetical or logical test. See also loop. boundary line A line on a graph that indicates the boundary at which an inequal- breadth A horizontal distance, usually ity in two dimensions holds. A common taken at right angles to a length. convention for drawing the boundary line is that if the inequality is of the form Briggsian logarithm /brig-zee-ăn/ See ‘greater than or equal to’ (or ‘less than or logarithm. equal to’), the line is solid, to indicate that points on the line are included. If the in- British thermal unit (Btu) A unit of en- equality is of the type ‘greater than’ (or ergy equal to 1.055 06 × 103 joules. It was ‘less than’) the line is dotted to show that formerly defined by the heat needed to points on the line are not included. raise the temperature of one pound of air- free water by one degree Fahrenheit at bounded function A real function f (x) standard presure. Slightly different ver- defined on a domain D for which there is a sions of the unit were in use depending on number M such that f(x) ≤ M for all x in the domain D. It is an important result in the temperatures between which the degree mathematical analysis that if f(x) is a con- rise was measured. tinuous function on a closed interval between a and b then it is also a bounded B-spline See Bezier curve. function on that interval. BTU See Board of Trade unit. bounded set A set that is bounded both above and below; i.e. the set has both an Btu See British thermal unit. upper BOUND and a lower bound. buffer store (buffer) A small area of the brackets In mathematical expressions, main STORE of a computer in which infor- brackets are used to indicate the order in mation can be stored temporarily before, which operations are to be carried out. For during, and after processing. A buffer can example: be used, for instance, between a peripheral 9 + (3 × 4) = 9 + 12 device and the CENTRAL PROCESSOR, which = 21 operate at very different speeds. (9 + 3) × 4 = 12 × 4 = 48 bug An error or fault in a computer pro- Brackets are also used in FACTORIZATION. gram. See debug. For example: 4x3y2 – 10x2y2 = 2x2y(2xy – 5y) buoyancy // The tendency of an object to float. The term is sometimes also used for branch 1. A section of a curve separated the upward force (UPTHRUST) on a body. from the remainder of the curve by discon- See center of buoyancy. See also tinuities or special points such as vertices, Archimedes’ principle.

22 byte bushel A unit of capacity usually used for one part of the world may be the cause of solid substances. In the USA it is equal to a tornado in another part of the world. See 64 US dry pints or 2150.42 cubic inches. In chaos theory. the UK it is equal to 8 UK gallons. byte /bÿt/ A subdivision of a WORD in butterfly effect Any effect in which a computing, often being the number of BITS small change to a system results in a dis- used to represent a single letter, number, or proportionately large disturbance. The other CHARACTER. In most computers a byte term comes from the idea that the Earth’s consists of a fixed number of bits, usually atmosphere is so sensitive to initial condi- eight. In some computers bytes can have tions that a butterfly flapping its wings in their own individual addresses in the store.

23 C

calculus /kal-kyŭ-lŭs/ (infinitesimal calcu- calibration /kal-ă-bray-shŏn/ The mark- lus) (pl. calculuses) The branch of mathe- ing of a scale on a measuring instrument. matics that deals with the DIFFERENTIATION For example, a thermometer can be cali- and INTEGRATION of functions. By treating brated in degrees Celsius by marking the continuous changes as if they consisted of freezing point of water (0°C) and the boil- infinitely small step changes, differential ing point of water (100°C). calculus can, for example, be used to find the rate at which the velocity of a body is calorie /kal-ŏ-ree/ Symbol: cal A unit of changing with time (acceleration) at a par- energy approximately equal to 4.2 joules. ticular instant. It was formerly defined as the energy Integral calculus is the reverse process, needed to raise the temperature of one that is finding the end result of known con- gram of water by one degree Celsius. Be- tinuous change. For example, if a car’s ac- cause the specific thermal capacity of water celeration a varies with time in a known changes with temperature, this definition is way between times t1 and t2, then the total not precise. The mean or thermochemical change in velocity is calculated by the inte- calorie (calTH) is defined as 4.184 joules. gration of a over the time interval t1 to t2. The international table calorie (calIT) is de- Integral calculus is also used to find the fined as 4.186 8 joules. Formerly the mean area under a curve. See differentiation; in- calorie was defined as one hundredth of the tegration. heat needed to raise one gram of water from 0°C to 100°C, and the 15°C calorie calculus of variations The branch of as the heat needed to raise it from 14.5°C mathematics that uses calculus to find min- to 15.5°C. imum or maximum values for a system. It has many applications in physical science cancellation Removing a common factor and engineering. For example, FERMAT’S in a numerator and denominator, or re- PRINCIPLE in optics can be regarded as an moving the same quantity from both sides application of the calculus of variations. of an algebraic equation. For example, See also variational principle. xy/yz can be simplified, by the cancellation

1 0 0 −3 0 0 multiply 3 2 0 3 2 0 row 1 0 0 2 0 0 2 by −3

−3 0 0 −3 0 0 add row 1 3 2 0 0 0 2 to row 2 0 0 2 0 0 2

Reduction of a matrix to canonical form.

24 cardinality of y, to x/z. The equation z + x = 2 + x is from a21, and hence the new number will simplified to z = 2 by canceling (subtract- not be equal to the second number on the ing) x from both sides. list. Continuing in this way we have a method of producing an infinite decimal candela Symbol: cd The SI base unit of lu- that must define a real number, but is not minous intensity, defined as the intensity equal to any of the real numbers in our list. (in the perpendicular direction) of the But we assumed that all real numbers oc- black-body radiation from a surface of curred somewhere in the list; hence there is 1/600 000 square meter at the temperature a contradiction, and so it cannot be true of freezing platinum and at a pressure of that the real numbers are countable. 101 325 pascals. Cantor’s diagonal argument is named for the Russian mathematician Georg Can- canonical form (normal form) In matrix tor (1845–1918). See also Cantor set. algebra, the DIAGONAL MATRIX derived by a series of transformations on another Cantor set /kan-ter/ The set obtained by SQUARE MATRIX of the same order. taking a closed interval in the real line, e.g. [0,1], and removing the middle third, i.e. Cantor’s diagonal argument /kan-terz/ the open interval (1/3, 2/3), and then doing An argument to show that the real num- the same to the two remaining closed inter- bers, unlike the rationals, are not count- vals [0,1/3] and [2/3,1], and so on ad in- able, and hence that there are more real finitum. The set generated in this way has numbers than rational numbers. The argu- the remarkable property of containing un- ment proceeds by assuming that the reals countably many points – i.e. the same num- are countable and showing that this leads ber of points as the whole real line – yet to a contradiction. being nowhere dense – i.e. for any point in Every real number can be expressed as the set one can always find a point not in an infinite decimal expansion. We suppose the set arbitrarily close to it. This set is also that all reals are expressed in this way and sometimes known as the Cantor discontin- since they are countable they can be uum arranged in order in a list as shown. We now define a real number b1 . b2 b3 b4 capital 1. The total sum of all the assets of … by saying that b1 must be any number a person or company, including cash, in- different from a1. Hence our new number vestments, household goods, land, build- will not be equal to the first real number in ings, machinery, and finished or unfinished our list. b2 must be any number different goods. 2. A sum of money borrowed or lent on which interest is payable or received. See compound interest; simple interest. 3. The total amount of money contributed a1 a11 a12 a13 a14 by the shareholders when a company is formed, or the amount contributed to a partnership by the partners. a2 a21 a22 a23 a24 carat /ka-răt/ (metric) A unit of mass used for precious stones. It is equal to 200 mil- a3 a31 a32 a33 a34 ligrams.

cardinality /kar-dă-nal-ă-tee/ Symbol

a4 a41 a42 a43 a44 n(A). The number of elements in a finite set A. The symbols |A| and #A are sometimes used to denote the cardinality of A. The cardinality is sometimes called the cardinal Cantor’s diagonal argument number of the set. If A, B, and C are all fi-

25 cardinal numbers nite subsets of a set S then the following re- point in the same direction as the z-axis. A lations between cardinal numbers hold: left-handed system is the mirror image of n(A∪B) = n(A) + n(B) – n(A∩B) this. In a rectangular system, all three axs and are mutually at right-angles. The Cartesian n(A∪B∪C) = n(A) + n(B) + n(C) – coordinate system is named for the French n(A∩B) – n(A∩C) – n(B∩C) + mathematician, philosopher, and scientist n(A∩B∩C). René du Perron Descartes (1596–1650). See also cylindrical polar coordinates; cardinal numbers Whole numbers that polar coordinates. are used for counting or for specifying a total number of items, but not the order in Cartesian product The Cartesian prod- which they are arranged. For example, uct of two sets A and B, which is written A when one says ‘three books’, the three is a × B, is the set of ordered pairs 〈x,y〉 where cardinal number. x belongs to A and y belongs to B. Two sets are said to have the same car- A × B = {〈x,y〉 | x ∈A and y ∈B} dinal number if their elements can be put into one-to-one correspondence with each catastrophe theory The study of the other. Compare ordinal numbers. See also ways in which discontinuities appear, both aleph. in mathematics and its physical applications. Catastrophe theory deals with prob- cardioid /kar-dee-oid/ An EPICYCLOID that lems that cannot be analyzed using has only one loop, formed by the path of a calculus since calculus is largely concerned point on a circle rolling round the circum- with continuity. Catastrophe theory can be ference of another that has the same radius. regarded as a branch of TOPOLOGY. It has been applied to several physical problems, Cartesian coordinates /kar-tee-zhăn/ A including problems in the theory of waves, method of defining the position of a point especially optics. It has also been applied to by its distance from a fixed point (origin) in the evolution of complex systems such as the direction of two or more straight lines. biological systems and (far more dubi- On a flat surface, two straight lines, called ously) to social and economic changes. the x-axis and the y-axis, form the basis of a two-dimensional Cartesian coordinate category A category consists of two system. The point at which they cross is the classes: a class of objects and a class of origin (O). An imaginary grid is formed by morphisms – i.e. mappings that are in some lines parallel to the axes and one unit sense structure-preserving. Associated with length apart. The point (2,3), for example, each pair of objects are a set of the mor- is the point at which the line parallel to the phisms and a law of composition for these y-axis two units in the direction of the x- morphisms. Category theory is the study of axis, crosses the line parallel to the x-axis such entities. It provides a model for many three units in the direction of the y-axis. situations where sets with certain struc- Usually the x-axis is horizontal and the y- tures are studied along with a class of map- axis is perpendicular to it. These are pings that preserve these structures. known as rectangular coordinates. If the Examples of categories are sets with func- axs are not at right angles, they are oblique tions and groups with homomorphisms. coordinates In three dimensions, a third axis, the z- catenary /kat-ĕ-nair-ee/ The plane curve axis, is added to define the height or depth of a flexible uniform line suspended from of a point. The COORDINATES of a point are two points. For example, an empty wash- then three numbers (x,y,z). A right-handed ing line attached to two poles and hanging system is one for which if the thumb of the freely between them follows a catenary. right hand points along the x-axis, the fin- The catenary is symmetrical about an axis gers of the hand fold in the direction in perpendicular to the line joining the two which the y-axis would have to rotate to points of suspension. In Cartesian coordi-

26 CD-ROM

P(a,b)

ordinate of P = b units

0 x

absicca of P = a units

Two-dimensional rectangular Cartesian coordinates showing a point P(a,b).

z z

x x left-handed right-handed

In three-dimensional Cartesian coordinates, a right-handed y system is the mirror image of a left-handed system.

Σ 2 ≤ Σ 2 Σ 2 nates, the equation of a catenary that has equality ( aibi) ( ai )( bi ). This in- its axis of symmetry lying along the y-axis equality only becomes an equality in the at y = a, is case when there are constants k and l that x/a –x/a y = (a/2)(e + e ) satisfy kai = lbi for all i, i.e. if the ai and the bi are all proportional. It is also called the catenoid /kat-ĕ-noid/ The curved surface Cauchy–Schwarz inequality. formed by rotating a catenary about its axis of symmetry. CD-ROM A type of compact disc with a read-only memory, which provides read- Cauchy’s inequality /koh-sheez, koh- only access to up to 640 megabytes of sheez/ For sums, the inequality has the memory for use on a computer. The disk form that if a1, a2, ..., an and b1, b2, ..., bn can carry data as text, images (video) and are two sets of real numbers then the in- sound (audio).

27 Celsius degree

Celsius degree /sel-see-ŭs/ Symbol: °C A In the case of a body having a uniform den- unit of temperature difference equal to one sity an integration must be used to obtain hundredth of the difference between the the position of the center of mass, which temperatures of freezing and boiling water coincides with the CENTROID. at one atmosphere pressure. It was formerly known as the degree centigrade and center of pressure For a body or surface is equivalent to 1 K. On the Celsius scale of a fluid, the point at which the resultant water freezes at 0°C and boils at 100°C. It of pressure forces acts. If a surface lies hor- is named for the Swedish astronomer An- izontally in a fluid, the pressure at all ders Celsius (1701–44). points will be the same. The resultant force will then act through the centroid of the center A point about which a geometric surface. If the surface is not horizontal, the figure is symmetrical. pressure on it will vary with depth. The resultant force will now act through a differ- center of buoyancy For an object in a ent point and the center of pressure is not fluid, the center of mass of the displaced at the centroid. volume of fluid. In order for a floating object to be stable the center of mass of the center of projection The point at which object must lie below the center of buoy- all the lines forming a central projection ancy; when the object is in equilibrium, meet. See central projection. the two lie on a vertical line. See also –2 Archimedes’ principle. centi- Symbol: c A prefix denoting 10 . For example, 1 centimeter (cm) = 10–2 meter (m). center of curvature See curvature. central conic A conic with a center of center of gravity See center of mass. symmetry; e.g. an ellipse or hyperbola. center of mass (barycentre) A point in a central enlargement A central projec- body (or system) at which the whole mass tion. See also scale factor. of the body may be considered to act. Often the term center of gravity is used. central force A force that acts on any af- This is, strictly, not the same unless the fected object along a line to an origin. For body is in a constant gravitational field. instance, the motion of electric forces be- The center of gravity is the point at which tween charged particles are central; fric- the weight may be considered to act. tional forces are not. The center of mass coincides with the center of symmetry if the symmetrical body central limit theorem A result of proba- has a uniform density throughout. In other bility theory that if X1, X2, ... Xn is a ran- cases the principle of moments may be used dom sample of size n from a distribution, to locate the point. For instance, two which is not necessarily a normal distribu- masses m1 and m2 a distance d apart have tion, in which the finite mean is µ and the a center of mass on the line between them. finite variance is σ2, then as the sample size If this is a distance d1 from m1 and d2 from increases the distribution is approximately m2 then m1d1 = m2d2 or: a normal distribution. This theorem_ can m1d1 = m2(d – d1) also be stated in the forms that X= (ΣXi)/n 2 d1 = m2d/(m1 + m2) is _approximately equal to N(µ,σ /n) and A more general relationship can be applied Σ(X – µ)/(σ/√n) is approximately equal to to a number of masses m1, m2,…, mi that N(0,1). The size that n needs to be for the are respectively distances r1, r2,…, ri from theorem to be useful depends on how close an origin. The distance r from the origin to the distribution is to a normal distribution. the center of mass is given by: If the distribution is close to a normal dis- ∑ ∑ r = rimi/ mi tribution then for n = 10 it is a reasonably 28 centripetal force good approximation. If it is a long way is the first plane, and the screen or print is from being a normal distribution it is not a the second. In this case the two planes are good approximation until well after n = usually parallel, but this is not always so in 100. central projection. central processor (central processing central quadric A quadric surface that is unit; CPU) A highly complex electronic denot a DEGENERATE QUADRIC and has a cen- vice that is the nerve center of a COMPUTER. ter of symmetry. This definition means that It consists of the control unit and the arith- a central quadric has to be either an ellip- metic and logic unit (ALU). Also some- soid or a hyperboloid. The hyperboloid times considered part of the central can have either one sheet or two. See coni- processor is the main store, or memory, coid. where a program or a section of a program is stored in binary form. centrifugal force A force supposed to act The control unit supervises all activities radially outward on a body moving in a within the computer, interpreting the in- curve. In fact there is no real force acting; structions that make up the program. Each centrifugal force is said to be a ‘fictitious’ instruction is automatically brought, in force, and the use is best avoided. The idea turn, from the main store and kept tem- arises from the effect of inertia on an object porarily in a small store called a register. moving in a curve. If a car is moving Electronic circuits analyze the instruction around a bend, for instance, it is forced in and determine the operation to be carried a curved path by friction between the out and the exact location or locations in wheels and the road. Without this friction store of the data on which the operation is (which is directed toward the center of the to be performed. The operation is actually curve) the car would continue in a straight performed by the ALU, again using elec- line. The driver also moves in the curve, tronic circuitry and a set of registers. It may constrained by friction with the seat, re- be an arithmetical calculation, such as the straint from a seat belt, or a ‘push’ from the addition of two numbers, or a logical oper- door. To the driver it appears that there is ation, such as selecting or comparing data. a force radially outward pushing his or her This process of fetching, analyzing, and ex- body out – the centrifugal force. In fact this ecuting instructions is repeated in the re- is not the case; if the driver fell out of the quired order until an instruction to stop is car he or she would move straight forward executed. at a tangent to the curve. It is sometimes The size of central processors has di- said that the centrifugal force is a ‘reaction’ minished considerably with advances in to the CENTRIPETAL FORCE – this is not true. technology. It is now possible to form a (The ‘reaction’ to the centripetal force is an central processor on a single silicon chip a outward push on the road surface by the few millimeters square in area. This tiny tires of the car.) device is known as a microprocessor. centripetal force A force that causes an central projection (conical projection) A object to move in a curved path rather than geometrical transformation in which a continuing in a straight line. The force is straight line from a point (called the center provided by, for instance: of projection) to each point in the figure is – the tension of the string, for an object continued to the point at which it passes whirled on the end of a string; through a second (image) plane. These – gravity, for an object in orbit round a points form the image of the original fig- planet or a star; ure. When a photographic image is created – electric force, for an electron in the shell from a film using an enlarger, this is the of an atom. kind of PROJECTION that takes place. The The centripetal force for an object of light source is at the center of projection, mass m with constant speed v and path ra- the light rays are the straight lines, the film dius r is mv2/r, or mω2r, where ω is angular

29 centroid speed. A body moving in a curved path has called the total derivative, is given by the an acceleration because the direction of the chain rule for partial differentiation, which velocity changes, even though the magni- is: ∂ ∂ tude of the velocity may remain constant. dz/dt = ( z/ x1)(dx1/dt) + ∂ ∂ This acceleration, which is directed toward ( z/ x2)(dx2/dt) + … the center of the curve, is the centripetal ac- 2 2 celeration. It is given by v /r or ω r. changing the subject of a formula Re- arranging a formula so that the single term centroid /sen-troid/ (mean center) The on the left-hand side (the subject) is re- point in a figure or solid at which the CEN- placed by another term from the formula. TER OF MASS would be if the figure or body For example, the formula for the area of a were of uniform-density material. The cen- sphere is A = 4πr2 (where A is the area and troid of a symmetrical figure is at the cen- r is the radius). Changing the subject of the ter of symmetry; thus, the centroid of a formula to r gives r = ½√(A/π). circle is at its center. The centroid of a triangle is the point at which the medians meet. channel A path along which information For non-symmetrical figures or bodies can travel in a computer system or com- integration is used to find the centroid. The munications system. centroid of a line, figure, or solid is the point that has coordinates that are the chaos theory The theory of systems that mean values of the coordinates of all the exhibit apparently random unpredictable points in the line, figure, etc. For a surface, behavior. The theory originated in studies the coordinates of the centroid are given of the Earth’s atmosphere and the weather. In such a system there are a number of vari- by: _ x = [∫∫xdxdy]/A, etc. ables involved and the equations describ- the integration being over the surface, and ing them are nonlinear. As a result, the A being the area. For a volume, a triple in- state of the system as it changes with time tegral is used to obtain the coordinates of is extremely sensitive to the original condi- the centroid:_ tions. A small difference in starting condi- x = [∫∫∫xdxdydz]/V, etc. tions may be magnified and produce a large variation in possible future states of c.g.s. system A system of units that uses the system. As a result, the system appears the centimeter, the gram, and the second as to behave in an unpredictable way and may the base mechanical units. Much early sci- exhibit seemingly random fluctuations entific work used this system, but it has (chaotic behavior). The study of such non- now almost been abandoned. linear systems has been applied in a number of fields, including studies of fluid chain A former unit of length equal to 22 dynamics and turbulence, random electri- yards. It is equivalent to 20.116 8 m. cal oscillations, and certain types of chem- chain rule A rule for expressing the de- ical reaction. See also attractor. rivative of a function z = f(x) in terms of another function of the same variable, character One of a set of symbols that can u(x), where z is also a function of u. That be represented in a computer. It can be a is: letter, number, punctuation mark, or a spe- dz/dx = (dz/du)(du/dx) cial symbol. A character is stored or ma- This is often called the ‘function of a func- nipulated in the computer as a group of tion’ rule. BITS (i.e. binary digits). See also byte; word; For a function z = f(x1,x2,x3,…) of sev- store. eral variables, in which each of the variables x1, x2, x3,… is itself a function of a characteristic 1. See logarithm. single variable, t, the derivative dz/dt, 2. See eliminant.

30 circular functions chi-square distribution /kÿ-skwair/ (χ2 the chord of contact is also known as the distribution) The distribution of the sum polar of P with respect to the conic and the of the squares of random variables with point P is known as the pole. For a general standard normal distributions. For exam- conic section described by the equation 2 2 ple, if x1, x2,…xi,… are independent vari- ax + 2hxy + by + 2gx + 2fy + c = 0, ables with standard normal distribution, the equation of the chord of contact for P then (p,q) is χ2 ∑ 2 = xi apx + h(py + qx) + bqy + g(p + x) + has a chi-square distribution with n de- f(q + y) + c = 0. χ 2 grees of freedom, written n . The mean and variance are n and 2n respectively. The circle The plane figure formed by a closed χ 2 α χ2 ≤χ2 α α values n ( ) for which P( n ( )) = curve consisting of all the points that are at are tabulated for various values of n. a fixed distance (the radius, r) from a particular point in the plane; the point is the chi-square test A measure of how well a center of the circle. The diameter of a circle theoretical probability distribution fits a is twice its radius; the circumference is 2πr; π 2 set of data. For i = 1, 2, … m the value xi and its area is r . In Cartesian coordi- occurs oi times in the data and the theory nates, the equation of a circle centred at the predicts that it will occur ei times. Provided origin is ≥ 2 2 2 ei 5 for all values of i (otherwise values x + y = r must be combined), then The circle is the curve that encloses the χ2 ∑ 2 = (oi – ei) /ei largest possible area within any given has a chi-square distribution with n de- perimeter length. It is a special case of an grees of freedom. See also chi-square distri- ellipse with eccentricity 0. bution. circular argument An argument that, choice, axiom of The axiom states that, tacitly or explicitly, assumes what it is try- given any collection of sets, one can form a ing to prove, and is consequently invalid. new set by choosing one element from each. This axiom may seem intuitively ob- circular cone A CONE that has a circular vious and it was presupposed in many clas- base. sical mathematical works. However, it has been a point of debate and controversy circular cylinder A CYLINDER in which since many of its consequences appeared to the base is circular. be paradoxical. An example is the Banach- Tarski theorem, which proves that it is pos- circular functions See trigonometry. sible to cut a solid sphere into a finite number of pieces and to reassemble these pieces to form two solid spheres the same γ size as the original sphere. Despite these x r apparent paradoxes the axiom is widely accepted. It has many equivalents including the well-ordering principle and ZORN’S γ LEMMA. r chord A straight line joining two points on a curve, for example, the line segment joining two points on the circumference of a circle. chord of contact The line that connects Circular measure: in a circle of radius r and cir- the points of contact between two tangents cumference 2πr, the angle γ radians subtends to a curve from a point P (p,q). For a conic an arc length γ×r.

31 circular measure

secant

d iameter • nt r e a ng ch d ta o i r u d s

segment

• sector

x D x Angles in the same segment of a circle are • equal. A x C

Properties of circles circular measure The measurement of an circumcenter /ser-kŭm-sen-ter/ See cir- angle in radians. cumcircle. circular mil See mil. circumcircle /ser-kŭm-ser-kăl/ (circumscribed circle) The circle that passes circular motion A form of periodic (or through all three vertices of a triangle or cyclic) motion; that of an object moving in through the vertices of any other cyclic a circular path. For this to be possible, a polygon. The figure inside the circle is said positive central force must act. If the object to be inscribed. The point in the figure that has a uniform speed v and the radius of the is the center of the circle is called the cir- circle is r, the angular velocity (ω) is v/r. cumcenter. For a triangle with side lengths There is an acceleration toward the center a, b, and c the radius r of the circumcircle of the circle (the centripetal acceleration) is given by: equal to v2/r or ω2r. See also centripetal r = abc/{4√[s(s – a)(s – b)(s – c)]} force; rotational motion. where s is (a + b + c)/2.

32 circumcircle

D P

B An angle that an arc∧ subtends∧ at the center of a circle is twice the angle that it subtends at the circumference: ACB = 2ADB ∧ ∧ An angle in a semicircle is a right angle: XPY (=½XCY) = 90o A

D C P

B Two tangents from an external point: (1) are equal, PA = PB ∧ ∧ (2) subtend equal angles at the center, PCA = PCB (3) the line from the point to the center bisects the line AB, AD = DB F A D

G P

A tangent and a secant from an external point: PC.PB = PA2 Two intersecting chords: FX.GX = DX.XE

Properties of circles

33 circumference circumference The boundary, or length a few microseconds (millionths of a sec- of the boundary, of a closed curve, usually ond). a circle. The circumference of a circle is equal to 2πr, where r is the radius of the clockwise Rotating in the same sense as circle. the hands of a clock. For example, the head of an ordinary screw is turned clockwise circumscribed Describing a geometric (looking at the head of the screw) to drive figure that is drawn around and enclosing it in. Looking at the other end the rotation another geometrical figure. For example, appears to be anticlockwise (counterclock- in a square, a circle can be drawn through wise). the vertices. This is called the circumscribed circle, and the square is called the closed Describing a set for which a given inscribed square of the circle. Similarly, a operation gives results in the same set. For regular polyhedron might have a circum- example, the set of positive integers is scribed sphere, and a rectangular pyramid closed with respect to addition and multi- a circumscribed cone. Compare inscribed. plication. Adding or multiplying any two members gives another positive integer. cissoid /sis-oid/ A curve that is defined by The set is not closed with respect to divi- 2 3 the equation y = x /2(a–x). It has the line sion since dividing certain integers does not x = 2a as an ASYMPTOTE and a CUSP at the give a positive integer (e.g. 4/5). The set of origin. positive integers is also not closed with respect to subtraction (e.g. 5–7 = –2). See 1. A grouping of data that is taken as class also closed interval; closed set. one item in a FREQUENCY TABLE or HIS- TOGRAM. closed curve (closed contour) A curve, 2. Often used simply as a synonym for set. such as a circle or an ellipse, that forms a However, in set theory it is sometimes de- complete loop. It has no end points. A sim- sirable, to avoid paradoxes, to allow the ple closed curve is a closed curve that does existence of collections that are not sets. not cross itself. Compare open curve. Such collections are known as classes or proper classes. For example, the collection A set consisting of the of all sets is a proper class not a set. closed interval numbers between two given numbers (end classical mechanics A system of me- points), including the end points. For ex- chanics that is based on Newton’s laws of ample, all the real numbers greater than or motion. Relativity effects and quantum equal to 2 and less than or equal to 5 con- theory are not taken into account in classi- stitute a closed interval. The closed interval cal mechanics. between two real numbers a and b is written [a,b]. On a number line the end points class mark See frequency table. are marked by a blacked-in circle. Com- pare open interval. See also interval. clock pulse One of a series of regular pulses produced by an electronic device closed set A set in which the limits that called a clock and are used to synchronize define the set are included. The set of ra- operations in a computer. Every instruc- tional numbers greater than or equal to 0 tion in a computer program causes a num- and less than or equal to ten, written {x: ber of operations to be done by the 0≤x≤10; x ∈R}, and the set of points on CENTRAL PROCESSOR of the computer. Each and within a circle are examples of closed of these operations, performed by the con- sets. Compare open sets. trol unit or arithmetic and logic unit, are triggered by one clock pulse and must be closed surface A surface that has no completed before the next clock pulse. The boundary lines or curves, for example a interval at which the pulses occur is usually sphere or an ellipsoid.

34 collinear closed system (isolated system) A set of the elements in a matrix is called the ad- one or more objects that may interact with joint of the matrix. each other, but do not interact with the world outside the system. This means that there is no net force from outside or energy a b c transfer. Because of this the system’s angular momentum, energy, mass, and linear A = d e f momentum remain constant. g h i closure See group. e f cluster point See accumulation point. a’ = = ei – hf h i coaxial /koh-aks-ee-ăl/ 1. Coaxial circles are circles such that all pairs of the circles have the same RADICAL AXIS. d f 2. Coaxial planes are planes that pass b’ = = di – gf through the same straight line (the axis). g i

COBOL /koh-bôl/ See program. d e coding The writing of instructions in a c’ = = dh – ge computer programming language. The per- g h son doing the coding starts with a written description or a diagram representing the task to be carried out by the computer. The cofactors a’, b’, and c’ of the elements a, b, and c in a 3 x 3 matrix A. This is then converted into a precise and ordered sequence of instructions in the language selected. See also flowchart; pro- a’ b’ c’ gram. The adjoint of A. d’ e’ f’ coefficient /koh-i-fish-ĕnt/ A multiplying g’ h’ i’ factor. For example, in the equation 2x2 + 3x = 0, where x is a variable, the coefficient of x2 is 2 and the coefficient of x is 3. Some- Cofactor times the value of the coefficients is not known, although they are known to stay constant as x changes, for example, ax2 + coherent units A system or subset of bx = 0. In this case a and b are constant co- units (e.g. SI units) in which the derived efficients. See also constant. units are obtained by multiplying or dividing together base units, with no numerical coefficient, binomial See binomial coef- factor involved. ficient. colatitude /koh-lat-ă-tewd/ See spherical coefficient of friction See friction. polar coordinates. coefficient of restitution See restitution, collinear /kŏ-lin-ee-er/ Lying on the same coefficient of. straight line. Any two points, for example, could be said to be collinear because there cofactor /koh-fak-ter/ The DETERMINANT is a straight line that passes through both. of the matrix obtained by removing the Similarly, two vectors are collinear if they row and column containing the element. are parallel and both act through the same The matrix formed by all the cofactors of point.

35 cologarithm cologarithm /koh-lôg-ă-rith-

commensurable /kŏ-men-shŭ-ră-băl/ x Able to be measured in the same way and in terms of the same units. For example, a y 30-centimeter rule is commensurable with z a 1-meter length of rope, because both can be measured in centimeters. Neither is commensurable with an area. The column vector that defines the displacement of a point (x, y, z) from the common denominator A whole number origin of a Cartesian coordinate system. that is a common multiple of the denomi- Column vector nators of two or more fractions. For example, 6 and 12 are both common denominators of 1/2 and 1/3. The lowest combination Any subset of a given set of (or least) common denominator (LCD) is objects regardless of the order in which the smallest number that is a common mul- they are selected. If r objects are selected tiple of the denominators of two or more from n, and each object can only be chosen fractions. For example, the LCD of 1/2, once, the number of different combina- 1/3, and 1/4 is 12. Fractions are put in tions is terms of the LCDs when they are to be n!/[r!(n–r)!] added or subtracted: written as nCr or C(n,r). For instance, if 1/2 + 1/3 + 1/4 = 6/12 + 4/12 + 3/12 there are 15 students in a class and only 5 = 13/12 books, then each book has to be shared by 3 students. The number of ways in which common difference The difference be- this can happen – i.e. the number of com- tween successive terms in an arithmetic se- binations of 3 from 15 – is 15!/3!12!, or quence or arithmetic series. 455. If each object can be selected more than once the number of different combi- common factor 1. A whole number that nations is n+r–1Cr. See also factorial, permu- divides exactly into two or more given tation. numbers. For example, 7 is a common factor of 14, 49, and 84. Since 7 is the largest combinatorics /kŏm-bÿ-nă-tor-iks, -toh- number that divides into all three exactly, riks/ (combinatorial analysis) The branch it is the highest common factor (HCF). See of mathematics that studies the number of also factor. possible configurations or arrangements of 2. A number or variable by which several a certain type. It forms the basis for the parts of an expression are multiplied. For

36 complementary angles example, in 4x2 + 4y2, 4 is a common fac- compasses An instrument used for draw- tor of x2 and y2, and from the distributive ing circles. It consists of two rigid arms law for multiplication and addition, joined by a hinge. At one end is a sharp 4x2 + 4y2 = 4(x2 + y2) point, which is placed at the center of the circle. At the other end is a pencil or other common fraction See fraction. marker, which traces out the circumference when the compasses are pivoted around common logarithm See logarithm. the point. In a beam compass, used for drawing large circles, the sharp point and common multiple A whole number that the marker are attached to opposite ends of is a multiple of each of a group of numbers. a horizontal beam. For example, 100 is a common multiple of 5, 25, and 50. The lowest (or least) com- compiler See program. mon multiple (LCM) is the smallest number that is a common multiple; in this case complement The set of all the elements it is 50. that are not in a particular set. If the set A = {1, 2, 3}, and the universal set, E, is taken common ratio The ratio of successive as containing all the natural numbers, then terms in a geometric sequence or geometric the complement of A, written A′ or , is {4, series. 5, 6, …}. See Venn diagram. common tangent A single line that forms a tangent to two or more separate curves. E The term is also used for the length of the line joining the two tangential points. commutative /kŏ-myoo-tă-tiv, kom-yŭ- tay-tiv/ Denoting an operation that is in- A dependent of the order of combination. A binary operation • is commutative if a•b = A’ b•a for all values of a and b. In ordinary arithmetic, multiplication and addition are commutative operations. This is sometimes The shaded area in the Venn diagram is the referred to as the commutative law of mul- complement A′ of the set A. tiplication and the commutative law of addition. Subtraction and division are not commutative operations. See also associa- complementary angles A pair of angles tive; distributive. that add together to make a right angle (90° or π/2 radians). Compare conjugate commutative group See Abelian group. angles; supplementary angles. compact A set S of real numbers is compact if, given any collection of open sets whose union contains S, we can find a finite subcollection of those open sets whose union also contains S. The concept can be generalized to any topological space, and also to mathematical logic. In logic a formal system is said to be compact if it is such that, when a given sentence is a logi- α cal consequence of a given set of sentences, β it is a consequence of some finite subset of them. Complementary angles: α + β = 90°

37 complementary function complementary function See auxiliary as to whether there are general ‘laws of equation. complexity’, which are applicable to all complex systems. complete In mathematical logic a formal system is said to be complete if every true complex number A number that has sentence in the system is also provable both a real part and an imaginary part. The within the system. Not all logical systems imaginary part is a multiple of the square have this property. See Gödel’s incomplete- root of minus one (i). Some algebraic equa- ness theorem. tions cannot be solved with real numbers. For example, x2 + 4x + 6 = 0 has the solu- completing the square A way of solving tions x = –2 + √(–2) and x = –2 – √(–2). If a QUADRATIC EQUATION, by dividing both the number system is extended to include i sides by the coefficient of the square term = √–1, all algebraic equations can be and adding a constant, in order to express solved. In this case the solutions are x = –2 the equation as a single squared term. For + i√2 and x = –2 – i√2. The real part is –2 example, to solve 3x2 + 6x + 2= 0: and the imaginary part is +i√2 or –i√2. x2 + 2x + ⅔ = 0 Complex numbers are sometimes repre- (x + 1)2 – 1 + ⅔ = 0 sented on an Argand diagram, which is x + 1 = +√⅓ or –√⅓ similar to a graph in Cartesian coordinates, x = –1 + √⅓ or –1 – √⅓ but with the horizontal axis representing the real part of the number and the vertical complex analysis The branch of analysis axis the imaginary part. The diagram is that is specifically concerned with COMPLEX named for the Swiss mathematician Jean FUNCTIONS. Some of the results and tech- Argand (1768–1822). niques of real analysis can be extended to Any complex number can also be writ- complex analysis but many results arise ten as a function of an angle θ, just as specifically because of the mathematical Cartesian coordinates can be converted structure imposed by dealing with complex into polar coordinates. Thus r(cosθ + functions. Complex analysis is used in the i sinθ) is equivalent to x + iy, where x = theory of ASYMPTOTIC SERIES and has many rcosθ and y = rsinθ. Here, r is the modulus applications in the physical sciences and in of the complex number and θ is the argu- engineering. ment (or amplitude). This can also be written in the exponential form r = eiθ. complex fraction See fraction.

iy complex function A FUNCTION in which the variable quantities are complex numbers. See complex analysis. 4 – 3 – • P complexity theory The mathematical theory of systems that can exhibit complex 2 – r behavior. It includes such topics as chaos 1 – theory, ideas about how organization and # 0 order can emerge in systems, and how | | | | | x emergent laws of phenomena arise at dif- 1 2 3 4 5 ferent levels of description. Complexity theory is a subject of great interest to math- The point P(4,3) on an Argand diagram repre- ematicians, physicists, chemists, biologists, sents the complex number z = 4 + 3i. In the polar form z = r (cos θ + i sin θ). and economists, amongst others. Comput- ers are used extensively in complexity theory since the systems being analyzed do not, in general, have exact solutions. At the component /kŏm-poh-nĕnt/ The resolved time of writing, there is not yet a consensus part of a VECTOR in a particular direction,

38 compound transformation often one of two components at right an- two resulting functions are not the same. In gles. general, composing functions in a different order will produce different results. component forces See component vectors. compound growth The growth of money that is associated with COMPOUND component vectors The components of INTEREST. Compound growth can be tabu- a given VECTOR (such as a force or velocity) lated in tables called compound growth ta- are two or more vectors with the same ef- bles, which give the multiplying factors for fect as the given vector. In other words the different periods of time (usually years) given vector is the resultant of the compo- and different percentage rates. These tables nents. Any vector has an infinite number of can be drawn in the form of a matrix, with sets of components. Some sets are more use the number of years increasing from left to than others in a given case, especially pairs right and the percentage rates increasing ° at 90 . The component of a given vector from top to bottom. Compound growth ta- (V) in a given direction is the projection of bles can also be used to record the growth θ the vector on to that direction; i.e. Vcos , in population with time for different popu- θ where is the angle between the vector and lation growth rates. the direction. compound interest The interest earned component velocities See component on capital, when the interest in each period vectors. is added to the original capital as it is earned. Thus the capital, and therefore the A function formed composite function interest on it, increases year by year. If P is by combining two functions f and g. There the principal (the original amount of are several notations for such a composite money invested), R percent the interest rate function: for example, g[f(x)], gof, g.f, gf. per annum, and n the number of interest For a composite function to exist it is nec- periods, then the compound interest is essary that the domain of g contains the P(1 + R/100)n range of f. The order of the functions f and This formula is a geometric progression g in a composite function has to be specified since, in general, g[f(x)] ≠ f[g(x)]. This whose first term is P (when n = 0) and means that to find the composite function whose common ratio is (1 + R/100). Com- g[f(x)], f(x) is found first and then g[f(x)] is pare simple interest. found. The non-commuting nature of composite functions can be seen by considering compound proposition See proposition. the functions f(x) = sinx and g(x) = x2 for which g[f(x)] = (sinx)2 and f[g(x)] = sin(x2). compound transformation A transformation on a figure that can be regarded as composite number An integer that has one transformation followed by another more than one prime factor. For example, transformation. For example, a compound 4 (= 2 × 2), 6 (= 2× 3), 10 (= 2 × 5) are com- transformation could consist of a rotation posite numbers. The prime numbers and followed by a translation through space. ±1 are not composite. Another example is a translation followed by a reflection. If the first and second trans- composition The process of combining formations on the object O are denoted T1 two or more functions to obtain a new one. and T2 respectively, then the compound For example the composition of f(x) and transformation is denoted by T2T1(0). In g(x), which is written f•g, is obtained by general, performing the two transforma- applying g(x) and then applying f(x) to the tions in different order leads to two differ- result. If f(x) = x – 2 and g(x) = x3 + 1 then ent compound transformations. This is f•g(x) = f(x3+1) = x3 – 1, whereas g•f(x) = consistent with the fact that transforma- g(x–2) = (x – 2)3 + 1. As can be seen, these tions can be described by matrices.

39 computability computability /kŏm-pyoo-tă-bil-ă-tee/ integrated circuits, which have become Intuitively, a problem or function is com- more and more complicated. As the elec- putable if it is capable of being solved by an tronic circuits used in the various devices in ideal machine (computer) in a finite time. a computer system have decreased in size In the 1930s it was discovered that some and increased in complexity, so the com- problems had no algorithmic solution and puters themselves have grown smaller, could not be solved by computers. This led faster, and more powerful. The microcom- many mathematicians to try to formulate a puter has been developed as a somewhat precise definition of the intuitive concept simpler version of the full-size mainframe of computability, and Turing, Gödel, and computer. Computers now have an im- Church independently came up with three mense range of uses in science, technology, very different abstract definitions, which industry, commerce, education, and many all turned out to define exactly the same set other fields. of functions. The definition given by Tur- ing is that a function f is computable if for computer graphics The creation and re- each element x of its domain, when some production of pictures, photographs, and representation of x is placed on the tape of diagrams using a computer. There are a TURING MACHINE, the machine stops in a many different formats for storing images finite time with a representation of f(x) on but they fall into two main classes. In raster the tape. graphics the picture is stored as a series of dots (or pixels). The information in the computer Any automatic device or ma- computer file is a stream of data indicating chine that can perform calculations and the presence or absence of a dot and the other operations on data. The data must be color if present. Images of this type are received in an acceptable form and is sometimes known as bitmaps. This format processed according to instructions. The is used for high-quality artwork and for most versatile and most widely used com- photographs. Diagrams are more conve- puter is the digital computer, which is usu- niently stored using vector graphics, in ally referred to simply as a computer. See which the information is stored as mathe- also analog computer; hybrid computer. matical instructions. For example, it is pos- A digital computer is an automatically sible to specify a circle by its center, its controlled calculating machine in which in- radius, and the thickness of the line form- formation, generally known as data, is rep- ing the circumference. More complicated resented by combinations of discrete curves are usually drawn using BEZIER electrical pulses denoted by the binary dig- CURVES. Vector images are easier to change its 0 and 1. Various operations, both arith- and take up less storage space than raster metical and logical, are performed on the images. data according to a set of instructions (a program). Instructions and data are fed computer modeling The development of into the main store or memory of the com- a description or mathematical representa- puter, where they are held until required. tion (i.e. a model) of a complicated process The instructions, coded like the data in bi- or system, using a computer. This model nary form, are analyzed and carried out by can then be used to study the behavior or the central processor of the computer. The control of the process or system by varying result of this processing is then delivered to the conditions in it, again with the aid of a the user. computer. The technology used in digital computers is so highly advanced that they can op- concave /kong-kayv, kong-kayv/ Curved erate at extremely high speeds and can inwards. For example, the inner surface of store a huge amount of information. The a hollow sphere is concave. Similarly in tube valves used in early computers were two dimensions, the inside edge of the cir- replaced by transistors; transistors, resis- cumference of a circle is concave. A con- tors, etc., were subsequently packed into cave polygon is a polygon that has one (or

40 cone

conditional convergence See absolute convergence.

conditional equation See equation. convex concave conditional probability See probability.

cone A solid defined by a closed plane curve (forming the base) and a point outside the plane (the vertex). A line segment from the vertex to a point on the plane Concave and convex curvatures curve generates a curved lateral surface as the point moves around the plane curve. The line is the generator of the cone and more) interior angles greater than 180°. the plane curve is its directrix. Any line seg- Compare convex. ment from the vertex to the directrix is an element of the cone. concentric Denoting circles or spheres If the directrix is a circle the cone is a that have the same center. For example, a circular cone. If the base has a center, a line hollowed out sphere consists of two con- from the vertex to this is an axis of the centric spherical surfaces. Compare eccen- cone. If the axis is at right angles to the tric. base the cone is a right cone; otherwise it is an oblique cone. The volume of a cone is conclusion The proposition that is as- one third of the base area multiplied by the serted at the end of an argument; i.e. what altitude (the perpendicular distance from the argument sets out to prove. the vertex to the base). For a right circular cone condition In logic, a proposition or state- V = πr2h/3 ment, P, that is required to be true in order where r is the radius of the base and h the that another proposition Q be true. If P is altitude. The area of the curved (lateral) a necessary condition then Q could not be surface of a right circular cone is πrs, where true without P. If P is a sufficient condi- s is the length of an element (the slant tion, then whenever P is true Q is also true, height). but not vice versa. For example, for a quadrilateral to be a rectangle it must satisfy the necessary condition that two of its sides be parallel, but this is not a sufficient condition. A sufficient condition for a quadrilateral to be a rhombus is that all its sides have a length of 5 centimeters, but this is not a necessary condition. For a rectangle to be a square it is both a necessary ellipse hyperbola and a sufficient condition that all its sides are of equal length. In formal terms, if P is a necessary condition for Q, then Q → P. If P is a sufficient condition, then P → Q. If P is a necessary and sufficient condition for Q then P ≡ Q. See also biconditional, symbolic logic. conditional (conditional statement; con- parabola ditional proposition) An if… then… state- The three sections of a cone – the ellipse, the ment. parabola, and the hyperbola.

41 confidence interval

If an extended line is used to generate 3. Three sides of one are equal to three the curved surface (i.e. extending beyond sides of the other. the directrix and beyond the vertex), an ex- In solid geometry, two figures are con- tended surface is produced with two parts gruent if they can be brought into coinci- (nappes) each side of the vertex. More dence in space. strictly, this is called a conical surface Sometimes the term directly congruent is used to describe identical figures; indi- confidence interval An interval that is rectly congruent figures are ones that are thought, with a preselected degree of con- mirror images of each other. Compare sim- fidence, to contain the value of a parame- ilar. ter being estimated. For example, in a 2. Two elements a and b of a ring are con- binomial experiment the α% confidence gruent modulo d if there exist elements in interval for the probability of success P lies the ring, p, q, and r, such that a = dp + r, b between P – a and P + a, where = dq + r. Intuitively, this means that they a = z√[P(1 – P)/N] N is the sample size, P the proportion of both leave the same remainder when di- successes in the sample, and z is given by a vided by d. table of area under the standard normal 3. Two square matrices A and B are con- curve. P will lie in this interval α times out gruent if A can be transformed into B by a of every 100. congruent transformation; i.e. there exists a nonsingular matrix C such that B = confocal conics /kon-foh-kăl/ Two or CTAC, where CT is the transpose of C. more conics that have the same focus. conic /kon-ik/ A type of plane curve de- conformable matrices See matrix. fined so that for all points on the curve the distance from a fixed point (the focus) has conformal mapping A geometrical a constant ratio to the perpendicular dis- transformation that does not change the tance from a fixed straight line (the direc- angles of intersection between two lines or trix). The ratio is the eccentricity of the curves. For example, Mercator’s projection conic, e; i.e. the eccentricity is the distance is a conformal mapping in which any angle from curve to focus divided by distance between a line on the spherical surface and from curve to directrix. a line of latitude or longitude will be the The type of conic depends on the value same on the map. of e: when e is less than 1 it is an ELLIPSE; when e equals 1 it is a PARABOLA; when e is /kong-groo-ĕns/ The property congruence greater than 1 it is a HYPERBOLA. A circle is of being congruent. a special case of an ellipse with eccentricity 0. congruent /kong-groo-ĕnt/ 1. Denoting The original definition of conics was as two or more figures that are identical in plane sections of a conical surface – hence size and shape. Two congruent plane fig- the name conic section. In a conical surface ures will fit into the area occupied by each θ other; i.e. one could be brought into coin- having an apex angle of 2 , the cross-sec- θ cidence with the other by moving it with- tion on a plane that makes an angle with out change of size. Two circles are the axis of the cone, (i.e. a plane parallel to congruent if they have the same radius. The the slanting edge of the cone) is a parabola. conditions for two triangles to be congru- A cross-section in a plane that makes an ent are: 1. Two sides and the included angle greater than θ with the axis is an el- angle of one are equal to two sides and the lipse. A cross-section in a plane making an included angle of the other. angle less than θ with the axis is a hyper- 2. Two angles and the included side of one bola and because this plane cuts both are equal to two angles and the included halves (nappes) of the cone, the hyperbola side of the other. has two arms.

42 conjunction

There are various ways of writing the conjugate angles A pair of angles that equation of a conic. In Cartesian coordi- add together to make a complete revolu- nates: tion (360° or 2π radians). Compare com- (1 – e2)x2 + 2e2qx + y2 = e2q plementary angles; supplementary angles. where the focus is at the origin and the directrix is the line x = q (a line parallel to the y-axis a distance q from the origin). The general equation of a conic (i.e. the general conic) is: ax2 + bxy + cy2 + dx + ey + f = 0 where a, b, c, d, e, and f are constants (here e is not the eccentricity). This includes de- α generate cases (degenerate conics), such as β a point, a straight line, and a pair of intersecting straight lines. A point, for example, α β is a section through the vertex of the coni- Conjugate angles: + = 360° cal surface. A pair of intersecting straight lines is a section down the axis of the surface. The tangent to the general conic at the conjugate axis See hyperbola. point (x ,y ) is: 1 1 Two com- ax x + b(xy + x y) + cy y + conjugate complex numbers 1 1 1 1 plex numbers of the form x + iy and x – iy, d(x + x ) + e(y + y ) + f = 0 1 1 which when multiplied together have a real product x2 + y2. If z = x + iy, the complex conical helix See helix. _ conjugate of z is z = x – iy. conical projection See central projec- conjugate diameter Two diameters AB tion. and CD of an ellipse are said to be conjugate diameters if CD is the diameter that conical surface See cone. contains the mid-points of the set of chords parallel to AB, where a diameter of an el- A type of surface in which sec- conicoid lipse is defined to be a locus of the mid- tions of the surface are conics. Conicoids points of a set of parallel chords. For an can have equations in three-dimensional equation for an ellipse of the form x2/a2 + Cartesian coordinates as follows: y2/b2 = 1 the product of the gradients of elliptic paraboloid two conjugate diameters is –b2/a2. This 2 2 2 2 x /a + y /b = 2z/c means that, in general, conjugate diameters hyperbolic paraboloid are not perpendicular (except in the case of 2 2 2 2 x /a – y /b = 2z/c a = b; i.e. a circle). hyperboloid 2 2 2 2 2 2 x /a + y /b – z /c = 1 conjugate hyperbola See hyperbola. ellipsoid x2/a2 + y2/b2 + z2/c2 = 1 conjunction /kŏn-junk-shŏn/ Symbol: ∧ Conicoids are also known as quadric In logic, the relationship and between two surfaces or quadrics. The simplest types are conicoids of revolution, formed by rotating a conic about an axis. A sphere is a special P Q P ∧ Q case of an ELLIPSOID in which all three axes are equal (just as a circle is a special case of T T T an ellipse). See also paraboloid. See illus- T F F tration overleaf. F T F F F F conic sections See conic. Conjunction

43 connected

hyperboloid of one sheet

hyperboloid of two sheets

elliptic paraboloid

ellipsoid

hyperbolic paraboloid

Conicoid surfaces

44 conservation of mass and energy or more propositions or statements. The celeration due to gravity is denoted g then: conjunction of P and Q is true when P is a = [(m2 – m1)g]/(m1 + m2) and T = true and Q is true, and false otherwise. The 2m1m2g/(m1 + m2). truth table definition for conjunction is shown in the illustration. Compare dis- connectivity The number of cuts needed junction. See also truth table. to break a shape in two parts. For example, a rectangle, a circle, and a sphere, all have connected Intuitively, a connected set is a a connectivity of one. A flat disk with a set with only one piece. More rigorously, a hole in it or a torus has a connectivity of set in a topological space is said to be con- two. See also topology. nected if it is not the union of two nonempty disjoint closed sets. For example, consequent In logic, the second part of a the set of rational numbers is not con- conditional statement; a proposition or nected since the set of all rational numbers statement that is said to follow from or be less than √3 and the set of all rational num- implied by another. For example, in the bers greater than √3 are both closed in the statement ‘if Jill is happy, then Jack is set of all rational numbers. However, the happy’, ‘Jack is happy’ is the consequent. set of all real numbers is connected since no Compare antecedent. See also implication. such decomposition is possible. A subset S of a topological space is path-connected if conservation law A law stating that the any two points in S can be joined by a path total value of some physical quantity is in S, where a path from a to b in S is a con- conserved (i.e. remains constant) through- tinuous map f:[0,1]→S such that f(0)=a out any changes in a closed system. The and f(1)=b. conservation laws applying in mechanics A simply connected set is a path-con- are the laws of constant mass, constant en- nected set such that any closed curve ergy, constant linear momentum, and con- within it can be deformed continuously to stant angular momentum. a point of the set without leaving the set. A path-connected set that is not simply con- conservation of angular momentum, nected is multiply connected and the con- law of See constant angular momentum; nectivity of the set is one plus the law of. maximum number of points that can be deleted if the set is a curve, or one plus the conservation of energy, law of See con- maximum number of closed cuts that can stant energy; law of. be made if the set is a surface, without separating the set so that it is no longer path- conservation of linear momentum, connected. For example, the region law of See constant linear momentum; between two concentric circles has connec- law of. tivity two, or is doubly connected, since one closed cut can be made which still conservation of mass, law of See con- leaves a connected region. stant mass; law of. connected particles Particles that are conservation of mass and energy The connected by a light inextensible string. If law that the total energy (rest mass energy particles are connected then their motions + kinetic energy + potential energy) of a are not independent. An example is the closed system is constant. In most chemical motion of two particles joined by a light in- and physical interactions the mass change extensible string which passes over a fixed is undetectably small, so that the measur- pulley. If the two particles have masses m1 able rest-mass energy does not change (it is and m2 then the heavier particle moves up. regarded as ‘passive’). The law then be- > If m2 m1, the acceleration down of m2 comes the classical law of conservation of (and the acceleration up of m1) = a, the ten- energy. In practice, the inclusion of mass in sion in the string is denoted T and the ac- the calculation is necessary only in the case

45 conservation of momentum, law of of nuclear changes or systems involving INITE INTEGRAL includes an arbitrary con- very high speeds. See also mass-energy stant (the constant of integration), which equation; rest mass. depends on the limits chosen. conservation of momentum, law of constant angular momentum, law of See constant linear momentum; law of. (law of conservation of angular momentum) The principle that the total angular conservative field A field such that the momentum of a system cannot change un- work done in moving an object between less a net outside torque acts on the system. two points in the field is independent of the See also constant linear momentum; law path taken. See conservative force. of. conservative force A force such that, if it constant energy, law of (law of conser- moves an object between two points, the vation of energy) The principle that the energy transfer (work done) does not de- total energy of a system cannot change un- pend on the path between the points. It less energy is taken from or given to the must then be true that if a conservative outside. See also mass-energy equation. force moves an object in a closed path (back to the starting point), the energy constant linear momentum, law of transfer is zero. Gravitation is an example (law of conservation of linear momentum) of a conservative force; friction is a non- The principle that the total linear momen- conservative force. tum of a system cannot change unless a net outside force acts. consistent Describing a theory, system, or set of propositions giving rise to no contra- constant mass, law of (law of conserva- dictions. Arithmetic, for example, is tion of mass) The principle that the total thought to be a consistent logical system mass of a system cannot change unless because none of its axioms nor any of the mass is taken from or given to the outside. theorems that are derived from these by the See also mass-energy equation. rules, are believed to be contradictory. See contradiction. constant momentum, law of See constant linear momentum; law of. consistent equations A set of equations that can be satisfied by at least one set of construct In geometry, to draw a figure, values for the variables. For example, the line, point, etc., meeting certain conditions; equations x + y = 2 and x + 4y = 6 are sat- e.g. a line that bisects a given line. Usually isfied by x = 2/3 and y = 4/3 and they are certain specific restrictions are imposed on therefore consistent. The equations x + y = the method used; e.g. using only a straight 4 and x + y = 9 are inconsistent. edge and compasses. There is an important class of problems concerning questions of constant A quantity that does not change whether certain things can be constructed its value in a general relationship between using given methods. Examples are two variables. For example, in the equation y = celebrated problems of whether it is possi- 2x + 3, where x and y are variables, the ble to construct two lines that trisect a numbers 2 and 3 are constants. In this case given angle, and to construct a square they are absolute constants because their equal in area to a given circle – in both values never change. Sometimes a constant cases using only a straight edge and com- can take any one of a number of values in passes. Both these constructions have been different applications of a general formula. shown to be impossible. In the general quadratic equation ax2 + bx + c = 0 constructive proof A proof that not only a, b, and c are arbitrary constants because shows that a certain mathematical entity, no values are specified for them. An INDEF- such as a root of an equation or a fixed-

46 continuum point of a transformation, exists, but also such as b0, b1, b2, etc., are called conver- explicitly produces it. Constructive proofs gents. are usually considerably longer and more complicated and harder to find than non- continued product Symbol: Π The prod- constructive proofs of the same results. uct of a number of related terms. For ex- Many results that have been proved non- ample, 2 × 4 × 6 × 8… is a continued constructively have yet to be given con- product, written: structive proofs. k Πan constructivist mathematics An ap- This means the product of k terms, with proach to mathematics that insists that the nth term, an = 2n. only constructive proofs are acceptable ∞ and rejects as meaningless nonconstructive Πan proofs. Constructivist mathematics is con- 1 siderably more restricted than classical has an infinite number of terms. mathematics and rejects many of its theorems. Different varieties of constructivism continuous function A function that has differ over what exactly counts as an ac- no sudden changes in values as the variable ceptably constructive proof. One of the increases or decreases smoothly. More pre- best known examples of mathematical cisely, a function f(x) is continuous at a constructivism is the intuitionism of point x = a if the limit of f(x) as x ap- Brouwer. proaches a is f(a). When a function does not satisfy this condition at a point, it is contact force A force between bodies in said to be discontinuous, or to have a dis- which the bodies are in contact. By con- continuity, at that point. For example, tanθ trast, in a non-contact force such as the has discontinuities at θ = π/2, 3π/2, 5π/2,… gravitational force between the Earth and A function is continuous in an interval of x the Sun, the two bodies are separated by if there are no points of discontinuity in ‘empty’ space. Examples of contact forces that interval. include tension in which a particle is hanging in equilibrium at the end of a string and continuous stationery A length of fan- thrust in which a particle is suspended by a folded paper with sprocket holes along spring underneath it. In the first case the tension in the string acts upwards on the each side for transporting it through the particle, as does the thrust in the spring in printer of a computer. It may be perforated the second case. for tearing it into separate sheets after printing; there may also be perforations continued fraction An expression that along the sides to tear off the sprocket can be written in the form: holes.

ϩ b0 a1 continuum /kŏn-tin-yoo-ŭm/ (pl. con- ϩ b1 a2 tinua) A compact connected set with at ϩ least two points. The conditions that the b2 a3 b ϩ ... set has at least two points and is connected 3 imply that the set has an infinite number of where all the a’s and b’s are numbers points. Any closed interval of the real num- (which are usually positive integers). If this bers is a continuum and the set of all real fraction terminates after a finite number of numbers is called the real continuum. terms it is said to be terminating or finite. The continuum hypothesis is the con- If it does not terminate it is said to be non- jecture that every infinite subset of the real terminating or infinite. The values of a con- continuum has the CARDINAL NUMBER ei- tinued fraction that are obtained by ther of the positive integers or of the entire truncating the fraction at a finite point set of real numbers. This is equivalent to

47 contour integral

height in meters

contour lines 50 40 30 20 10 50 40 30 cross-section along the dotted line 20 10 0

map

A hill shown as contour lines on a map and as a cross-section.

' the statement that 2 0 is the lowest cardi- to obtain a specific type of desired dynam- nal number greater than '. See aleph. ical behavior from a physical system. Some device is added to the system to affect its contour integral See line integral. behavior, and the concept of feedback is applied very widely in control systems. contour line A line on a map joining There are many important applications of points of equal height. Contour lines are control theory, including industrial ma- usually drawn for equal intervals of height, chinery, robots, chemical processes, and so that the steeper a slope, the closer to- the stability of cars, trains, and aircraft. See gether the contour lines. See illustration also cybernetics. overleaf. control unit See central processor. contradiction In LOGIC, a proposition, statement, or sentence that both asserts convergent sequence A sequence in something and denies it. It is a form of which the difference between each term words or symbols that cannot possibly be and the one following it becomes smaller true; for example, ‘if I can read the book throughout the SEQUENCE; i.e. the differ- then I cannot read the book’ and ‘he is ence between the nth term and the (n + 1)th coming and he is not coming’. Compare term decreases as n increases. For example, tautology. {1, ½, ¼, ⅛, …} is a convergent sequence, but {1, 2, 4, 8, …} is not. A convergent se- contradiction, law of See laws of quence has a limit; i.e. a value towards thought. which the nth term tends as n becomes infinitely large. In the first example here the contrapositive /kon-tră-poz-ă-tiv/ In limit is 0. Compare divergent sequence. See logic, a statement in which the antecedent also convergent series; geometric sequence. and consequent of a conditional are reversed and negated. The contrapositive of convergent series An infinite SERIES a1 + → ∼ →∼ A B is B A (not B implies not A), a2 + … is convergent if the partial sums a1 and the two statements are logically equiv- + a2 + … + an tend to a limit value as n alent. See implication. See also bicondi- tends to infinity. For example, the series S tional. = 1 + ½ + ¼ + ⅛ + … is a convergent series with sum 2, since 2 is the limit approached control theory A branch of applied by the sum of the first n terms, namely mathematics that is concerned with trying 1–(1/2n) as n tends to infinity. The series 1

48 coplanar

+ (–1) + 1 + (–1) + 1 + … is not convergent. from the graph. For example, air pressure Compare divergent series. See also conver- depends on height above sea level. A stan- gent sequence; geometric series. dard curve of altitude against air pressure may be plotted on a graph. An air-pressure converse A logical IMPLICATION taken in measurement can then be converted to an the reverse order. For example, the con- indication of height by reading the appro- verse of priate value from the graph. if I am under 16, then I go to school is convex /kon-veks, kon-veks/ Curved out- if I go to school, then I am under 16. wards. For example, the outer surface of a The converse of an implication is not al- sphere is convex. Similarly, in two dimen- ways true if the implication itself is true. sions, the outside of a circle is its convex There are a number of theorems in mathe- side. A convex polygon is one in which no matics for which both the statement and interior angle is greater than 180°. Com- the converse are true. For example, the the- pare concave. orem: if two chords of a circle are equidistant coordinate geometry See analytical from the center, then they are equal geometry. has a true converse: if two chords of a circle are equal, then coordinates Numbers that define the po- they are equidistant from the center of the sition of a point, or set of points. A fixed circle. point, called the origin, and fixed lines, called axs, are used as a reference. For ex- conversion factor The ratio of a meas- ample, a horizontal line and a vertical line urement in one set of units to the equiva- drawn on a page might be defined as the x- lent numerical value in other units. For axis and the y-axis respectively, and the example, the conversion factor from inches point at which they cross as the origin (O). to centimeters is 2.54 because 1 inch = 2.54 Any point on the page can then be given centimeters (to two decimal places). two numbers – its distance from O along the x-axis and its distance from O along conversion graph A graph showing a re- the y-axis. Distances to the right of the ori- lationship between two variable quantities. gin for x and above the origin for y are pos- If one quantity is known, the correspond- itive; distances to the left of the origin for x ing value of the other can be read directly and below the origin for y are negative. These two numbers would be the x and y coordinates of the point. This type of coordinate system is known as a rectangular height in meters Cartesian coordinate system. It can have 4000 – above sea level two axs, as on a flat surface, such as a map, or three axs, when depth or height also 3000 – have to be specified. Another type of coordinate system (POLAR COORDINATES) ex- 2000 – presses the position of a point as radial distance from the origin (the pole), with its direction expressed as an angle or angles 1000 – (positive when anticlockwise) between the radius and a fixed axis (the polar axis). See

0 – – – – – also Cartesian coordinates. 0.06 0.07 0.08 0.09 0.10 air pressure in megapascals (MPa) coplanar /koh-play-ner/ Lying in the same A conversion graph for finding altitude from plane. Any set of three points, for example, air pressure measurements. (Standard air could be said to be coplanar because there pressure at sea level is 0.101325 megapascals.) is a plane in which they all lie. Two vectors

49 coplanar forces are coplanar if there is a plane that con- where cov and var denote covariance and tains both. variance respectively. It satisfies –1 ≤ r ≤ 1 and is a measurement of the interdepen- coplanar forces Forces in a single plane. dence between random variables, or their If only two forces act through a point, they tendency to vary together. If r ≠ 0 then X must be coplanar. So too are two parallel and Y are said to be correlated: they are forces. However, nonparallel forces that correlated positively if 0 < r ≤ 1 and nega- do not act through a point cannot be copla- tively if –1 ≤ r < 0. If r = 0 then X and Y are nar. Three or more nonparallel forces act- said to be uncorrelated. ing through a point may not be coplanar. If For two sets of numbers (x1,…xn) and a set of coplanar forces act on a body, their (y1,…yn) the correlation coefficient is n algebraic sum must be zero for the body to Σ - - (xi – x) (yi – y) be in equilibrium (i.e. the resultant in one i=1 direction must equal the resultant in the r = ––––––––––––––––––––––––n n Σ - 2 Σ - 2 opposite direction). In addition there must [ (xi – x) (yi – y) ] be no couple on the body (the moment of i=1 i=1 the forces about a point must be zero). where x- and y- are the corresponding Coriolis force /kor-ee-oh-lis, ko-ree-/ A means. It measures how near the points ‘fictitious’ force used to describe the mo- (x1,y1)…(xn,yn) are to lying on a straight tion of an object in a rotating system. For line. If r = 1 the points lie on a line and the instance, air moving from north to south two sets of data are said to be in perfect correlation. over the surface of the Earth would, to an observer outside the Earth, be moving in a correspondence See function. straight line. To an observer on the Earth the path would appear to be curved, as the corresponding angles A pair of angles Earth rotates. Such systems can be de- on the same side of a line (the transversal) scribed by introducing a tangential Corio- that intersects two other lines; they are be- lis ‘force’. The idea is used in meteorology tween the transversal and the other lines. If to explain wind directions. It is named for the intersected lines are parallel, the corre- the French physicist Gustave-Gaspard sponding angles are equal. Compare alter- Coriolis (1792–1843). nate angles. corollary /kŏ-rol-ă-ree/ A result that fol- cos /koz/ See cosine. lows easily from a given theorem, so that it is not necessary to prove it as a separate cosec /koh-sek/ See cosecant. theorem. cosecant /koh-see-kănt, -kant/ (cosec; csc) correction A quantity added to a previ- A trigonometric function of an angle equal ously obtained approximation to yield a to the reciprocal of its sine; i.e. cosecα = better approximation. When using loga- 1/sinα. See also trigonometry. rithmic or trigonometric tables the correction is the number added to a logarithm or cosech /koh-sech, -sek/ A hyperbolic cose- to a trigonometric function in the table to cant. See hyperbolic functions. give the logarithm or trigonometric function of a number or angle that is not in the cosh /kosh, kos-aych/ A hyperbolic cosine. table. See hyperbolic functions. correlation In statistics, the correlation cosine /koh-sÿn/ (cos) A trigonometric coefficient of two random variables X and function of an angle. The cosine of an angle Y is defined by α (cosα) in a right-angled triangle is the r(X,Y) = cov(X,Y)/√(var(X)var(Y) ratio of the side adjacent to it, to the hy-

50 countable

y C

+1 α β π π π π 0 2 3 4 x

–1 Cosine: The graph of y = cos x. # A mnD B potenuse. This definition applies only to Cotangent rule angles between 0° and 90° (0 and π/2 radians). More generally, in rectangular Carte- cotangent rule can also be expressed in the sian coordinates, the x-coordinate of any form: (m + n) cot θ = n cot A–m cot B, point on the circumference of a circle of ra- where A is the angle at A and B the angle dius r centred on the origin is rcosα, where at B. α is the angle between the x-axis and the An important special case of the cotangent radius to that point. In other words, the co- rule is that in which D is the mid-point of sine function is the horizontal component AB. In this case the rule becomes 2 cot θ = of a point on a circle. Cosα varies periodi- cot α – cot β. cally in the same way as sinα, but 90° The cotangent rule can be used to analyze ahead. That is: cosα is 1 when α is 0°, falls some problems in mechanics involving to zero when α = 90° (π/2) and then to –1 equilibrium, particularly in the case when when α = 180° (π), returning to zero at α = D is the mid-point of AB. 270° (3π/2) and then to +1 again at α = 360° (2π). This cycle is repeated every com- coth /koth/ A hyperbolic cotangent. See plete revolution. The cosine function has hyperbolic functions. the following properties: cosα = cos(α + 360°) = sin(α + 90°) coulomb /koo-lom/ Symbol: C The SI unit cosα = cos(–α) of electric charge, equal to the charge cos(90° + α) = –cosα transported by an electric current of one The cosine function can also be defined ampere flowing for one second. 1 C = 1 A as an infinite series. In the range from +1 to s. The unit is named for the French physi- –1: cist Charles Augustin de Coulomb (1736– cosx = 1 – x2/2! + x4/4! – x6/6! + … 1806). cosine rule In any triangle, if a, b, and c countable (denumerable) A set is count- are the side lengths and γ is the angle opable if it can be put in one-one correspon- posite the side of length c, then dence with the integers. The set of rational c2 = a2 + b2 – 2abcosγ numbers, for example, is countable whereas the set of real numbers is not (see cot /kot/ See cotangent. Cantor’s diagonal argument). To show that the rationals are countable we need to cotangent /koh-tan-jĕnt/ (cot) A trigono- show how they can be arranged in a series metric function of an angle equal to the such that every rational number will be in- reciprocal of its tangent; i.e. cotα = 1/tanα. cluded somewhere. See also trigonometry. If we consider the array in the diagram it is clear that every rational number will cotangent rule A rule for triangles that occur in it somewhere. But by starting with states that if the side AB of a triangle is di- 1/1 and following the path indicated we vided into the ratio m:n by a point D then can enumerate every number in the array. (m + n) cot θ = m cot α – n cot β, where θ, If we reduce each fraction to its lowest α, and β are defined in the diagram. The terms and then remove any that has al-

51 counterclockwise

1 2 3 4 5 ... can be written uniquely as x = A–1 = b. Cramer’s rule states that the xi can be writ- 1 1/1 1/2 1/3 1/4 1/5 ... ten as: xi = (b1A1i + b2A2i + ...bnAni)/(det A), 2 2/1 2/2 2/3 2/4 2/5 ... for all i between 1 and n, where the bj are the entries in the b column and the Aji are 3 3/1 3/2 3/3 3/4 3/5 ... the cofactors of A. This is the case because A–1 = adjA/detA, where adjA is the adjoint 4 4/1 4/2 4/3 4/4 4/5 ... of the matrix A. In turn, this means that x = (adj A) b/detA. In the case of two linear 5 5/1 5/2 5/3 5/4 5/5 ... equations with two unknowns x , and x . . . . . 1 2 . . . . . of the form ax1 + bx2 = l, Countable cx1 + dx2 = m, with ad – bc ≠ 0, Cramer’s rule leads to the result: ready occurred in the list it is clear that this x1 = (ld–bm)/(ad–bc), will give a list in which each rational num- x2 = (am–lc)/(ad–bc). ber occurs once and only once. critical damping See damping. counterclockwise See anticlockwise. critical path The sequence of operations couple A pair of equal parallel forces in that should be followed in order to com- opposite directions and not acting through plete a complicated process, task, etc., in a single point. Their linear resultant is zero, the minimum time. It is usually determined but there is a net turning effect (moment). by using a computer. The net turning effect T (the torque) is given by: critical region See acceptance region. T = Fd1 + Fd2 F being the magnitude of each force and d1 cross multiplication A way of simplify- and d2 the distances from any point to the ing an equation in which one or both terms lines of action of each force. This is equiv- are fractions. The product of the numera- alent to: tor on the left-hand side of the equation T = Fd and the denominator on the right-hand where d is the distance between the forces. side equals the product of the denominator on the left-hand side and the numerator on covariance /koh-vair-ee-ăns/ A statistic the right-hand side. For example, cross that measures the association between two multiplying the equation variables. If for x and y there are n pairs of 4x 3y values (xnyi), then the covariance is defined — = — as 3 2 ∑ ′ ′ [1/(n – 1)] (xi – x )(yi – y ) gives where x′ and y′ are the mean values. 4x × 2 = 3y × 3 or CPU /see-pee-yoo/ See central processor. 8x – 9y

Cramer’s rule /kray-merz/ A rule that cross product See vector product. gives the solution of a set of linear equations in terms of a matrix. If a set of n lin- cross-section (section) A plane cutting ear equations for n unknown variables x1, through a solid figure or the plane figure x2, ... xn can be written in the form of a ma- produced by such a cut. For example, the trix equation Ax = b, where A is an invert- cross-section through the middle of a ible matrix the solution for the equations sphere is a circle. A vertical cross-section

52 cubic graph through an upright cone and off the axis is The cube roots of unity satisfy the follow- a hyperbola. ing relations: _ ω = ω2 crystallographic group /kris-tă-lŏ-graf- ω2 + ω + 1 = 0 ik/ A POINT GROUP that is compatible with the symmetry of a crystal. This restriction cubic equation A polynomial equation in means that there are 32 possible point which the highest power of the unknown groups. Crystallographic groups can only variable is three. The general form of a have twofold, threefold, fourfold, or six- cubic equation in a variable x is fold rotational symmetry since any other ax3 + bx2 + cx + d = 0 rotational symmetry would be incompati- where a, b, c, and d are constants. It is also ble with the translational symmetry exist- sometimes written in the reduced form 3 2 ing in a crystal. However, fivefold and x + bx /a + cx/a + d/a = 0 icosahedral symmetries are possible in In general, there are three values of x QUASICRYSTALLINE SYMMETRY. The SPACE that satisfy a cubic equation. For example, 3 2 GROUP of a crystal is characterized by the 2x – 3x – 5x + 6 = 0 combination of the crystallographic point can be factorized to group and the translational symmetry of (2x + 3)(x – 1)(x – 2) = 0 the crystal lattice. and its solutions (or roots) are –3/2, 1, and 2. On a Cartesian coordinate graph, the curve crystallographic symmetry The symme- 3 2 try that is associated with the regular three- y = 2x – 3x – 5x + 6 crosses the x-axis at x = –3/2, x = +1, and x dimensional structure of a crystal. The set = +2. of crystallographic symmetry operations of a crystal makes up the SPACE GROUP of the cubic graph A graph of the equation crystal. This consists of the symmetry op- y = ax3 + bx2 + cx + d, erations of the CRYSTALLOGRAPHIC GROUP where a, b, c, and d are constants. As x be- and the symmetry operations associated comes very large y→ax3. This means that if with the translational symmetry of the a > 0 then y→∞ as x→∞ and y→–∞ as crystal. x→–∞ and that if a < 0 then y→–∞ as x→∞ and y→∞ as x→–∞. Since dy/dx = 3ax2 + csc See cosecant. 2bx + c the nature of the stationary points of the cubic graph are determined by the cube 1. The third power of a number or nature of the solutions of the quadratic × × 3 variable. The cube of x is x x x = x (x equation 3ax2 +2bx + c = 0. If the qua- cubed). dratic equation has two real roots that are 2. In geometry, a solid figure that has six distinct then the curve has two turning 3 square faces. The volume of a cube is l , points that are distinct, one of which is a where l is the length of a side. maximum and one of which is a minimum. If the quadratic equation has two real roots cube root An expression that has a third that are equal the curve has a point of in- power equal to a given number. The cube flexion. If the quadratic equation does not root of 27 is 3, since 33 = 27. have any real roots the curve does not have any real stationary points. The curve is cube root of unity A complex number z continuous since y does not tend to infinity for which z3 = 1. There are three cube roots for any finite value of x. A cubic curve can of unity, which can be denoted as 1, ω, and either: (1) cross the x-axis three times; (2) ω2, where ω and ω2 are defined by cross and touch the x-axis; (3) cross the x- ω = exp(2πi/3) = cos(2π/3) + i sin (2π/3) axis once; or (4) touch the x-axis at the = + 1/2 –√3i/2, point of inflexion. This means that there ω2 = exp(4πi/3) = cos(4π/3) + i sin has to be at least one point of intersection (4π/3) = –1/2 – √3i/2. with the x-axis. This result means, in turn,

53 cuboid that a CUBIC EQUATION ax3 + bx2 + cx + d = (d2y/dx2)/[1 + (dy/dx)2]3/2 0 must have at least one real root. curve A set of points forming a continu- cuboid A box-shaped solid figure ous line. For example, in a graph plotted in bounded by six rectangular faces. The op- Cartesian coordinates, the curve of the posite faces are congruent and parallel. At equation y = x2 is a parabola. A curved sur- each of the eight vertices, three faces meet face may similarly represent a function of at right angles to each other. The volume of two variables in three-dimensional coordi- a cuboid is its length, l, times its breadth, b, nates. times its height, h. The surface area is the sum of the areas of the faces, that is curve sketching Sketching the graph of a 2(l × b) + 2(b × h) + 2(l × h) function y = f(x) in such a way as to indi- In the special case in which l = b = h, all cate the main features of interest of that the faces are square and the cuboid is a curve. This usually involves determining cube of volume l3 and surface area 6l2. the general shape of the curve and investigating how it behaves at points of special cumulative distribution See distribution interest. To be more specific, symmetry of function. the curve about the x-axis and the y-axis are investigated, as is symmetry about the cumulative frequency The total fre- origin. The behavior as x and y become quency of all values up to and including the very large is investigated in both the posi- upper boundary of the class interval under tive and negative directions. The points at consideration. See also frequency table. which the curve cross the x-axis and the y- axis are established. The problems of curl (Symbol ∇) A vector operator on a whether there are any values of x for which vector function that, for a three-dimen- y is infinite and any values of y for which x sional function, is equal to the sum of the is infinite are investigated. The nature of vector product (cross product) of the unit any stationary points is also investigated. vectors and partial derivatives in each of There are also a number of other features the component directions. That is: that can be looked at, including: establish- curl F = ∇F = i×∂F/∂x + ing the intervals for which the function is j×∂F/∂y + k×∂F/∂z always decreasing or always increasing; the where i, j, and k are the unit vectors in the concave or convex nature of the curve; and x, y, and z directions respectively. In all the asymptotes of the curve. Usually physics, the curl of a vector arises in the re- curve sketching is done using Cartesian co- lationship between electric current and ordinates. It is a way of visualizing how magnetic flux, and in the relationship be- functions behave without calculating the tween the velocity and angular momentum exact values. of a moving fluid. See also div; grad. curvilinear integral /ker-vă-lin-ee-er/ See curvature The rate of change of the slope line integral. of the tangent to a curve, with respect to distance along the curve. For each point on cusp A sharp point formed by a disconti- a smooth curve there is a circle that has the nuity in a curve. For example, two semicir- same tangent and the same curvature at cles placed side by side and touching form that point. The radius of this circle, called a cusp at which they touch. the radius of curvature, is the reciprocal of the curvature, and its center is known as cut See Dedekind cut. the center of curvature. If the graph of a function y = f(x) is a continuous curve, the cybernetics /sÿ-ber-net-iks/ The branch of slope of the tangent at any point is given by science concerned with control systems, es- the derivative dy/dx and the curvature is pecially with regard to the comparisons be- given by: tween those of machines and those of

54 cycloid human beings and other animals. In a se- cyclic function See periodic function. ries of operations, information gained at one stage can be used to modify later per- cyclic group A group in which each el- formances of that operation. This is known ement can be expressed as a power of any as feedback and enables a control system to other element. For example, the set of all check and possibly adjust its actions when numbers that are powers of 3 could be required. written as {…31/3, 31/2, 3, 32, 33, …} or {… 91/6, 91/4, 91/2, 9, 93/2, …}, etc. See also Abelian group. y cyclic polygon A polygon for which there is a circle on which all the vertices lie. All triangles are cyclic. All regular polygons are cyclic. All squares and rectangles are cyclic quadrilaterals. However, not all quadrilaterals are cyclic. Convex quadri- O x laterals are cyclic if the opposite angles are supplementary. For a cyclic quadrilateral with sides of length a, b, c, and d (in order) the expression (ac + bd) is equal to the product of the diagonals. This is known as Ptolemy’s theorem, named for the Egypt- ian astronomer Ptolemy (or Claudius Ptolemaeus) (fl. 2nd century AD).

In this graph a cusp occurs at the origin O. cyclic quadrilateral A four-sides figure whose corners (vertices) lie on a circumscribed circle. The opposite angles are supplementary, i.e. they add to 180°. See cycle A series of events that is regularly re- circumcircle; supplementary angles. peated (e.g. a single orbit, rotation, vibration, oscillation, or wave). A cycle is a cycloid /sÿ-kloid/ The curve traced out by complete single set of changes, starting a point on a circle rolling along a straight from one point and returning to the same line, for example, a point on the rim of a point in the same way. wheel rolling along the ground. For a circle

P r 2r # x O 2πr

Cycloid

55 cylinder

P(r, #, z) of radius r along a horizontal axis, the cycloid produced is a series of continuous arcs that rises from the axis to a height 2π and fall to touch the axis again at a cusp point, where the next arc begins. The horizontal distance between successive cusps is 2πr, the circumference of the circle. The length of the cycloid between adjacent z cusps is 8r. If θ is the angle formed by the radius to a point P (x,y) on the cycloid and the radius to the point of contact with the x-axis, the parametric equations of the cycloid are: θ θ x = r( – sin ) # y = r(1 – cosθ) O r cylinder A solid defined by a closed plane curve (forming a base) with an identical curve parallel to it. Any line segment from A point P(r,θ,z) in cylindrical polar coordinates. a point on one curve to a corresponding point on the other curve is an element of the cylinder. If one of these elements moves cylindrical polar coordinates A method parallel to itself round the base it sweeps of defining the position of a point in space out a curved lateral surface. The line is a by its horizontal radius r from a fixed ver- generator of the cylinder and the plane tical axis, the angular direction θ of the ra- closed curve forming the base is called the dius from an axis, and the height z above a directrix fixed horizontal reference plane. Starting If the bases are circles the cylinder is a at the origin O of the coordinate system, circular cylinder. If the bases have centers the point P(r,θ,z) is reached by moving out the line joining them is an axis of the cylin- along a fixed horizontal axis to a distance der. A right cylinder is one with its axis at r, following the circumference of the hori- right angles to the base; otherwise it is an zontal circle radius r centred at O through oblique cylinder. The volume of a cylinder an angle θ, and then moving vertically up- is Ah, where A is the base area and h the al- ward by a distance z. For a point P(r,θ,z), titude (the perpendicular distance between the corresponding rectangular CARTESIAN the bases). For a right circular cylinder, the COORDINATES (x,y,z) are: curved lateral surface area is 2πrh, where r x = rcosθ is the radius. y = rsinθ If the generator is an indefinitely ex- z = z tended line it sweeps out an extended sur- Compare spherical polar coordinates. face – a cylindrical surface See also coordinates; polar coordinates. cylindrical helix See helix. cylindrical surface See cylinder.

56 D

d’Alembertian /dal-ahm-bair-ti-ăn/ Sym- damped oscillation An oscillation with bol ٗ2. An operator that acts on the func- an amplitude that progressively decreases tion u(x,y,z,t) for the displacement of a with time. See damping. wave. It is related to the LAPLACIAN operator ∇2 by: damping The reduction in amplitude of a -x2 + ∂2/∂y2 + ∂2/∂z2 – (1/c2)∂2/∂t2 vibration with time by some form of resis∂/2∂ = 2ٗ = ∇2 – (1/c2) ∂2/∂t2, tance. A swinging pendulum will at last where c is the speed of the wave. Using this come to rest; a plucked string will not vi- operator the WAVE EQUATION can be writ- brate for long – in both cases internal -ten very concisely as ٗ2u = 0. Sometimes and/or external resistive forces progres the d’Alembertian is taken to refer specifi- sively reduce the amplitude and bring the cally to the wave equation for electromag- system to equilibrium. netic waves, with c being the speed of light In many cases the damping force(s) will in that case. It is named for the French be proportional to the object’s speed. In mathematician, encyclopedist, and phil- any event, energy must be transferred from osopher Jean Le Rond d’Alembert (1717– the vibrating system to overcome the resis- 83). tance. Where damping is an asset (as in bringing the pointer of a measuring instru- d’Alembert’s principle /dal-ahm-bairz/ ment to rest), the optimum situation occurs A principle that combines Newton’s sec- when the motion comes to zero in the ond and third laws of motion into the shortest time possible, without vibration. equation F–ma = 0. Using this principle, all This is critical damping. If the resistive problems involving forces can be treated as force is such that the time taken is longer equilibrium problems, with the –ma being than this, overdamping occurs. Con- regarded as an inertial reaction force. For versely, underdamping involves a longer bodies in motion this type of equilibrium is time with vibrations of decreasing ampli- called dynamic equilibrium or kinetic equi- tude. librium to distinguish it from static equilibrium. The application of d’Alembert’s data /day-tă/ (now often used as a singular principle can be used to simplify many noun) (sing. datum) The facts that refer to problems in Newtonian mechanics. or describe an object, idea, condition, situation, etc. In computing, data can be re- d’Alembert’s ratio test /dal-ahm-bairz/ garded as the facts on which a PROGRAM (generalized ratio test) A method of show- operates as opposed to the instructions in ing whether a series is convergent or diver- the program. It can only be accepted and gent. The absolute value of the ratio of processed by the computer in binary form. each term to the one before it is taken: Data is sometimes considered to be numer- |un+1/un| ical information only. If the LIMIT of this is l as n tends to infinity and l is less than 1, then the series is con- data bank A large collection of organized vergent. If l is greater than 1, the series is computer data, from which particular divergent. If l is equal to 1, the test fails and pieces of information can be readily ex- some other method has to be used. tracted. See also database.

57 database database A large collection of organized decahedron /dek-ă-hee-drŏn/ A polyhe- data providing a common pool of informa- dron that has ten faces. See polyhedron. tion for users, say, in the various sections of a large organization. Information can be deci- Symbol: d A prefix denoting 10–1. added, deleted, and updated as required. For example, 1 decimeter (dm) = 10–1 The management of a database is very meter (m). complicated and costly so that computer programs have been developed for this decibel /des-ă-bel/ Sybol: dB A unit of purpose. These programs allow the infor- power level, usually of a sound wave or mation to be extracted in many different electrical signal, measured on a logarithmic ways. For example, a request could be put scale. The threshold of hearing is taken as in for an alphabetical list of people over a 0 dB in sound measurement. Ten times this certain age and living in a specified area, in power level is 10 dB. The fundamental unit which their employment and income is the bel, but the decibel is almost exclu- should be given. Alternatively the request sively used (1 dB = 0.1 bel). could be for an alphabetical list of people A power P has a power level in decibels over a certain age and income level in given by: which their address and form of employ- 10 log10(P/P0) ment should be given. where P0 is the reference power. data processing The sequence of opera- decimal Denoting or based on the number tions performed on data in order to extract ten. The numbers in common use for information or to achieve some form of counting form a decimal number system. A order. The term usually means the process- decimal fraction is a rational number writing of data by computers but can also in- ten as units, tenths, hundredths, thou- clude its observation and collection. sandths, and so on. For example, ¼ is 0.25 in decimal notation. This type of decimal data transmission Any method of trans- fraction (or decimal) is a finite decimal be- ferring data from one computer to another cause the third and subsequent digits after or from an outstation (such as a cash-point the decimal point are 0. Some rational machine) to a central computer. numbers, such as 5/27 (= 0.185 185 185…) debug To detect, locate, and correct er- cannot be written as an exact decimal, but rors or faults (bugs) that occur in computer result in a number of digits that repeat in- programs or in pieces of computer equip- definitely. These are called repeating deci- ment. Since PROGRAMS and equipment are mals. All rational numbers can be written often highly complicated, debugging can as either finite decimals or repeating deci- be a tedious and lengthy job. Programming mals. A decimal that is not finite and does errors may result from the incorrect coding not repeat is an irrational number and can of an instruction (known as a syntax error) be quoted to any number of decimal places, π or from using instructions that will not give but never exactly. For example, to an ac- the required solution to a problem (a logic curacy of six decimal places is 3.141 593 error). Syntax errors can usually be de- and to seven decimal places is 3.141 592 7. tected and located by the compiler; logic A decimal measure is any measuring errors can be more difficult to find. system in which larger and smaller units are derived by multiplying and dividing the deca- Symbol: da A prefix denoting 10. basic unit by powers of ten. See also metric For example, 1 decameter (dam) = 10 me- system. ters (m). decision box See flowchart. decagon /dek-ă-gon/ A plane figure with ten straight sides. A regular decagon has decomposition 1. The process of break- ten equal sides and ten equal angles of 36°. ing a fraction up into partial fractions.

58 degree

2. Decomposition of a vector v is the g(x1) – g(x2) process of writing it in the form Compare indefinite integral. See also inte- v = grad(φ) + curl(A) gration. where φ is a scalar and A is a vector. Every vector may be decomposed in this form. definition In a measurement, the ACCU- RACY with which the instrument reading decreasing function A real function f(x) reflects the true value of the quantity being ≥ for which f(x1) f(x2) for all x1 and x2 in measured. < an interval I when x1 x2. If the stronger > < inequality f(x1) f(x2) holds when x1 x2 deformation A geometrical transforma- then f(x) is said to be a strictly decreasing tion that stretches, shrinks, or twists a function. shape but does not break up any of its lines or surfaces. It is often called, more precisely, a continuous deformation. See also decreasing sequence A sequence a1, a2, ≥ topology; transformation. a3, ... for which ai ai+1 for all i in the se- > quence. If ai ai+1 for all i in the sequence the sequence is said to be a strictly decreas- degeneracy The occurrence of two differing sequence. ent EIGENFUNCTIONS of an eigenvalue problem that have the same eigenvalue. An Dedekind cut /day-dĕ-kint/ A method of important example of degeneracy is given defining the real numbers, starting from in quantum mechanical systems such as the rational numbers. A Dedekind cut is a nuclei, atoms and molecules, where degen- division of the rational numbers into two eracy occurs when different quantum states have the same energy. The degener- disjoint sets, A and B, which are nonempty acy of a system is closely associated with its and which satisfy the conditions: (1) if x symmetry. ∈A and y ∈B then x

59 degrees of freedom

2. A unit of temperature. See Celsius de- and gree; Fahrenheit degree; kelvin. g(x) = (x – 3) 3. The exponent of a variable. For instance gives 3x3 has a degree of 3. If there are several F(x) = f(x)/g(x) terms, the sum of the exponents is used; The limit of F(x) as x→3 is indeterminate 2 3 5xy z has a degree of 6. (since x – 3 = 0). It can be obtained by using 4. The highest power of a variable in a the limit of 3 polynomial. For example in x + 2x + 1, the f′(x)/g′(x) = 2x degree is 3. as x→3. Thus the limit is 6. 5. The highest power of an equation. For If f′(x)/g′(x) also gives an indeterminate example, the degree of the equation x4 + 2x form at x = a, De l’Hôpital’s rule can be ap- = 0 is 4. plied again, differentiating as many times 6. The highest power to which the highest- as is necessary. order derivative is raised in a DIFFERENTIAL The rule is named for the French math- EQUATION. For example, the degree of (d2y/dx2)3 + dy/dx = 0 ematician Guillaume François Antoine, is three. The degree of Marquis De l’Hôpital (1661–1704). d3y/dx3 + 2y(d2y/dx2)2 = 0 is one. delta function See Dirac delta function. degrees of freedom The number of inde- De Moivre’s theorem /dĕ-mwahvrz/ A pendent parameters that are needed to formula for calculating a power of a com- specify the configuration of a system. For plex number. If the number is in the polar example, an atom in space has three inde- form pendent coordinates needed to specify its z = r(cosθ + i sinθ) position. A molecule with two atoms (e.g. then zn = rn(cosnθ + i sinnθ) 02) has additional degrees of freedom be- The theorem is named for the French cause it can also vibrate and rotate. In fact, mathematician Abraham De Moivre a diatomic molecule of this type has six de- (1667–1754). grees of freedom. It is usual in physics to interpret the number of degrees of freedom De Morgan’s laws /dĕ-mor-gănz/ Two as the number of independent ways in laws governing the relation between com- which the system can store energy. See also plementation, intersection, and union of phase space. sets. If represents the complement of the set A (i.e. the set of all things not in A) then De l’Hôpital’s rule /dĕ-loh-pee-tahlz/ De Morgan’s laws state that: –––– - - The rule stating that the limit of the ratio of (1) (A ∪ B) = A ∩ B and two functions of the same variable (x) as x –––– (2) (A ∩ B) = A ∪ B. approaches a value a, is equal to the limit Analogous laws are true for any finite in- of the ratios of their derivatives with re- tersection or union of sets. Parallel laws spect to x. That is, the limit of f(x)/g(x) as exist in other areas, e.g. in propositional x→a is the limit of f′(x)/g′(x) as x→a. De l’Hôpital’s rule can be used to find logic the equivalences ≡ ∨ the limits of f(x)/g(x) at points at which (p & q) p q and ∨ ≡ both f(x) and g(x) are zero and the ratio is (p q) p & q therefore indeterminate. Any function that are also known as De Morgan’s laws. The gives rise to an indeterminate form and laws are named for the English mathemati- that can be expressed as a ratio of two cian Augustus De Morgan (1806–71). functions, can be dealt with in this way. For example, in denominator The bottom part of a frac- F(x) = (x2 – 3)/(x –3) tion. For example, in the fraction ¾, 4 is writing the denominator and 3 is the numerator. f(x) = (x2 – 3) The denominator is the divisor.

60 desktop publishing dense A set S is said to be dense in another 5. A set of vectors, matrices, or other ob- set T if every point of T either belongs to S jects {x1,…xn} is said to be linearly depend- or is a limit point of S. If it is obvious what ent if there exists a linear relation the other set is, then a set is sometimes sim- a1x1 + a2x2 + … + anxn = 0 ply said to be dense; e.g. if we are consid- with at least one of the coefficients ering sets of real numbers. For example, nonzero. A set of objects is linearly inde- the set of rational numbers is dense (on the pendent if it is not linearly dependent. It real line) because every point of the real should be noted that the dependence is rel- line is either a rational number or is the ative to the set from which we may pick the π limit of a sequence of rational numbers. coefficients a1,…an. For example, 2 and Another way of putting this is to say that are linearly independent with respect to the any interval on the real line will always rational numbers but linearly dependent contain infinitely many rationals. with respect to the real numbers. This is the π case since a relation of the form a12 + a2 density The amount of matter per unit = 0 does not exist if a1 and a2 are rational volume; mass divided by volume. numbers but if a1 and a2 are allowed to be π irrational numbers we may take a1 = , a2 denumerable set /di-new-mĕ-ră-băl/ A = 2. SET in which the elements can be counted. For example, the set of prime numbers, al- dependent variable See variable. though infinite, can be counted, as can the set of positive even integers. These are deposit A sum of money paid by a buyer, known as denumerably infinite sets. The either to reserve goods or property that he set of real numbers, on the other hand, is wishes to buy at a later date or as the first not denumerable because between any two of a series of installments in an installment elements there can always be found a third. plan. If the buyer fails to complete the in- See countable. stallments the deposit is normally forfeited. dependent 1. An equation is dependent depth The distance downward from a ref- on a set of equations if every set of values erence level or backward from a reference of the unknowns that satisfy the set of plane. For example, the distance below a equations also satisfies this equation. Oth- water surface and the distance between a erwise the equation is independent of the wall surface and the back of an alcove in set of equations. the wall, are both called depths. 2. Two events are independent if the occurrence or nonoccurrence of one of them derivative The result of DIFFERENTIATION. does not affect the probability of the occurrence of the other. Otherwise the events derived unit A unit defined in terms of are dependent. If A and B are events whose base units, and not directly from a stan- probabilities are P(A) and P(B), then A and dard value of the quantity it measures. For B are independent if and only if P(A and B) example, the newton is a unit of force de- = P(A)P(B). For example, each toss of a fined as a kilogram meter seconds–2 coin is an independent event. (kg m s–2). See also SI units. 3. A set of functions {f1,…fn} are dependent if one can be expressed as a function of the desktop publishing A system that uses a others or equivalently there exists an ex- microcomputer with word-processing fa- ≡ ∂ ∂ pression F(f1,…fn) 0 with not all F/ fi = cilities linked to a laser printer to produce 0. Otherwise they are independent. For ex- multiple copies of a document. The word ample, the functions 2x + y and 4x + 2y + processor can provide various type fonts 6 are dependent since 4x + 2y + 6 = 2(2x + and a scanner can be included to add illus- y) + 6. trations (graphics). The result can rival the 4. See variable. quality of conventional printing.

61 determinant

a1 b1 = a1b2 – a2b1 a2 b2

The second-order determinant of a 2 x 2 matrix.

a1 b1 c1 = a b c – a b c + a b c – a b c a2 b2 c2 1 2 3 1 3 2 2 3 1 2 1 3

a3 b3 c3 + a3b1c2 – a3b2c1

The third-order determinant of a 3 x 3 matrix.

a1 b1 c1 b2 c2 a2 c2 a2 b2 = a – b + c1 a2 b2 c2 1 1 b3 c3 a3 c3 a3 b3 a3 b3 c3

= a1a1’ – b1b1’ + c1c1’

= a1a1’ – a2a2’ + a3a3’

+ – + – + – + – + Determinants: A third-order determinant is equal to the sum along any row, or down any column, of the product of each element with its cofactor. The cofactors are given alternate positive and negative signs in the pattern shown. Fourth- and higher order determinants can be calculated in a similar way.

determinant /di-ter-mă-nănt/ A function y = [(5 × 2) – (1 × 6)]/–2 = –2 of a square matrix derived by multiplying A third-order determinant has three rows and adding the elements together to obtain and columns and arises in a similar way in a single number. For example, in a 2 × 2 sets of three simultaneous equations in matrix the determinant is a1b2 – a2b1. This three variables. is written as a square array in vertical lines, The determinant of a transpose of a ma- symbol D2, and is called a second-order de- trix, |Ã|, is equal to the determinant of the terminant. Determinants occur in simulta- matrix, |A|. If the position of any of the neous equations. The solution of rows or columns in the matrix is changed, a1x + b1y + c1 = 0 the determinant remains the same. and a2x + b2y + c2 = 0 determinism The idea that if the present is state of a system is known exactly and the x = (b1c2 – c1b2)/D2 law governing the evolution of that system and with time is known then the subsequent y = (c1a2 – a1c2)/D2 evolution of the system can be determined If a1, a2, b1, b2, c1, and c2 are 1, 2, 3, 4, 5, exactly. Classical mechanics is governed by and 6 respectively, then D2 = –2 and determinism. Two developments in the x = [(3 × 6) – (5 × 4)]/–2 = 1 twentieth century undermined this simple and picture. In quantum mechanics, the uncer-

62 difference equation tainty principle of the German physicist ameter of a circle or a sphere is twice the Werner Heisenberg (1901–76) showed radius. that it is not possible to know the state of a system exactly since the position and the diametral /dÿ-am-ĕ-trăl/ Denoting a line momentum of a particle cannot be deter- or plane that forms a diameter of a figure. mined exactly simultaneously. Even within For example, a cross-section through the classical mechanics itself the development center of a sphere is a diametral plane. of CHAOS THEORY led to the realization that the word exactly is very important both in dichotomy, principle of In logic, the principle and in practice because there are principle that a proposition is either true or many systems for which slightly different false, but not both. For example, for two initial conditions result in widely different numbers x and y either x = y or x ≠ y, but results for the state of the system over a pe- not both. riod of time. Consequently, even systems that are described in a deterministic way diffeomorphism /diff-ee-oh-mor-fiz-ăm/ can, in practice, behave in ways that ap- A continuous transformation of a space pear random or chaotic. that moves the points of a space around but preserves those relationships between developable surface A surface that can them that are used to define which points be rolled out flat onto a plane. The lateral are close to one another. The set of diffeo- surface of a cone, for example, is devel- morphisms of a space is called the diffeo- opable. A spherical surface is not. morphism group of that space. The diffeomorphism group underlies discus- deviation See mean deviation; standard sions of invariance in RIEMANNIAN GEOME- deviation. TRY and general relativity theory. diagonal Joining opposite corners. A di- difference The result of subtracting one agonal of a square, for example, cuts it into quantity or expression from another. two congruent right angled triangles. In a solid figure, usually a polyhedron, a diago- difference between two squares A spe- nal plane is one that passes through two cial case of factorization involving two edges that are not adjacent. numbers or expressions that are squares. In general terms, diagonal matrix A square matrix in x2 – y2 = (x + y)(x – y) which all the elements are zero except those on the leading diagonal, that is, the difference equation An equation that ex- first element in the first row, the second el- presses a relation between finite differences ement in the second row, and so on. Diag- ∆, where ∆ is a difference operator which onal matrices, unlike most others, are operates on the rth term Ur of a sequence commutative in matrix multiplication. U1, U2, ... Un to produce the (r + 1)th term Ur+1. The definition of the difference op- ∆ ∆ erator means that Ur = Ur+1, U1 = ∆ U2–U1, U2 = U3–U2, etc. The expressions a11 0 0 just given are first finite differences of U1, ∆ 0 a22 0 U2, ... Un since only operates on them 0 0 a once. A second finite difference is defined 33 by operating with ∆ a second time. This × ∆2 A 3 3 diagonal matrix. leads to the results U1 = U3 – 2U2 + U1, ∆2 U2 = U4 – 2U3 + U2, etc. Higher-order ∆3 ∆4 differences Ur, Ur, ... can be defined in an analogous way. diameter The distance across a plane fig- The order of a difference equation is the ure or a solid at its widest point. The di- order of the highest-order finite difference

63 difference set

∆ in it. For example, Ur–aUr = b, where a See also differentiation. and b are constants is a first-order differ- ∆2 ∆ ence equation, while Ur–a Ur + bUr = differential equation An equation that f(r), where a and b are constants and f(r) is contains derivatives. An example of a sim- a function of r, is a second-order difference ple differential equation is: equation. There are analogies between dif- dy/dx + 4x + 6 = 0 ference equations and differential equa- To solve such equations it is necessary to tions. use integration. The equation above can be rearranged to give: difference set The set that is made up of dy = –(4x + 6)dx all the elements of set A that are not el- integrating both sides: ements of set B. In terms of a VENN DIA- ∫dy = ∫–(4x + 6)dx GRAM the difference set can be shown as a which gives: shaded region for two overlapping sets. y = –2x2 – 6x + C where C is a constant of integration. The differentiable function /dif-ĕ-ren-shă- value of C can be found if particular values băl/ A function f for which the derivative of x and y are known: for example, if y = 1 of f exists. More formally, if f is a real func- when x = 0 then C = 1, and the full solution tion of one variable x then the function is is differentiable in some interval if the limit of y = –2x2 – 6x + 1 [f(x + δx]/δx exists as δx→0 for all values Note that the solution to a differential of x in the interval. Thus, a function is dif- equation is itself an equation. Differentiat- ferentiable at a point if the gradient of the ing the solution gives the original equation. graph y = f(x) can be defined as that point, Equations like that above, which contain and therefore that the tangent to the curve only first derivatives (dy/dx) are said to be can be defined at that point. In the case of first order, if they contain second deriva- a function of a complex variable the func- tives they are second order; in general, the tion f(z) is differentiable at a particular order of a differential equation is the high- point if the limit [f(z + δz)–f(z)]/δz exists as est derivative in the equation. The degree δz→0 for that point and is independent of of a differential equation is the highest the way in which that point is approached. power of the highest order derivative. The differential equation in the exam- differential /dif-ĕ-ren-shăl/ An infinitesi- ple given is a first order and first degree mal change in a function of one or more equation. It is an example of a type of variables, resulting from a small change in equation solvable by separating the vari- the variables. For example, if f(x) is a func- ables onto both sides of the equation, so tion of x, and f changes by ∆f as a result of that each can be integrated (the variables a change ∆x in x, the differential df, is de- separable method of solution). Another fined as the limit of ∆f as ∆x becomes infi- type of first-order first-degree equation is nitely small. That is, df = f′(x)dx, where one of the form: f′(x) is the derivative of f with respect to x. dy/dx = f(y/x) This is a total differential, because it takes Such equations are known as homoge- into account changes in all of the variables, neous differential equations. An example is just one in this case. the equation: For a function of two variables, f(x,y) dy/dx = (x2 + y2)/x2 the rate of change of f with respect to x is To solve homogeneous equations a substi- the partial derivative ∂f/∂x. The change in f tution is made, y = mx, where m is a func- resulting from changing x by dx and keep- tion of x. Then: ing y constant is the partial differential, dy/dx = m + xdm/dx (∂f/∂y).dy. For any function, the total dif- and ferential is the sum of all the partial differ- (x2 + y2)/x2 = (x2 + m2x2)/x2 entials. For f(x,y): So the equation becomes: df = (∂f/∂x).dx + (∂f/∂y).dy m + xdm/dx = (x2 + m2x2)/x2

64 differentiator or: it. More generally, if an integral is over n xdm/dx = 1 + m2 – m dimensions then there is an n-form associ- The equation can now be solved by sepa- ated with that integral. There are many rating the variables. problems in mathematics and its applica- An equation of the form: tions to physical sciences and engineering dy/dx + P(x)y = Q(x) that can be analyzed using differential where P(x) and Q(x) are functions of x (but forms. not y), is a linear differential equation. Equations of this type can be put in a solv- differentiation /dif-ĕ-ren-shee-ay-shŏn/ A able form by multiplying both sides by the process for finding the rate at which one expression: variable quantity changes with respect to ∫ exp( P(x)dx) another. For example, a car might travel This is known as an integrating factor. For along a road from position x1 to position example, the differential equation x in a time interval t to t . Its average dy/dx + y/x = x2 2 1 2 speed is (x – x )/(t – t ), which can be is a linear first-order differential equation. 2 1 2 1 written ∆x/∆t, where ∆x represents the The function P(x) is 1/x, so the integrating change in x in the time ∆t. However, the factor is: car might accelerate or decelerate in this in- exp(∫dx/x) terval and it may be necessary to know the which is exp(logx); i.e. x. Multiplying both sides of the equation by x gives: speed at a particular instant, say t1. In this ∆ xdy/dx + y = x3 case the time interval t is made infinitely The left-hand side of the equation is equal small, i.e. t2 can be as close as necessary to ∆ ∆ ∆ to d(xy)/dx, so the equation becomes t1. The limit of x/ t as t approaches zero d(xy)/dx = x3 is the instantaneous velocity at t1. The re- Integrating both sides gives: sult of differentiation (i.e. the derivative) of xy = x4/4 + C a function y = f(x) is written dy/dx or f′(x). where C is a constant. On a graph of f(x), dy/dx at any point is the slope of the tangent to the curve y = f(x) at differential form Any entity that is under that point. See also integration. an integral sign. For example, the integral ∫Adx has a one-form Adx associated with differentiator /dif-ĕ-ren-shee-ay-ter/ An it. Similarly, a volume integral ∫∫∫Bdxdydz analog computer device whose output has a three-form Bdxdydz associated with (which is variable) is proportional to the

y = f(x)

Differentiation of a function y = f(x). The derivative dy/dx is the limit of y/x as y and y become infinitely small.

x O x Differentiation

65 digit time differential of the input (also vari- sional. In more abstract studies n-dimen- able). See differential. sional spaces can be used. 2. The size of a plane figure or solid. The digit A symbol that forms part of a num- dimensions of a rectangle are its length and ber. For example, in the number 3121 width; the dimensions of a rectangular par- there are four digits. The ordinary (deci- allelepiped are its length, width, and mal) number system has ten digits (0–9), height. whereas the binary (base two) system 3. One of the fundamental physical quanti- needs only two, 0 and 1. ties that can be used to express other quantities. Usually, mass [M], length [L], and digital Using numerical digits. For exam- time [T] are chosen. Velocity, for example, ple, a digital watch shows the time in num- has dimensions of [L][T]–1 (distance di- bers of hours and minutes and not as the vided by time). Force, as defined by the position of hands on a dial. In general, dig- equation: ital devices work by some kind of counting F = ma process, either mechanical or electronic. where m is mass and a acceleration, has di- The abacus is a very simple example. Early mensions [M][L][T]–2. See also dimen- calculating machines counted with me- sional analysis. chanical relays. Modern calculators use 4. Of a matrix, the number of rows or electronic switching circuits. number of columns. A matrix with 4 rows and 5 columns is 4 × 5 matrix. digital/analog converter A device that converts digital signals, usually from a dig- dimensional analysis The use of the di- ital computer, into continuously varying mensions of physical quantities to check electrical signals for inputting to an ANA- relationships between them. For instance, LOG COMPUTER. See computer. Einstein’s equation E = mc2 can be checked. The dimensions of speed2 are digital computer See computer. ([L][T]–1)2, i.e. [L]2[T]–2, so mc2 has di- 2 –2 dihedral /dÿ-hee-drăl/ Formed by two in- mensions of [M][L] [T] . Energy also has tersecting planes. Two planes intersect these dimensions since it is force –2 along a straight line (edge). The dihedral [M][L][T] multiplied by distance [L]. Di- angle (or dihedron) between the planes is mensional analysis is also used to obtain the angle between two lines (one in each the units of a quantity and to suggest new plane) drawn perpendicular to the edge equations. from a point on the edge. The dihedral angle of a polyhedron is the angle between Diophantine equation /dÿ-ŏ-fan-tÿn, - two faces. teen, -tin/ See indeterminate equation. dihedron /dÿ-hee-drŏn/ See dihedral. Dirac delta function /di-rak/ Symbol δ(x). A mathematical symbol which is used dilatation /dil-ă-tay-shŏn, dÿ-lă-/ A geo- to represent a sudden pulse. Strictly speak- metrical mapping or projection in which a ing, the Dirac delta function is not a proper figure is ‘stretched’, not necessarily by the mathematical function. It can be defined as same amount in each direction. A square, the generalization to continuous variables δ for example, may be mapped into a rectan- of the Kronecker delta ij which is defined δ gle by dilatation, or a cube into a cuboid. for discrete variables i and j by: ij = 1, if i = j, and =0 if i≠j. The Direct delta function δ dimension 1. The number of coordinates can also be defined∞ by the ∞properties: (x) = 0, if x≠0, ∫ δ(x)dx = 1, ∫ f(x)δ(x) = f(0). needed to represent the points on a line, –∞ –∞ shape, or solid. A plane figure is said to be It is named for the British physicist Paul two-dimensional; a solid is three-dimen- Dirac (1902–84).

66 direction angle

P (x, y, z )

␥

␤

O y ␣

Direction angles: α, β, and γ are made by the line OP with the x-, y-, and z-axes respectively.

directed Having a specified positive or gin and axes. The direction of a curve at a negative sign, or a definite direction. A di- point is the angle from the x axis to the tan- rected number usually has one of the signs gent at the point. + or – written in front of it. A directed angle is measured from one specified line to directional derivative The rate of the other. If the direction were reversed, change of a function with respect to dis- the size of the angle would be a negative tance s in a particular direction, or along a number. specified curve. Going from a point P(x,y,z) in the direction that makes angles directed line A straight line in which a di- α, β, and γ with the x, y, and z axs respec- rection along the line is specified. The di- tively, the directional derivative of a func- rection specified is called the positive tion f(x,y,z) is direction: the opposite direction is called df/ds = (∂f/∂x)cosα + (∂f/∂y)cosβ the negative direction. It is also possible to + (∂f/∂z)cosγ indicate the direction on a directed line by If there is a direction for which the direc- specifying a point 0 on the line to be the tional derivative is a maximum, then this origin, with the positive direction from this derivative is the gradient of f (grad f) at origin being directed towards the end x of point P. See also grad. the line. In this notation the directed line is given by 0x. direction angle The angle between a line and one of the axs in a rectangular Carte- directed line-segment The combination sian coordinate system. In a plane system, of a straight line and a specific direction it is the angle, α, that the line makes with along that line. For example if P and Q are the positive direction of the x-axis. In three two points on a straight line then PQ is the dimensions, there are three direction an- directed line-segment from P to Q, and QP gles, α, β, and γ, for the x, y, and z axes re- is the directed line-segment from Q to P. spectively. If two direction angles are The concept of a directed line-segment is known, the third can be calculated by the closely related to that of a VECTOR. relationship: cos2α + cos2β + cos2γ = 1 direction A property of vector quantities, Cosα, cosβ, and cosγ are called the direc- usually defined in reference to a fixed ori- tion cosines of the line, sometimes given

67 direction cosines the symbols l, m, and n. Any three numbers discontinuous See continuous function. in the ratio l:m:n are called the direction numbers or the direction ratio of the line. If discount 1. The difference between the a line joins the point A (x1,y1,z1) and the issue price of a stock or share and its nom- point B (x2,y2,z2) and the distance between inal value when the issue price is less than A and B is D, then the nominal value. Compare premium. l = (x2 – x1)/D 2. A reduction in the price of an article or m = (y2 – y1)/D commodity for payment in cash (cash dis- n = (z2 – z1)/D count), or for a large order (bulk discount), or for a retailer who will be selling the direction cosines See direction angle. goods on to members of the public (trade discount). direction numbers See direction angle. discrete Denoting a set of events or num- direction ratio See direction angle. bers in which there are no intermediate levels. The set of integers, for example, is director circle The locus of points P(x,y) discrete but the set of rational numbers is from which two perpendicular tangents are not. Between any two rational numbers, no drawn to an ellipse of the form x2/a2 + matter how close, there can always be y2/b2 = 1 as the points of contact vary. The found another rational number. The re- equation of a director circle defined in this sults of tossing dice form a discrete set of way is x2 = a2 + b2, as can be shown by ap- events, since a die has to land on one of its plying coordinate geometry to the equa- six faces. Putting the shot, on the other tions of the two tangents from P(x,y) to the hand, does not have a discrete set of out- ellipse. comes, since it may travel for any distance in a continuous range of lengths. direct proof A logical argument in which the theorem or proposition being proved is discriminant /dis-krim-ă-nănt/ The ex- the conclusion of a step-by-step process pression (b2 – 4ac) in a QUADRATIC EQUA- based on a set of initial statements that are TION of the form ax2 + bx + c = 0. If the known or assumed to be true. Compare in- roots of the equation are equal, the dis- direct proof. criminant is zero. For example, in x2 – 4x + 4 = 0 directrix /dă-rek-triks/ 1. A straight line b2 – 4ac = 0 and the only root is 2. If the associated with a conic, from which the discriminant is positive, the roots are dif- shortest distance to any point on the conic ferent and real. For example, in maintains a constant ratio with the dis- x2 + x – 6 = 0 tance from that point to the focus. See also b2 – 4ac = 25 and the roots are 2 and –3. If conic. the discriminant is negative, the roots of 2. A plane curve defining the base of a cone the equation are complex numbers. For ex- or cylinder. ample, the equation: x2 + x + 1 = 0 discontinuity See continuous function. has roots [–½ – (√3)/2]i and [–½ + (√3)/2]i.

P Q P ∨ Q P Q P ∨ Q

T T F T T T T F T T F T F T T F T T F F F F F F

Disjunction (exclusive) Disjunction (inclusive)

68 distance ratio disjoint Two sets are said to be disjoint if lowing instructions from the central they have no members in common; i.e. if A processor. The time to reach a specified lo- ∩ B = 0 then A and B are disjoint. cation is very short. This factor, together with the immense storage capacity, makes disjunction Symbol: ∨ In logic, the rela- the disk unit a major backing store in a tionship or between two propositions or computer system. Compare drum; mag- statements. Disjunction can be either inclu- netic tape. See also floppy disk; hard disk. sive or exclusive. Inclusive disjunction (sometimes called alternation) is the one diskette See floppy disk. most commonly used in mathematical logic, and can be interpreted as ‘one or the dispersion A measure of the extent to other or both’. For two propositions P and which data are spread about an average. ∨ Q, P Q is false if P and Q are both false, The range, the difference between the and true in all other cases. The more rarely largest and smallest results, is one measure. used exclusive disjunction can be inter- If Pr is the value below which r% of the re- preted as ‘either one or the other but not sults occur, then the range can be written both’. With this definition P ∨ Q is false as (P100 – P0). The interquartile range is when P and Q are both true, as well as (P75 – P25). The semi-interquartile range is when they are both false. The truth-table (P –P )/2. The mean deviation of X , X , definitions for both types of disjunction are 75 25 1 2 …, Xn measures the spread about the MEAN shown in the illustrations. Compare con- X and is junction. See also truth table. n Σ |x – x|/n disk A device that is widely used in com- j If values X , X , …, X occur with fre- puter systems to store information. It is a 1 2 k quencies f1, f2, …fk it becomes flat circular metal plate coated usually on n both sides with a magnetizable substance. Σ Σ fj |xj – x|/ fj Information is stored in the form of small magnetized spots, which are closely packed displacement Symbol: s The vector form in concentric tracks on the coated surfaces of distance, measured in meters (m) and in- of the disk. The spots are magnetized in one of two directions so that the informa- volving direction as well as magnitude. tion is in binary form. The magnetization pattern of a group of spots represents a let- dissipation The removal of energy from a ter, digit (0–9), or some other character. system to overcome some form of resistive One disk can store several million charac- force. Without resistance (as in motion in a ters. Information can be altered or deleted vacuum) there can be no dissipation. Dissi- as necessary by magnetic means. Disks are pated energy normally appears as thermal usually stacked on a common spindle in a energy. single unit known as a disk pack. Disk packs storing 200 million characters are distance Symbol: d The length of the path common. between two points. The SI unit is the Information can be recorded on a disk meter (m). Distance may or may not be using a special typewriter; this method is measured in a straight line. It is a scalar; known as key-to-disk. The information is the vector form is displacement. fed into a computer using a complex device called a disk unit. The disk pack is rotated distance formula The formula for the at very great speed in the disk unit. Small distance between two points (x1,y1) and electromagnets, known as read–write (x2,y2) in Cartesian coordinates. It is: √ 2 2 heads, move radially in and out over the [(x1 – x2) + (y1 – y2) ] surfaces of the rotating disks. They extract (read) or record (write) items of informa- distance ratio (velocity ratio) For a MA- tion at specified locations on a track, fol- CHINE, the ratio of the distance moved by

69 distribution function the effort in a given time to the distance (heat source, electric charge, etc.) within moved by the load in the same time. the volume, then div F = 0 and the total flux entering the volume equals the total distribution function For a random vari- flux leaving. See also grad. able x, the function f(x) that is equal to the probability of each value of x occurring. If divergence theorem (Gauss’s theorem; all values of x between a and b are equally Ostrogradsky’s theorem) A basic theorem likely, x has a uniform distribution in this in vector calculus that states that the inte- interval and a graph of the distribution gral of the divergence of a vector over a function f(x) against x is a horizontal line. volume enclosed by a surface is equal to the For example, the probability of the results integral of the normal component of the 1 to 6 when throwing dice is a uniform dis- vector over the closed surface. This can be tribution. Continuous random variables expressed by the equation usually have a varying distribution func- ∫∫F.ndS = ∫∫∫divFdV, tion with a maximum value x and in m where F is the vector, n is the normal to S, which the probability of x decreases as x and V is the volume. The divergence theo- moves away from xm. The cumulative distribution function F(x) is the probability of rem has important physical applications, a value less than or equal to x. For the dice particularly in electrostatics (where it is example, F(x) is a step function that in- known as Gauss’s law). creases from zero to one in six equal steps. For continuous functions, F(x) is often an divergent sequence A SEQUENCE in which s-shaped curve. In both cases F(x) is the the difference between the nth term and the area under the curve of f(x) to the left of x. one after it is constant or increases as n increases. {1,2,4,8,…} is divergent. A diver- distributive Denoting an operation that is gent sequence has no limit. Compare independent of being carried out before or convergent sequence. See also divergent se- after another operation. For two opera- ries; geometric sequence. tions • and °, • is distributive with respect to ° if a•(b°c) = (a•b)°(a•c) for all values of divergent series A SERIES in which the a, b, and c. In ordinary arithmetic, multi- sum of all the terms after the nth term does plication is distributive with respect to ad- not decrease as n increases. A divergent se- dition [a(b + c) = ab + ac] and to ries, unlike a convergent series, has no sum subtraction. to infinity. An infinite series a + a + … is ∩ 1 2 In set theory intersection ( ) is distrib- divergent if the partial sums a + a + … + ∪ 1 2 utive with respect to union ( ): a tend to infinity as n tends to infinity. For ∩ ∪ ∩ ∪ ∩ n [A (B C) = (A B) (A C)] example, the series 1 + 2 + 3 + 4 + … is di- See also associative; commutative. vergent. Compare convergent series. See also divergent sequence; geometric series. div /div/ (divergence) Symbol: ∇. A scalar operator that, for a three-dimensional vec- dividend 1. The number into which an- tor function F(x,y,z), is the sum of the other number (the divisor) is divided to scalar products of the unit vectors and the ÷ partial derivatives in each of the three com- give a quotient. For example, in 16 3, 16 ponent directions. That is: is the dividend and 3 is the divisor. div F = ∇.F = i.∂F/∂x + 2. A share of the profits of a corporation j.∂F/∂y + k.∂F/∂z paid to shareholders. The rate of dividend In physics, div F is used to describe the ex- paid will depend on profits in the preceding cess flux leaving an element of volume in year. It is expressed as a percentage of the space. This may be a flow of liquid, a flow nominal value of the shares. For example, of heat in a field of varying temperature, or a 10% dividend on a 75c share will pay an electric or magnetic flux in an electric or 7.5c per share (independent of the market magnetic field. If there is no source of flux price of the share). See also yield.

70 double integral dividers A drawing instrument, similar to D operator The differential operator compasses, but with sharp points on both d/dx. The derivative df/dx of a function ends. Dividers are used for measuring f(x) is often written as Df. This notation is lengths on a drawing, or for dividing used in solving DIFFERENTIAL EQUATIONS. A straight lines. second derivative, d2f/dx2, is written as D2f, a third derivative, d3f/dx3, as D3f, and division Symbol: ÷ The binary operation so on. In some ways, the D operator can be of finding the quotient of two quantities. treated like an ordinary algebraic quantity, Division is the inverse operation to MULTI- despite the fact that it has no numerical PLICATION. In arithmetic, the division of value. For example, the differential equa- two numbers is not commutative (2 ÷ 3 ≠ 3 tion 2 2 ÷ 2), nor associative [(2 ÷ 3) ÷ 4 ≠ 2 ÷ (3 ÷ d y/dx + 2xdy/dx + dy/dx + 2x = 0 4)]. The identity element for division is one or 2 only when it comes on the right hand side D y + 2xD + D + 2x = 0 (5 ÷ 1 = 5 but 1 ÷ 5 ≠ 5). can be factorized to (D + 2x)(D + 1) = 0. In this case, (D + 2x) then operates on the division of fractions See fractions. function (D + 1). divisor The number by which another dot product See scalar product. number (the dividend) is divided to give a quotient. For example, in 16 ÷ 3, 16 is the double-angle formulae See addition for- dividend and 3 is the divisor. See also fac- mulae. tor. double integral The result of integrating the same function twice, first with respect documentation Written instructions and to one variable, holding a second variable comments that give a full description of a constant, and then with respect to the sec- computer program. The documentation ond variable, holding the first variable con- describes the purposes for which the pro- stant. For example, if f(x,y) is a function of gram can be used, how it operates, the the variables x and y, then the double inte- exact form of the inputs and outputs, and gral, first with respect to x and then with how the computer must be operated. It al- respect to y, is: lows the program to be amended when ∫∫f(x,y)dydx necessary or to be converted for use on dif- This is equivalent to summing f(x,y) over ferent types of machines. intervals of both x and y, or to finding the volume bounded by the surface represent- dodecagon /doh-dek-ă-gon/ A plane figure with 12 sides and 12 interior angles. The sum of the interior angles is 1800°. dodecahedron /doh-dek-ă- hee-drŏn/ A solid figure with 12 faces. The faces of a regular dodecahedron are all regular pentagons. domain A set of numbers or quantities on which a mapping is, or may be, carried out. In algebra, the domain of the function f(x) is the set of values that the independent variable x can take. If, for example, f(x) represents taking the square root of x, then the domain might be defined as all the pos- Dodecahedron: a regular dodecahedron has itive rational numbers. See also range. regular pentagonal faces.

71 double point

y y

node cusp

x x O • O•

tacnode y

x O • isolated double point

x O•

Four types of double point at the origin of a two-dimensional Cartesian coordinate system.

ing f(x,y). The integral is not affected by gential to itself. There are several types of the order in which the integrations are car- double point. At a node the curve crosses ried out if they are definite integrals. An- over itself forming a loop. In this case it has other kind of double integral is the result of two distinct tangents. At a cusp it double integrating twice with respect to the same back on itself and has only one tangent. At variable. For example, if a car’s accelera- an acnode two arcs of a curve touch each tion a increases with time t in a known other and have the same tangent but, un- way, then the integral like a cusp, the arcs continue through the ∫adt singular point to form four arms. An iso- is the velocity (v) expressed as a function of lated double point may also occur. This time; the double integral satisfies the equation of the curve but does ∫∫adt2 = ∫v.dt = x not lie on the main arc of the curve. See where x is the distance traveled as a func- also isolated point; multiple point. tion of time. drum A metal cylinder coated with a mag- double point A singular point on a curve netizable substance and used in a computer at which the curve crosses itself or is tan- system to store information.

72 dyne dual See duality. spectively, 12 would be written as 10 and 22 as 1A. Duodecimal numbers are of little duality The principle in mathematics use, but some duodecimal units (1 foot = whereby one true theorem can be obtained 12 inches) are still in use. Compare binary; from another merely by substituting cer- decimal; hexadecimal; octal. tain words in a systematic way. In a plane, ‘point’ and ‘line’ are dual elements and, for duplication of the cube A classic prob- example, drawing a line thrugh a point and lem of ancient Greek geometry, to find a marking a point on a line are dual opera- way, using only a straight edge and com- tions. Theorems that can be obtained from pass, to find the side of a cube the volume one another by replacing each element and of which is exactly double that of a given operation by its dual are dual theorems. cube. It is now known that this cannot be Dual theorems feature prominently in done. This is the case because it is equiva- PROJECTIVE GEOMETRY. In logic, ‘implied’ lent to the problem of finding the cube root and ‘is implied by’ are dual relations and of 2 using a ruler and compass. This is im- may be interchanged together with the log- possible because geometrical methods in- ical connectives ‘and’ and ‘or’. In set volving a ruler and compass can only lead theory, the relations ‘is contained in’ and to length in the cases of addition, subtrac- ‘contains’ are dual relations and can be in- tion, multiplication, division and square terchanged along with the ‘union’ and ‘in- roots but definitely not cube roots. tersection’. In general, the principle of duality is found where the structure under duty A tax levied on certain kinds of consideration is a LATTICE. transactions. Examples include taxes on importing and exporting goods and the tax dummy variable A symbol that can be levied on alcohol and tobacco products. replaced by any other symbol without changing the meaning of the expression in dynamic friction See friction. which it occurs. dynamics The study of how objects move duodecahedron /dew-ŏ-dek-ă-hee-drŏn/ under the action of forces. This includes See dodecahedron. the relation between force and motion and the description of motion itself. See also duodecimal /dew-ŏ-dess-ă-măl/ Denoting mechanics. or based on twelve. In a duodecimal number system there are twelve different digits dyne /dÿn/ Symbol: dyn The former unit instead of ten. If, for example, ten and of force used in the c.g.s. system. It is equal eleven were given the symbols A and B re- to 10–5 N.

73 E

e A fundamental mathematical constant ecliptic /i-klip-tik/ The apparent path defined by the series: along which the Sun moves each year. It is e = lim(1 + 1/n) n →∞ the great circle formed by the intersection or by of the plane of the Earth’s orbit with the ce- e = 1 + 1/1! + 1/2! … + 1/n! + … lestial sphere. e is the base of natural logarithms. It is both irrational and transcendental. It is re- edge A straight line where two faces of a lated to π by the formula eiπ = –1. The func- solid meet. A cube has twelve edges. tion ex has the property that it is its own derivative, that is, dex/dx = ex. efficiency Symbol: η A measure used for processes of energy transfer; the ratio of eccentric Denoting intersecting circles, the useful energy produced by a system or spheres, etc., that do not have the same device to the energy input. For example, center. Compare concentric. the efficiency of an electric motor is the ratio of its mechanical power output to the eccentricity A measure of the shape of a electrical power input. There is no unit of conic. The eccentricity is the ratio of the efficiency; however efficiency is often distance of a point on the curve from a quoted as a percentage. In practical sys- fixed point (the focus) to the distance from tems some dissipation of energy always oc- a fixed line (the directrix). For a parabola, curs (by friction, air resistance, etc.) and the eccentricity is 1. For a hyperbola, it is the efficiency is less than 1. For a machine, greater than 1, and for an ellipse, it is be- the efficiency is the force ratio divided by tween 0 and 1. A circle has an eccentricity the distance ratio. of 0. effort The force applied to a MACHINE. echelon matrix A matrix in which all the rows of a matrix in which all the entries are eigenfunction /ÿ-gĕn-fung-shŏn/ See zero come beneath the rows in which not eigenvalue. all the entries are zero and in which the first non-zero entry in a row with non-zero eigenvalue /ÿ-gĕn-val-yoo/ (from German values is 1, with this entry being in a col- eigen = ‘allowed’) An eigenvalue for a lin- umn which is to the right of the first 1 in ear transformation L on a vector space V is the row above it. It is possible to convert a scalar λ for which there is a nonzero so- any matrix into an echelon matrix by using lution vector v in V such that Lv = λv and the technique of GAUSSIAN ELIMINATION. v satisfies any given boundary conditions. The vector v is an eigenvector (or charac- 13 8−27 teristic vector) belonging to the eigenvalue 0153−4 λ. An eigenvector for a linear operator on 00016 a vector space whose vectors are functions 00000 is also called an eigenfunction. As an example, consider the equation –y″ = λy with Echelon matrix boundary conditions y = 0 when x = a or b.

74 ellipse

A nonzero solution exists only if λ = n2π2 where n is an integer. The solution then is a1x + b1y + c1 = 0 π y = C[sinn (x – a)]/(b – a) a2x + b2y + c2 = 0 where C is an arbitrary constant. The a3x + b3y + c3 = 0 eigenvalues of the equation are given by n2π2 (where n is an integer). The corre- the eliminant is given by the matrix of co- sponding eigenfunctions (or eigenvectors) efficients: are given by y = C[sin(x – a)]/(b – a) | a1 b1 c1 | | a2 b2 c2 | eigenvector /ÿ-gĕn-vek-ter/ See eigen- | a b c | value. 3 3 3 elimination Removing one of the un- A collision for which the elastic collision knowns in an algebraic equation, for ex- restitution coefficient is equal to one. Ki- ample, by the substitution of variables or netic energy is conserved during an elastic by cancellation. collision. In practice, collisions are not perfectly elastic as some energy is transferred A CONIC with an eccentricity be- to internal energy of the bodies. See also ellipse restitution; coefficient of. tween 0 and 1. An ellipse has two foci. A line through the foci cuts the ellipse at two electronvolt /i-lek-tron-vohlt/ Symbol: eV vertices. The line segment between the ver- A unit of energy equal to 1.602 191 7 × tices is the major axis of the ellipse. The 10–19 joule. It is defined as the energy re- point on the major axis mid-way between quired to move the charge of an electron the vertices is the center of the ellipse. A across a potential difference of one volt. It line segment through the center perpendic- is normally used only to measure energies ular to the major axis is the minor axis. Ei- of elementary particles, ions, or states. ther of the chords of the ellipse through a focus parallel to the minor axis is a latus element /el-ĕ-mĕnt/ 1. A single item that rectum. The area of an ellipse is πab, where belongs to, or is a member of, a set. ‘Feb- a is half the major axis and b is half the ruary’, for example, is an element of the set minor axis. (Note that for a circle, in which {month in the year}. The number 5 is an el- the eccentricity is zero, a = b = r and the ement of the set of integers between 2 and area is πr2.) 10. In set notation this is written as The sum property of an ellipse is that ∈ 5 {2,3,4,5,6,7,8,9,10} for any point on the ellipse the sum of the 2. A line segment forming part of the distances from the point to each focus is a curved surface of a surface, as of a cone or constant. The ellipse also has a reflection cylinder. property; for a given point on the ellipse 3. A small part of a line, surface, or volume the two lines from each focus to the point summed by integration. make equal angles with a tangent at that 4. (of a matrix) See matrix. point. A drawing that shows the ap- In Cartesian coordinates the equation: elevation 2 2 2 2 pearance of a solid object as viewed from x /a + y /b = 1 the front, back or side. See plan. See also represents an ellipse with its center at the angle of elevation. origin. The major axis is on the x-axis and the minor axis on the y-axis. The major eliminant /i-lim-ă-nănt/ (characteristic; re- axis is 2a and the minor axis is 2b. The foci sultant) The relationship between coeffi- of the ellipse are at the points (+ea,0) and cients that results from eliminating the (–ea,0), where e is the eccentricity. The two variable from a set of simultaneous equa- directrices are the lines x = a/e and x = –a/e. tions. For example, in the equations The length of the latus rectum is 2b2/a.

75 ellipsoid

y directrix directrix C S P

A F2 O F1 B x

Ellipse: AB is the major axis and CD is the minor axis. F1 and F2 are the foci. Lines from these to any point P make equal angles with a tangent ST.

ellipsoid /i-lip-soid/ A solid body or elliptic paraboloid A CONICOID that is curved surface in which every plane cross- described by the equation x2/a2 + y2/b2 = section is an ellipse or a circle. An ellipsoid 2z/c, where a, b and c are constants. The xz has three axes of symmetry. In three-di- and yz planes are planes of reflection sym- mensional Cartesian coordinates, the equa- metry. Cross-sections of an elliptic parabo- tion of an ellipsoid with its center at the loid formed by planes that satisfy the origin is: equation z = k, where k is a non-negative x2/a2 + y2/b2 + z2/c2 = 1 number are, in general, ellipses. In the where a, b, and c are the points at which it special case in which a = b, the ellipses crosses the x, y, and z axes respectively. In become circles. If k is a negative number this case the axes of symmetry are the co- then the plane does not intersect the ellip- ordinate axes. A prolate ellipsoid is one tic paraboloid. Cross-sections of an ellip- generated by rotating an ellipse about its tic paraboloid formed by planes that are major axis. An oblate ellipsoid is generated parallel to either the xz or yz plane are by rotation about the minor axis. parabolas.

oblate ellipsoid prolate ellipsoid

An ellipsoid can be generated by rotating an ellipse about one of its axes. An oblate ellipsoid is generated by rotation about the minor axis and a prolate ellipsoid is generated by rotation about a major axis.

76 envelope empirical /em-pi-ră-kăl/ Derived directly entailment In logic, the relationship that from experimental results or observations. holds between two (or more) propositions when one can be deduced from the other. If empirical probability The probability of conclusion C is deducible from premisses A an event occurring as determined empiri- and B, then A and B are said to entail C. cally by carrying out a large number of tri- See deduction. als in which the event could take place. The number of times the event occurs is entropy /en-trŏ-pee/ A physical quantity counted, as is the number of trials. The that originated in thermodynamics and sta- value of the number of times the event oc- tistical mechanics but can be defined for curred divided by the number of trials (n) is general dynamical systems and also de- then calculated. The empirical probability fined in terms of information theory. The is defined to be the limiting value of this definition of entropy in statistical mechan- ratio as n →∞. In practice, taking n →∞ ics, which provides a molecular picture for means that the number of trials has to be- entropy in thermodynamics, means that come very large. Exactly how large de- entropy can be regarded as a measure of pends on the nature of the problem being the disorder of a system. In thermodynam- examined. It can also happen that empiri- ics, entropy is a measure of a quantity that cal probability gives a value that is differ- increases in a system as the ability of the ent from the value theoretically expected. energy of that system to do work decreases. For example, if a coin is tossed one might The entropy S of a system in thermody- expect that the probability of it landing namics is defined by dS = dQ/T, where dS is an infinitesimal change in the entropy of head up is exactly 0.5 but it might be the a system, dQ the infinitesimal amount of case that some bias in the coin causes a dif- heat absorbed by the system and T is the ferent empirical probability. thermodynamic temperature. In statistical mechanics entropy is given by S = klnW + empty set Symbol: ∅ (null set) The set B, where k is a constant of statistical me- that contains no elements. For example, chanics known as the Boltzmann constant, the set of ‘natural numbers less than 0’ is W is the statistical probability for the sys- an empty set. This could be written as {m: ∈ < ∅ tem, i.e. the number of distinguishable m N; m 0} = . ways the system can be realized, and B is another constant. Here, lnW is the natural energy Symbol: W A property of a system logarithm of W. The definition of entropy – its capacity to do work. Energy and work used in general dynamical systems leads to have the same unit: the joule (J). It is con- a non-zero value of entropy being associ- venient to divide energy into KINETIC EN- ated with chaos. In information theory, en- ERGY (energy of motion) and POTENTIAL tropy can be regarded as a measure of the ENERGY (‘stored’ energy). Names are given uncertainty in our knowledge about a sys- to many different forms of energy (chemi- tem. cal, electrical, nuclear, etc.); the only real difference lies in the system under discus- enumerable /i-new-mĕ-ră-băl/ See count- sion. For example, chemical energy is the able. kinetic and potential energies of electrons in a chemical compound. See also mass–en- envelope Consider a one-parameter fam- ergy equation. ily of curves in three dimensions – i.e. a family of curves that can be represented in engineering notation See standard form. terms of a common parameter that is constant along each curve, but is changed from enlargement A geometrical PROJECTION curve to curve. The envelope of this family that produces an image larger (smaller if of curves is the surface traced out by these the scale factor is less than 1) than, but sim- curves. This surface is tangent to every ilar to, the original shape. curve of the family. Its equation is obtained

77 epicycle

P b

a ␪ x O

Epicycloid: the epicycloid traced out by a point P on a circle of radius b that rolls round a large circle of radius a.

by eliminating the parameter between the circles is θ, the epicycloid is defined by the equation of the curve and the partial deriv- parametric equations: ative of this equation with respect to the x = (a + b) cosθ – a cos[(a + b)θ/a] parameter. For example, the envelope of y = (a + b) sinθ – a sin[(a + b)θ/a] See also the family of paraboloids given by x2 + y2 hypocycloid. = 4a(z – a) is the equation obtained by eliminating a from the equations x2 + y2 = equal Describing quantities that are the 4a(z – a) and z – 2a = 0, i.e. the circular same. For example, the quantities on the cone x2 + y2 = z2. left-hand side of an equation are equal to the quantities on the right-hand side of the epicycle /ep-ă-sÿ-kăl/ A circle that rolls equation; two sides of an isosceles triangle around the circumference of another, trac- are equal in length. Equality is often denoted by the equals sign =. ing out an EPICYCLOID.

epicycloid /ep-ă-sÿ-kloid/ The plane equality Symbol: = The relationship between two quantities that have the same curve traced out by a point on a circle or value or values. If two quantities are not epicycle rolling along the outside of an- equal, the symbol ≠ is used. For example, x other fixed circle. For example, if a small ≠ 0 means that the variable x cannot take cog wheel turns on a larger stationary the value zero. When the equality is only wheel, then a point on the rim of the approximate, the symbol ≅ is used. For ex- smaller wheel traces out an epicycloid. In a ample, if ∆x is small compared to x then x two-dimensional Cartesian-coordinate sys- + (∆x)2 ≅ x. When two expressions are ex- tem that has a fixed circle of radius a cen- actly equivalent the symbol ≡ is used. For tred at the origin and another of radius b example sin2α≡1 – cos2α because it ap- rolling around the circumference, the plies for all values of the variable α. See epicycloid is a series of continuous arcs also equation; inequality. that move away from the first circle to a distance 2b and then return to touch it equation A mathematical statement that again at a cusp point where the next arc one expression is equal to another, that is, begins. The epicycloid has only one arc if a two quantities joined by an equals sign. An = b, two if a = b/2, and so on. If the angle algebraic equation contains unknown or between the radius from the origin to the variable quantities. It may state that two moving point of contact between the two quantities are identical for all values of the

78 equilibrium

2 variables, and in this case the identity sym- v2 = v1 + at bol ≡ is normally used. For example: s = (v1 + v2)t/2 2 2 x – 4 ≡ (x – 2)(x + 2) s = v1t + at /2 2 is an identity because it is true for all values s = v2t – at /2 2 2 that x might have. The other kind of alge- v2 = v1 + 2as braic equation is a conditional equation, which is true only for certain values of the equator On the Earth’s surface, the circle variables. To solve such an equation, that formed by the plane cross-section that per- is, to find the values of the variables at pendicularly bisects the axis of rotation. which it is valid, it often has to be re- The plane in which the circle lies is called arranged into a simpler form. In simplify- the equatorial plane. A similar circle on ing an equation, the same operation can be any sphere with a defined axis is also called carried out on the expressions on both an equator, or equatorial circle. sides of the equals sign. For example, 2x – 3 = 4x + 2 equatorial See equator. can be simplified by adding 3 to both sides to give equiangular /ee-kwee-ang-gyŭ-ler, ek- 2x = 4x + 5 wee-/ Having equal angles. then subtracting 4x from both sides to give –2x = 5 equidistant /ee-kwă-dis-tănt, ek-wă-/ At and finally dividing both sides by –2 to ob- the same distance. For example, all points tain the solution x = –5/2. on the circumference of a circle are equidistant from the center. This kind of equation is called a linear equation, because the highest power of the equilateral /ee-kwă-lat-ĕ-răl, ek-wă-/ variable x is one. It could also be written in Having sides of equal length. For example, the form –2x – 5 = 0. On a Cartesian coor- an equilateral triangle has three sides of dinate graph, equal length (and equal interior angles of y = –2x – 5 60°). is a straight line that crosses the x-axis at x = –5/2. equilibrant /i-kwil-ă-brănt/ A single force Performing the same operation on both that is able to balance a given set of forces sides of an equation does not necessarily and thus cause equilibrium. It is equal and give an equation exactly equivalent to the opposite to the resultant of the given original. For example, starting with x = y forces. and squaring both sides gives x2 = y2, which means that x = y or x = –y. In this equilibrium /ee-kwă-lib-ree-ŭm, ek-wă-/ ⇒ case the symbol is used between the A state of constant momentum. An object equations, meaning that the first implies is in equilibrium if: 1. its linear momentum the second, but the second does not imply does not change (it moves in a straight line the first. That is, at constant speed and has constant mass, x = y ⇒ x2 = y2 or is at rest); Where the two equations are equiva- 2. its angular momentum does not change lent, the symbol ⇔ is used, for example, (its rotation is zero or constant). 2x = 2 ⇔ x = 1 For these conditions to be met: 1. the resultant of all outside forces acting on the equations of motion Equations that de- object must be zero (or there are no outside scribe the motion of an object with con- forces); stant accleration (a). They relate the 2. there is no resultant turning-effect (mo- velocity v1 of the object at the start of tim- ment). ing to its velocity v2 at some later time t An object is not in equilibrium if any of and to the object’s displacement s. They the following are true: 1. its mass is chang- are: ing;

79 equivalence

2. its speed is changing; with all the numbers less than or equal to 3. its direction is changing; √n. Only prime numbers will remain. This 4. its rotational speed is changing. method in number theory is named for the See also stability. Greek astronomer Eratosthenes of Cyrene (c. 276 BC–c. 194 BC). equivalence /i-kwiv-ă-lĕns/ See biconditional. erg A former unit of energy used in the c.g.s. system. It is equal to 10–7 joule. equivalence class The set of elements [x] in a set S that is equivalent to x by an Erlangen program /er-lang-ĕn/ A view of EQUIVALENCE RELATION on that set. The dis- geometry in which each type of geometry is tinct equivalence classes of a set are said to considered to be characterized by a group be a partition of that set, with each element of transformations and the invariants of of the set belonging to only one of the such transformations. For example, the equivalence classes. geometry of a crystal lattice is characterized by the SPACE GROUP of that lattice. The equivalence principle See relativity; Erlangen program is so called because the theory of. eminent German mathematician Felix Klein put forward this idea when he be- equivalence relation A binary relation R came a member of the faculty of Erlangen defined on a set S is said to be an equiva- University in 1872. Not all types of geom- lence relation if it satisfies the following etry can be incorporated into the Erlangen three properties: (1) xRx for every x in S – program. For example, RIEMANNIAN GEOM- this is the property of reflexivity, (2) if xRy ETRY cannot be incorporated directly, al- then yRx – this is the property of symme- though there are some ways of generalizing try, and (3) if xRy and yRz then xRz – this the Erlangen program that can include Rie- is the property of transitivity. Such rela- mannian geometry. tions are especially important because they partition the set on which they are defined error The uncertainty in a measurement into disjoint classes, known as equivalence or estimate of a quantity. For example, on classes. a mercury thermometer, it is often possible to read temperature only to the nearest de- equivalent fractions /i-kwiv-ă-lĕnt/ Two gree Celsius. A temperature of 20°C should or more fractions that represent the same then be written as (20 ± 0.5)°C because it rational number; they can be canceled until really means ‘between 19.5°C and 20.5°C’. they are the same fraction (see cancella- There are two basic types of error. Ran- tion). For example, 10/16, 15/24, 45/72 dom error occurs in any direction, cannot and 65/104 are equivalent fractions (equal be predicted, and cannot be compensated to 5/8). for. It includes the limitations in the accuracy of the measuring instrument and the erase head The part of any magnetic limitations in reading it. Systematic error recording device – tape or disk – that erases arises from faults or changes in conditions data (recorded signals) before the write that can be corrected for. For example, if a head records new data. See write head. 1 gram weight used on a balance is 2 milligrams underweight, every measurement Eratosthenes, sieve of /e-ră-tos-th’ĕ-neez/ taken with it will be 2 milligrams less than A method of finding prime numbers. To the correct value. find all the prime numbers less than a given number n one first goes through all the escape speed (escape velocity) The mini- numbers from 2 to n removing all those mum initial speed (velocity) that an object that are multiples of 2. Then all those after must have in order to escape from the sur- 3 are examined and all the multiples of 3 face of a planet (or moon) against the grav- are removed. One proceeds in this way itational attraction. The escape speed is

80 Euler’s formula equal to √(2GM/r), where G is the gravita- system is essentially that used today for tional constant, M is the mass of the planet, ‘pure’ geometry. and r is the radius of the planet. The con- One important postulate in Euclid’s cept also applies to the escape of the object system is that concerned with parallel lines from a distant orbit. (the parallel postulate). Its modern form is that if a point lies outside a straight line estimate A rough calculation, usually in- only one straight line can be drawn volving one or more approximations, through that point parallel to the other made to give a preliminary answer to a line. See non-Euclidean geometry. problem. Euler characteristic /oi-ler/ A topological ether /ee-th’er/ (aether) A hypothetical property of a curve or surface. For a curve, fluid, formerly thought to permeate all the Euler characteristic is the number of space and to be the medium through which vertices minus the number of closed con- electromagnetic waves were propagated. tinuous line segments between. For exam- See relativity; theory of. ple, any polygon has an Euler characteristic of zero. For a surface, the Euclidean algorithm /yoo-klid-ee-ăn/ A Euler characteristic is equal to the number method of finding the highest common fac- of vertices plus the number of faces minus tor of two positive integers. The smaller the number of edges. For example, a cube number is divided into the larger. The re- has an Euler characteristic of 2, and a mainder is then divided into the smaller cylinder, a Möbius strip, and a Klein bottle number, obtaining a second remainder. have an Euler characteristic of zero. The This second remainder is then divided into Euler characteristic is named for the Swiss the first remainder, to give a third remain- mathematician Leonhard Euler (1707–83). der. This is divided into the second, and so on, until a zero remainder is obtained. The Euler line A straight line on which the remainder preceding this is the highest CENTROID G, the CIRCUMCENTER C and the common factor of the two numbers. For ORTHOCENTER O of a triangle all lie. The example, the numbers 54 and 930. Divid- ratio of CG to OG on this line is always ing 54 into 930 gives 17 with a remainder 1/2. of 12. Dividing 12 into 54 gives 4 with a remainder of 6. Dividing 6 into 12 gives 2 Euler’s constant Symbol: γ A fundamen- with a remainder of 0. Thus 6 is the high- tal mathematical constant defined by the est common factor of 54 and 930. The al- limit of: gorithm is named for the Greek 1 + 1/2 + 1/3 + … + 1/n – log n mathematician Euclid (c. 330 BC–c. 260 as n →∞. To six figures its value is BC). 0.577 216. It is not known whether Euler’s constant is irrational or not. Euclidean geometry A system of geometry described by the Greek mathematician Euler’s formula 1. (for polyhedra) The Euclid in his book Elements (c. 300 BC). It formula that relates the number of vertices is based on a number of definitions – point, v, faces f, and edges e in a polyhedron, that line, etc. – together with a number of basic is: assumptions. These were axioms or ‘com- v + f – e = 2 mon notions’ – for example, that the whole For example, a cube has eight vertices is greater than the part – and postulates six faces and twelve edges: about geometric properties – for example, 8 + 6 – 12 = 2 that a straight line is determined by two Using the theorem it can be shown that points. Using these basic ideas a large num- there are only five regular polyhedrons. ber of theorems were proved using formal 2. The definition of the function eiθ for any deductive arguments. The basic assump- real value of θ, where i is the square root of tions of Euclid have been modified, but the –1, is

81 even

eiθ = cosθ + i sinθ excluded middle, law of See laws of Any complex number z = x + iy can be thought. written in this form. x = rcosθ and y = rsinθ are real, with r and θ representing z on an exclusive disjunction (exclusive OR) See Argand diagram. Note that putting θ = π disjunction. gives eiπ = –1 and θ = 2π gives e2πi = 1. exclusive OR gate See logic gate. even Divisible by two. The set of even numbers is {2,4,6,8,…}. Compare odd. existence theorem A theorem that proves that one or more mathematical en- even function A function f(x) of a vari- tities of a certain kind exists; e.g. that a able x, for which f(–x) = f(x). For example, function has a zero or a fixed point. An ex- cosx and x2 are even functions of x. Com- istence proof may be indirect and show pare odd function. that a certain entity must exist without giving any information about it or how to find event In probability, an event is any sub- it. set of the possible outcomes of an experiment. The event is said to occur if the existential quantifier In mathematical outcome is a member of the subset. For ex- logic, a symbol meaning ‘there is (are)’. It is ample, if two dice are thrown, an event is a usually written ∃. The quantifier is fol- subset of all ordered pairs (m,n) where m lowed by a variable that it is said to bind. and n are each one of the integers 1, 2, 3, Thus (∃x)F(x) means ‘There is something 4, 5, 6. Thus {(1,3),(2,2),(3,1)} is an event, that has property F’. which may also be described as ‘obtaining a sum of four’. See also dependent. expansion A quantity expressed as a sum of a series of terms. For instance, the ex- evolute /ev-ŏ-loot/ The evolute of a given pression: curve is the locus of the centers of curva- (x + 1)(x + 2) ture of all the points on the curve. The evo- can be expanded to: lute of a surface is another surface formed x2 + 3x + 2 by the locus of all the centers of curvature Often a function can be written as an of the first surface. infinite series that is convergent. The function can then be approximated to any re- exact differential equation A differen- quired accuracy by taking the sum of a tial equation of the form P(x,y)dx + sufficient number of terms at the beginning Q(x,y)dy = 0, which can be matched to the of the series. There are general formulae differential dφ, where dφ = (dφ/dx)dx + for expanding some types of expression. (dφ/dy)dy, where φ(x,y) is a function of x For example, the expansion of (1 + x)n is and y. If this is the case, then φ(x,y) = C, 1 + nx + [n(n – 1)/2!]x2 + where C is a constant and dφ = 0. If this [n(n – 1)(n – 2)/3!] x3 + … function φ(x,y) exists then: P(x,y)dx + where x is a variable between –1 and +1, Q(x,y) = (dφ/dx)dx + (dφ/dy)dy, and P(x,y) and n is an integer. See binomial expan- = dφ/dx, Q(x,y) = dφ/dy. The properties of sion; determinant; Fourier series; Taylor an exact differential equation can be used series. to find solutions of that equation. For a differential equation to be exact it is necessary expectation See expected value. and sufficient that the second mixed partial derivatives of φ(x,y) do not depend on the expected value (expectation) The value order of differentiation, i.e. ∂p(x,y)/∂y = of a variable quantity that is calculated to ∂2φ/(∂y∂x) = ∂2φ/(∂x∂y) = ∂Q(x,y)/∂x. For be most likely to occur. If x can take any of example, the equation (x + 2y)dx + (2x + the set of discrete values y)dy = 0 is an exact differential equation. {x1,x2,…xn} 82 extrapolation which have corresponding probabilities This series is convergent for all real- {p1,p2,…pn}, then the expected value is number values of the variable x. E(x) = x1p1 + x2p2 + … + xnpn Replacing x by –x gives an alternating If x is a continuous variable with a series for e–x: probability density function f(x), then 1 – x + x2/2! – x3/3! … ∞ Series for sinhx and coshx can be obtained by combining series for ex and e–x. E(x) = –∞ xf(x)dx explicit Denoting a function that contains expression A combination of symbols no dependent variables. Compare implicit. (representing numbers of other mathematical entities) and operations; e.g. 3x2, √(x2 exponent /eks-poh-nĕnt/ A number or + 2), ex – 1. symbol placed as a superscript after an expression to indicate the power to which it exterior angle The angle formed on the is raised. For example, x is an exponent in outside of a plane figure between the ex- yx and in (ay + b)x. tension of one straight edge beyond a ver- The laws of exponents are used for tex, and the outer side of the other straight combining exponents of numbers as fol- edge at that vertex. In a triangle, the exte- lows: rior angle at one vertex equals the sum of Multiplication: the angles on the insides of the other two xaxb = xa+b vertices, i.e. the sum of the interior oppo- Division: site angles. Compare interior angle. xa/xb = xa–b Power of a power: α (xa)b = xab Negative exponent: x–a = 1/xa β γ δ Fractional exponent: xa/b = b√xa The exterior angle δ = 180° – γ = α + β. A number raised to the power zero is 0 equal to 1; i.e. x = 1. extraction The process of finding a root of a number. exponential /eks-pŏ-nen-shăl/ A function or quantity that varies as the power of an- The process of estimating x extrapolation other quantity. In y = 4 , y is said to vary a value outside a known range of values. exponentially with respect to x. The func- For example, if the speed of an engine is x tion e (or expx), where e is the base of nat- controlled by a lever, and depressing the ural logarithms, is the exponential of x. lever by two, four, and six centimeters The infinite series gives speeds of 20, 30, and 40 revolutions 2 3 n 1 + x + x /2! + x /3! + … + x /n! + … per second respectively, then one can ex- x is equal to e and is known as the expo- trapolate from this information and as- nential series. The exponential form of a sume that depressing it by a further two complex number is centimeters will increase the speed to 50 θ rei = r(cosθ + i sinθ) revolutions per second. Extrapolation can See also complex number; Euler’s for- be carried out graphically; for example, a mula; Taylor series. graph can be drawn over a known range of values and the resulting curve extended. exponential series The infinite power se- The further from the known range, the ries that is the expansion of the function ex, greater will be the uncertainty in the ex- namely: trapolation. The case in which the graph of 1 + x + x2/2! + x3/3! + … the behavior is a straight line is a linear ex- + xn/n! + … trapolation. Compare interpolation.

83 F

face A flat surface on the outside of a solid Fahrenheit scale (TF) to a temperature on figure. A cube has six identical faces. the Celsius scale (TC) the following formula is used: TF = 9TC/5 + 32. The scale is facet A FACE, or flat side, of a many-sided named for the German physicist (Gabriel) object. Daniel Fahrenheit (1686–1736). factor (divisor) A number by which an- fallacy See logic. other number is divided. See also common factor. family A set of related curves or figures. For example, the equation y = 3x + c rep- factorial The product of all the whole resents a family of parallel straight lines. numbers less than or equal to a number. For example, factorial 7, written 7!, is farad /fa-răd, -rad/ Symbol: F The SI unit equal to 7× 6 × 5 × 4 × 3 × 2 × 1. Factorial of capacitance. When the plates of a capac- zero is defined as 1. itor are charged by one coulomb and there is a potential difference of one volt between The process of changing al- them, then the capacitor has a capacitance factorization –1 gebraic or numerical expressions from a of one farad. 1 F = 1 CV , 1 farad = 1 sum of terms into a product. For example, coulomb per volt. The unit is named for the British physicist and chemist Michael Fara- the left side of the equation 4x2 – 4x – 8 = day (1791–1867). 0 can be factorized to (2x + 2)(2x – 4) making it easy to solve for x. As the product of Farey sequence /fair-ee/ The sequence of the two factors is 0 when either of the fac- all fractions in lowest terms whose denom- tors is 0, it follows that (2x + 2) = 0 and (2x inators are less than n, where n is the order – 4) = 0 will provide solutions, i.e. x = –1 of the sequence, listed in increasing size. and x = 2. For example, the Farey sequence of order 5 is 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5. factor theorem The condition that (x – a) The sequence is named for the English is a factor of a polynomial f(x) in a variable mathematician John Farey (1766–1826). x if and only if f(a) = 0. For example, if f(x) 2 = x + x – 6, f(2) = 4 + 2 – 6 = 0 and f(–3) fathom A unit of length used to measure = 9 – 3 – 6 = 0, so the factors of f(x) are (x depth of water. It is equal to 6 feet – 2) and (x + 3). The factor theorem is de- (1.8288 m). rived from the remainder theorem. F distribution The statistical distribution Fahrenheit degree /fa-rĕn-hÿt/ Symbol: followed by the ratio of variances, s 2/s 2, ° 1 2 F A unit of temperature difference equal of pairs of random samples, size n1 and n2, to one hundred and eightieth of the differ- taken from a normal distribution. It is used ence between the temperatures of freezing to compare different estimates of the same and boiling water. On the Fahrenheit scale variance. water freezes at 32°F and boils at 212°F. To convert from a temperature on the feedback See cybernetics.

84 first moment femto- Symbol: f A prefix denoting 10–15. field 1. A set of entities with two opera- For example, 1 femtometer (fm) = 10–15 tions, called addition and multiplication. meter (m). The entities form a commutative group under addition with 0 as the identity el- Fermat’s last theorem /fair-mahz/ The ement. If 0 is omitted, the entities form a theorem that the equation commutative group under multiplication. xn + yn = zn Also the distribution law, a(b + c) = ab + where n is an integer greater than 2, can ac, applies for all a, b, and c. An example have no solution for x, y, and z. Fermat of a field is the set of rational numbers. wrote in the margin of a book on equations 2. A region in which a particle or body ex- that he had discovered a ‘truly wonderful’ erts a force on another particle or body through space. In a gravitational field a proof of the theorem but that the margin mass is supposed to affect the properties of was too small to write it down. It is now the surrounding space so that another mass generally thought that he was deluding in this region experiences a force. The re- himself; a proof was finally found in 1995 gion is thus called a ‘field of force’. Electric, using advanced mathematics which did not magnetic, and electromagnetic fields can exist at the time of Fermat (the seventeenth be similarly described. The concept of a century). The theorem is named for the field was introduced to explain action at a French mathematician and physicist Pierre distance. de Fermat (1601–65). figure A shape formed by a combination Fermat’s principle A fundamental prin- of points, lines, curves, or surfaces. Circles, ciple of optics that states that the path a squares, and triangles are plane figures. light ray takes is always the path that takes Spheres, cubes, and pyramids are solid fig- the least time. Fermat’s principle can be ures. used to derive the laws describing the reflection and refraction of light. It is an ex- finite decimal See decimal. ample of a VARIATIONAL PRINCIPLE. finite sequence See sequence. fermi /fer-mee/ A unit of length equal to 10–15 meter. It was formerly used in atomic finite series See series. and nuclear physics. The unit is named for the Italian–American physicist Enrico finite set A set that has a fixed countable Fermi (1901–54). number of elements. For example, the set of ‘months in the year’ has 12 members Fibonacci numbers /fee-bŏ-nah-chee/ and is therefore a finite set. Compare infi- The infinite sequence in which successive nite set. numbers are formed by adding the two pre- first moment For an area A about an vious numbers, that is: axis, the product of A and the distance of 1, 1, 2, 3, 5, 8, 13, 21, … C, the centroid of A, from the axis. Thus, if The Fibonacci sequence of numbers is A is in the xy plane and the centroid C of named for the Italian mathematician the area has coordinates which are denoted Leonardo Fibonacci (c. 1170–c. 1250). by (x,y) then the first moment of A about the x-axis is Ay and the first moment of A fictitious force A force in a system that about the y-axis is Ax. For a volume V arises because of the frame of reference of about the axis of rotation of a volume of the observer. Such ‘forces’ are said to be revolution, the product of V and the dis- ‘fictitious’ because they do not actually tance of C, the centroid of the volume, exist; they can be removed by transfer to a from the axis. Thus, if a volume V which is different frame of refernce. Examples are generated by rotating an area about the x- centrifugal force and Coriolis force. axis has a centroid C which has coordi-

85 first-order differential equation nates denoted by (x,0) then the first mo- two of these digits being after the decimal ment of V about the x-axis is zero and the point, it is written as 2538.90. first moment of V about the y-axis is Vx. floppy disk (diskette) A device that can first-order differential equation A DIF- be used to store data in electronic form, FERENTIAL EQUATION in which the highest consisting of a flexible plastic DISK with a derivative of the dependent variable is a magnetic coating on one or both sides. It is first derivative. permanently encased in a stiff envelope inside which it can be made to rotate. A fixed-point iteration An approximate read–write head operates through a slot in method for finding the root of an equation the envelope. f(x) = 0. The first step in the procedure is to write the equation as x = g(x). The next flotation, law of An object floating in a step is to take a value x0 to be an approxi- fluid displaces its own weight of fluid. This mation to a true root. Subsequent approx- follows from Archimedes’ principle for the imate values of the root are found by using special case of floating objects. (A floating the equation xi+1 = g(xi). If the values of xi object is in equilibrium, its only support tend to a limiting value a as i increases then coming from the fluid. It may be totally or a = g(a). This means that a is a root of the partly submerged.) equation f(x) = 0. flowchart A diagram on which can be fixed-point notation See floating point. represented the major steps in a process used, say, in industry, or a problem to be A theorem that fixed-point theorem investigated, or a task to be performed. A demonstrates that a function leaves one flowchart is built up from a number of point in its domain unchanged, i.e. for boxes connected by arrowed lines. The which f(x) = x. One celebrated example is boxes, of various shapes, have a label at- Brouwer’s fixed point theorem, which tached showing for example the operation states that any continuous transformation or calculation to be done at each step. At a of a circular disk onto itself must have a decision box a question is asked. The an- fixed point. swer, either yes or no, determines which of flip-flop See bistable circuit. two possible paths to take. Computer programs are often written by first drawing a floating objects, law of See flotation; flowchart of the problem or task in hand. law of. See also program. floating-point A notation used to de- fluctuation A deviation from the average scribe real numbers, particularly in com- value of some quantity. If there are many puters. In this notation, a number is particles in a system the fluctuations in the written in the form a × bn, where a is a value of a quantity Q can usually be de- number between 0.1 and 1, b is the number scribed by the equation base being used (usually 10 or 2) and n is – 2 = 1/ an integer. The numbers a, b and n are where denotes the average value of called the mantissa, the base, and the ex- a quantity. If it is proportional to the num- ponent respectively. For example, the number of particles in the system then the fluc- ber 2538.9 can be written as 0.25389 × tuations are usually very small. However, 104. Floating-point notation is contrasted there are many problems in which fluc- with fixed-point notation, in which all tuations are important, a notable ex- numbers have both a fixed number of dig- ample being a system near to a phase tran- its and a fixed number of digits after the sition. decimal point. If the number in the example given is expressed by six digits, with fluid ounce See ounce.

86 focus flywheel A large heavy wheel (with a focal radius A line from the focus of a large moment of inertia) used in mechani- conic to a point on the conic. cal devices. Energy is used to make the wheel rotate at high speed; the inertia of focus (pl. focuses or foci) A point associ- the wheel keeps the device moving at con- ated with a CONIC. The distance between stant speed, even though there may be fluc- the focus and any point on the curve is in a tuations in the torque. A flywheel thus acts fixed ratio (the eccentricity) to the distance as an ‘energy-storage’ device. between that point and a line (the directrix). An ellipse has two foci. The sum of focal chord A chord of a conic that passes the distances to each focus is the same for through a focus. all points on the ellipse.

set F at A set B at end of Z

open book between F and B

does NO does NO LL follow RR precede word? word?

YES YES

put B on put F on preceding following page page

does LR follow word?

NO YES

word on word on left-hand right-hand page page

A flowchart for finding which page a word is on in this dictionary (assuming that it is in). F is a front marker, B a back marker, LL is the first word on a left-hand page, RR the last word on a right- hand page, and LR the first word on the right-hand page.

87 foot foot Symbol: ft The unit of length in the eral expression that can be applied to sev- f.p.s. system (one third of a yard). It is eral different values of the quantities in equal to 0.304 8 meter. question. For example, the formula for the area of a circle is πr2, where r is the radius. force Symbol: F That which tends to change an object’s momentum. Force is a Foucault pendulum /foo-koh/ A simple vector; the unit is the newton (N). In SI, pendulum consisting of a heavy bob on a this unit is so defined that: long string. The period is large and the F = d(mv)/dt plane of vibration rotates slowly over a pe- from Newton’s second law. riod of time as a result of the rotation of the Earth below it. The apparent force forced oscillation (forced vibration) The causing this movement is the Coriolis oscillation of a system or object at a fre- force. The pendulum is named for the quency other than its natural frequency. French physicist Jean Bernard Léon Fou- Forced oscillation must be induced by an cault (1819–68). external periodic force. Compare free oscillation. See also resonance. four-color problem A problem in topology concerning the division of the surface force ratio (mechanical advantage) For a of a sphere into regions. The name comes MACHINE, the ratio of the output force from the coloring of maps. It appears that (load) to the input force (effort). There is in coloring a map it is not necessary to use no unit; the ratio is, however, sometimes more than four colors to distinguish re- given as a percentage. It is quite possible gions from each other. Two regions with a for force ratios far greater than one to be common line boundary between them need obtained. Indeed many machines are de- different colors, but two regions meeting at signed for this so that a small effort can a point do not. This was proved by Appel overcome a large load. However the effi- and Haken in 1976 with the extensive use ciency cannot be greater than one and a of computers. On the surface of a torus large force ratio implies a large distance ratio. only seven colors are necessary to distinguish regions. forces, parallelogram (law) of See parallelogram of vectors. Fourier series /foo-ree-ay, -er/ A method of expanding a function by expressing it as forces, triangle (law) of See triangle of an infinite series of periodic functions vectors. (sines and cosines). The general mathematical form of the Fourier series is: formalism A program for studying the f(x) = a0/2 + (a1cosx + b1sinx) + foundations of mathematics in which the (a2cos2x + b2sin2x) + completeness and consistency of mathe- (a3cos3x + b3sin3x) + matical systems is examined. This program (ancosnx + bnsinnx) + … was dealt a fatal blow by the discovery of The constants a0, a1, b1, etc., called Fourier GÖDEL’S INCOMPLETENESS THEOREM. coefficients, are obtained by the formulae: π π formal logic See symbolic logic. a0 = (1/ ) f(x)dx π-π π format The arrangement of information an = (1/ ) f(x)cosnxdx -π on a printed page, on a punched card, in a π computer storage device, etc., that must or π bn = (1/ ) f(x)sinnxdx should be used to meet with certain re- -π quirements. The series is named for the French mathematician Baron (Jean Baptiste) Joseph formula (pl. formulas or formulae) A gen- Fourier (1768–1830).

88 fraction

Fractal: generation of the snowflake curve.

fourth root of unity A complex number curve of this type in the limit has a dimen- z which satisfies the relation z4 = 1. This re- sion that lies between 1 (a line) and 2 (a lation is satisfied by four values of z, surface). The snowflake curve actually has namely 1, –1, i, and –i. a dimension of 1.26. One important aspect of fractals is that f.p.s. system A system of units that uses they are generated by an iterative process the foot, the pound, and the second as the and that a small part of the figure contains base units. It has now been largely replaced the information that could produce the by SI units for scientific and technical work whole figure. In this sense, fractals are said although it is still used to some extent in to be ‘self-similar’. The study of fractals the USA. has applications in chaos theory and in certain scientific fields (e.g. the growth of fractal /frak-tăl/ A curve or surface that crystals). It is also important in computer has a fractional dimension and is formed graphics, both as a method of generating by the limit of a series of successive opera- striking abstract images and, because of tions. A typical example of a fractal curve the self-similarity property, as a method of is the snowflake curve (also known as the compressing large graphics files. Koch curve and named for the Swedish See also Mandelbrot set. mathematician Helge von Koch (1870– 1924)). This is generated by starting with fraction A number written as a quotient; an equilateral triangle and dividing each i.e. as one number divided by another. For side into three equal parts. The center part example, in the fraction ⅔, 2 is known as of each of these sides is then used as the the numerator and 3 as the denominator. base of three smaller equilateral triangles When both numerator and denominator erected on the original sides. If the center are integers, the fraction is known as a sim- parts are removed, the result is a star- ple, common, or vulgar fraction. A com- shaped figure with 12 sides. The next stage plex fraction has another fraction as is to divide each of the 12 sides into three numerator or denominator, for example and generate more triangles. The process is (2/3)/(5/7) is a complex fraction. A unit continued indefinitely with the resulting fraction is a fraction that has 1 as the nu- generation of a snowflake-shaped curve. A merator. If the numerator of a fraction is

89 frame of reference less than the denominator, the fraction is of degrees of freedom is reduced by con- known as a proper fraction. If not, it is an straints on the system since the number of improper fraction. For example, 5/2 is an independent quantities necessary to deter- improper fraction and can be written as mine the system is also reduced. For exam- 2½. In this form it is called a mixed num- ple, a point in space has three degrees of ber. freedom since three coordinates are needed In adding or subtracting fractions, the to determine its postion. If the point is con- fractions are put in terms of their lowest strained to lie on a curve in space, it has common denominator. For example, then only one degree of freedom, since only 1/2 + 1/3 = 3/6 + 2/6 = 5/6 one parameter is needed to specify its posi- In multiplying fractions, the numerators tion on the curve. are multiplied and the denominators multiplied. For example: free oscillation (free vibration) An oscil- 2/3 × 5/7 = (2 × 5)/(3 × 7) = 10/21 lation at the natural frequency of the sys- In dividing fractions, one fraction is in- tem or object. For example, a pendulum verted thus: can be forced to swing at any frequency by 2/3 ÷ 1/2 = 2/3 × 2/1 = 4/3 applying a periodic external force, but it See also ratio. will swing freely at only one frequency, which depends on its length. Compare frame of reference A set of coordinate forced oscillation. See also resonance. axes with which the position of any object may be specified as it changes with time. free variable In mathematical logic, a The origin of the axes and their spatial di- variable that is not within the scope of any rections must be specified at every instant quantifier. (If it is within the scope of a of time for the frame to be fully deter- quantifier it is bound.) In the formula F(y) mined. → (∃x)G(x), y is a free variable. framework A collection of light rods that French curve A drawing instrument con- are joined together at their ends to give a sisting of a rigid piece of plastic or metal rigid structure. In a framework the rods are sheeting with curved edges. The curvature said to be light if their weights are very of the edges varies from being almost much smaller than the loads they are bear- straight to very tight curves, so that a part ing. When a framework has external forces of the edge can be chosen to guide a pen or acting on it each rod in the framework can pencil along any desired curvature. An- either prevent the structure from collapsing other instrument that serves the same pur- or stop the joints connecting the rods from pose consists of a deformable strip of lead becoming separated. If a rod is stopping a bar cut into short sections and surrounded collapse it exerts a push at both ends and is by a thick layer of plastic. This is bent to said to be in compression or in thrust. If a form any required curvature. rod is stopping the structure from becoming separated it exerts a pull at both ends and is said to be in tension. If the whole of the framework is in equilibrium then the forces at each joint have to be in equilibrium. This means that in equilibrium the external forces on the framework are in equilibrium with the internal forces since the forces in the rods occur as equal and opposite forces. French curve freedom, degrees of The number of independent quantities that are necessary to frequency Symbol: f, ν The number of cy- determine an object or system. The number cles per unit time of an oscillation (e.g. a

90 function pendulum, vibrating system, wave, alter- These integrals occur in the theory of dif- nating current, etc.). The unit is the hertz fraction and have been extensively tabu- (Hz). The symbol f is used for frequency, lated. The Fresnel integrals can be although ν is often employed for the fre- combined to give: ∫ π 2 quency of light or other electromagnetic C(x) – i S(x) = x0 exp[–i( u /2)] du, ∫ π 2 radiation. C(x) + i S(x) = x0 exp[i( u /2)] du. Angular frequency (ω) is related to frequency by ω = 2πf. friction A force opposing the relative mo- The frequency of an event is the number tion of two surfaces in contact. In fact, of times that it has occurred, as recorded in each surface applies a force on the other in a FREQUENCY TABLE. the opposite direction to the relative motion; the forces are parallel to the line of frequency curve A smoothed FREQUENCY contact. The exact causes of friction are POLYGON for data that can take a continu- still not fully understood. It probably re- ous set of values. As the amount of data is sults from minute surface roughness, even increased and the size of class interval de- on apparently ‘smooth’ surfaces. Frictional creased, the frequency polygon more forces do not depend on the area of con- closely approximates a smooth curve. Rel- tact. Presumably lubricants act by separat- ative frequency curves are smoothed rela- ing the surfaces. For friction between two tive frequency polygons. See also skewness. solid surfaces, sliding friction (or kinetic friction) opposes friction between two frequency function 1. The function that moving surfaces. It is less than the force of gives the values of the frequency of each re- static (or limiting) friction, which opposes sult or observation in an experiment. For a slip between surfaces that are at rest. large sample that is representative of the Rolling friction occurs when a body is whole population, the observed frequency rolling on a surface: here the surface in function will be the same as the probability contact is constantly changing. Frictional DISTRIBUTION FUNCTION f(x) of a population force (F) is proportional to the force hold- variable x. ing the bodies together (the ‘normal reac- 2. See random variable. tion’ R). The constants of proportionality (for different cases) are called coefficients frequency polygon The graph obtained of friction (symbol: µ): when the mid-points of the tops of the rec- µ = F/R tangles in a HISTOGRAM with equal class in- Two laws of friction are sometimes stated: tervals are joined by line segments. The 1. The frictional force is independent of the area under the polygon is equal to the total area of contact (for the same force holding area of the rectangles. the surfaces together). 2. The frictional force is proportional to frequency table A table showing how the force holding the surfaces together. In often each type (class) of result occurs in a sliding friction it is independent of the rel- sample or experiment. For example, the ative velocities of the surfaces. daily wages received by 100 employes in a company could be shown as the number in frustum /frus-tŭm/ (pl. frustums or frusta) each range from $50.00 to $74.99, $75.00 A geometric solid produced by two parallel to $99.99, and so on. In this case the rep- planes cutting a solid, or by one plane par- resentative value of each class (the class allel to the base. mark) is $(50 + 74.99)/2, etc. See also histogram. fulcrum /fûl-krŭm/ (pl. fulcrums or ful- cra) The point about which a lever turns. Fresnel integrals /fray-nel/ Two integrals C(x) and S(x) defined by: function (mapping) Any defined pro- ∫ π 2 C(x) = x0 cos( u /2)du cedure that relates one number, quantity, ∫ π 2 S(x) = x0 sin( u /2)du. etc., to another or others. In algebra, a 91 functional function of a variable x is often written as extensively investigated in the twentieth f(x). If two variable quantities, x and y, are century, with applications to differential related by the equation y = x2 + 2, for ex- equations, integral equations, and quan- ample, then y is a function of x or y = f(x) tum mechanics. These enabled the founda- = x2 + 2. The function here means ‘square tions of these subjects to be laid on a sound the number and add two’. x is the indepen- axiomatic basis. dent variable and y is the dependent variable. The inverse function – the one that fundamental The simplest way (mode) in expresses x in terms of y in this case – which an object can vibrate. The funda- ±√ would be x = (y –2), which might be mental frequency is the frequency of this written as x = g(y). vibration. The less simple modes of vibra- A function can be regarded as a relation are the higher harmonics; their fre- tionship between the elements of one set quencies are higher than that of the (the range) and those of another set (the fundamental. domain). For each element of the first set there is a corresponding element of the sec- fundamental theorem of algebra Every ond set into which it is ‘mapped’ by the polynomial equation of the form: function. For example, the set of numbers a zn + a zn–1 + a zn–2 + … {1,2,3,4} is mapped into the set {1,8,27,64} 0 1 2 a z + a = 0 by taking the cube of each element. A func- n–1 n in which a , a , a , etc., are complex num- tion may also map elements of a set into 0 1 2 bers, has at least one complex root. See others in the same set. Within the set {all also polynomial. women}, there are two subsets {mothers} and {daughters}. The mapping between them is ‘is the mother of’ and the inverse is fundamental theorem of calculus The ‘is the daughter of’. theorem used in calculating the value of a DEFINITE INTEGRAL. If f(x) is a continuous ≤ ≤ functional A function in which both the function of x in the interval a x b, and domain and range can be sets of functions. if g(x) is any INDEFINITE INTEGRAL of f(x), Roughly speaking, a functional can be con- then: sidered to be a function of a function. b b Functionals are used extensively in analysis a f(x)dx = [g(x)a] = g(b) – g(a) and physics, particularly for problems involving many degrees of freedom. A functional F of a function f is denoted by F[f]. fundamental units The units of length, See also functional analysis. mass, and time that form the basis of most systems of units. In SI, the fundamental functional analysis A branch of analysis units are the meter, the kilogram, and the that deals with mappings between classes second. See also base unit. of functions and the OPERATORS that bring about such mappings. In functional analy- furlong A unit of length equal to one sis a function can be regarded as a point in eighth of a mile. It is equivalent to 201.168 an abstract space. Functional analysis was m.

92 G

gallon A unit of capacity usually used to gate See logic gate. measure volumes of liquids. In the USA it is defined as 231 cubic inches and is equal to gauss /gows/ Symbol: G The unit of mag- 3.785 4 × 10–3 m3. In the UK it is defined netic flux density in the c.g.s. system. It is as the space occupied by 10 pounds of pure equal to 10–4 tesla. water and is equal to 4.546 1 × 10–3 m3. 1 UK gallon is equal to 1.2 US gallons. Gaussian distribution /gow-see-ăn/ See normal distribution. game theory A mathematical theory of the optimal behavior in competitive situa- Gaussian elimination A technique used tions in which the outcomes depend not in solving a set of linear equations for sev- only on the participants’ choices but also eral unknown quantities. The set of equa- on chance and the choices of others. A tions is expressed in terms of a matrix that game may be defined as a set of rules de- is formed from the coefficients and con- scribing a competitive situation involving a stants of the equations then converted into number of competing individuals or ECHELON FORM by elementary row opera- groups of individuals. These rules give tions, i.e. by multiplying a row by a number, adding a row which has been their permissible actions at each stage of multiplied by a number to another row or the game, the amount of information avail- swapping two rows. The set of solutions able, the probabilities associated with the corresponding to an equation that has been chance events that might occur, the cir- transformed so as to give values of the un- cumstances under which the competition known quantities directly is the same set of ends, and a pay-off scheme specifying the solutions for the original untransformed amount each player pays or receives at equations. such a conclusion. It is assumed that the players are rational in the sense that they Gauss’s theorem See divergence theo- prefer better rather than worse outcomes rem. and are able to place the possible outcomes in order of merit. Game theory has appli- general conic See conic. cations in military science, economics, pol- itics, and many other fields. general form (of an equation) A formula that defines a type of relationship between gamma function The integral function variables but does not specify values for ∞ constants. For example, the general form Γ x–1 –t (x) = t e dt of a polynomial equation in x is 0 axn + bxn–1 + cxn–2 + … = 0 If x is a positive integer n, then Γ(n) = n! If a, b, c, etc., are constants and n is the high- x is an integral multiple of ½, the function est integer power of x, called the degree of is a multiple of √π. the polynomial. Similarly, the general form Γ(½) = √π of a quadratic equation is Γ(3/2) = (½)√π ax2 + bx + c = 0 etc. See also equation; polynomial.

93 general theory general theory See relativity; theory of. In the example, the first term, a, is 1, the common ratio, r, is 2, and so the nth term generator A line that generates a surface; arn equals 2n. If r is greater than 1, the se- for example, in a cone, cylinder, or solid of ries will not be convergent. If –1 < r < 1 and revolution. the sum of all the terms after the nth term can be made as small as required by mak- geodesic /jee-ŏ-dess-ik/ A line on a surface ing n large enough, then the series is con- between two points that is the shortest dis- vergent. This means that there is a finite tance between the points. On a plane a ge- sum even when n is infinitely large. The odesic is a straight line. On a spherical sum to infinity of a convergent geometric surface it is part of a great circle of the series is a/(1 – r). Compare arithmetic se- sphere. ries. See also convergent series; divergent series; geometric sequence. geometric distribution The distribution of the number of independent Bernoulli trials before a successful result is obtained; geometry The branch of mathematics for example, the distribution of the num- concerned with points, lines, curves, and ber of times a coin has to be tossed before surfaces – their measurement, relation- a head comes up. The probability that the ships, and properties that are invariant number of trials (x) is k is under a given group of transformations. P(x=k) = qk–1p For example, geometry deals with the The mean and variance are 1/p and q/p2 re- measurement or calculation of angles be- spectively. The moment generating func- tween straight lines, the basic properties of tion is etp/(1 – qet). circles, and the relationship between lines and points on a surface. See analytical geometric mean See mean. geometry; Euclidean geometry; non-Eu- clidean geometry; topology. geometric progression See geometric sequence. giga- Symbol: G A prefix denoting 109. For example, 1 gigahertz (GHz) = 109 hertz geometric sequence (geometric progres- (Hz). sion) A SEQUENCE in which the ratio of each term to the one after it is constant, for gill /gil/ A unit of capacity equal to one example, 1, 3, 9, 27, … The general for- quarter of a pint. A US gill is equivalent to mula for the nth term of a geometric se- 1.182 9 × 10–4 m3 and a UK gill is equiva- n quence is un = ar . The ratio is called the lent to 1.420 × 10–4 m3. See pint. common ratio. In the example the first term, a, is 1, the common ratio, r, is 3, and glide A symmetry that can occur in crys- so u equals 3n. If a geometric sequence is n tals. It consists of the combination of a re- convergent, r lies between 1 and –1 (exclu- flection and a translation. See also space sive) and the limit of the sequence is 0. group. That is, un approaches zero as n becomes infinitely large. Compare arithmetic se- goh- quence. See also convergent sequence; di- Gödel’s incompleteness theorem / vergent sequence; geometric series. dĕlz/ A fundamental result of mathematical logic showing that any formal system geometric series A SERIES in which the powerful enough to express the truths of ratio of each term to the one after it is con- arithmetic must be incomplete; that is that stant, for example, 1 + 2 + 4 + 8 + 16 + … it will contain statements that are true but . The general formula for a geometric series cannot be proved using the system itself. is The incompleteness theorem is named for 2 n Sn = a + ar + ar + … + ar = the Austrian–American mathematician a(rn – 1)/(r – 1) Kurt Gödel (1906–78).

94 gram

Goldbach conjecture /golt-bahkh/ The y conjecture that every even number other 6 – than 2 is the sum of two prime numbers; so far, unproved. The conjecture is named for 5 – • the Prussian mathematician and historian 4 – Christian Goldbach (1690–1764). 3 – golden rectangle A rectangle in which the adjacent sides are in the ratio (1 + 2 – • √ 5)/2. 1 –

golden section The division of a line of – – – – – – x length l into two lengths a and b so that l/a 1 2 3 4 5 6 = a/b, that is, a/b = (1 + √5)/2. Proportions based on the golden section are particu- The gradient of the curve at the point (2,2) is 2, larly pleasing to the eye and occur in many and at the point (5,5) is 1/2. paintings, buildings, designs, etc. goniometer /gon-ee-om-ĕ-ter/ An instru- maximum exists. In the Earth’s gravita- ment for measuring the angles between ad- tional field this would be radially toward joining flat surfaces, such as the faces of a the center of the Earth (downward). In a crystal. magnetic field, ∇F would point along the lines of force. governor A mechanical device to control the speed of a machine. One type of simple grade Symbol: g A unit of plane angle governor consists of two loads attached to equal to one hundredth of a right angle. 1g a shaft so that as the speed of rotation of is equal to 0.9°. the shaft increases, the loads move farther outward from the center of rotation, while gradient 1. (slope) In rectangular Carte- still remaining attached to the shaft. As sian coordinates, the rate at which the y- they move outward they operate a control coordinate of a curve or a straight line that reduces the rate of fuel or energy input changes with respect to the x-coordinate. to the machine. As they reduce speed and The straight line y = 2x + 4 has a gradient move inward they increase the fuel or en- of +2; y increases by two for every unit in- ergy input. Thus, on the principle of nega- crease in x. The general equation of a tive feedback, the speed of the machine is straight line is y = mx + c, where m is the kept fairly constant under varying condi- gradient and c is a constant. (0,c) is the tions of load. point at which the line cuts the y-axis, i.e. grad (gradient) Symbol: ∇ A vector oper- the intercept. If m is negative, y decreases ator that, for any scalar function f(x,y,z), as x increases. has components in the x, y, and z direc- For a curve, the gradient changes con- tions equal to the PARTIAL DERIVATIVES with tinuously; the gradient at a point is the gra- respect to x, y, and z in that order. It is de- dient of the straight line that is a tangent to fined as: the curve at that point. For the curve y = grad f = ∇f = i∂f/∂x + j∂f/∂y + k∂f/∂z f(x), the gradient is the DERIVATIVE dy/dx. where i, j, and k are the unit vectors in the For example, the curve y = x2 has a gradi- x, y, and z directions. In physics, ∇F is ent given by dy/dx = 2x, at any particular often used to describe the spatial variation value of x. in the magnitude of a force F in, for exam- 2. See grad. ple, a magnetic or gravitational field. It is a vector with the direction in which the rate gram Symbol: g A unit of mass defined as of change of F is a maximum, if such a 10–3 kilogram.

95 gram-atom gram-atom See mole. gravitational constant Symbol: G The constant of proportionality in the equation gramme An alternative spelling of gram. that expresses NEWTON’S LAW OF UNIVERSAL GRAVITATION: 2 gram-molecule See mole. F = Gm1m2/r where F is the gravitational attraction be- graph 1. A drawing that shows the rela- tween two point masses m1 and m2 sepa- tionship between numbers or quantities. rated by a distance r. The value of G is 6.67 Graphs are usually drawn with coordinate × 10–11 Nm2 kg–2. It is regarded as a uni- axes at right angles. For example, the versal constant, although it has been sug- heights of children of different ages can be gested that the value of G may be changing shown by making the distance along a hor- slowly owing to the expansion of the Uni- izontal line represent the age in years and verse. (There is no current evidence that G the distance up a vertical line represent the has changed with time.) height in meters. A point marked on the graph ten units along and 1.5 units up rep- gravitational field The region of space in resents a ten-year-old who is 1.5 meters which one body attracts other bodies as a tall. Similarly, graphs are used to give a result of their mass. To escape from this geometric representation of equations. The field a body has to be projected outward graph of y = x2 is a parabola. The graph of with a certain speed (the escape speed). The y = 3x + 10 is a straight line. Simultaneous strength of the gravitational field at a point equations can be solved by drawing the is given by the ratio force/mass, which is graphs of the equations, and finding the equivalent to the acceleration of free fall, g. This may be defined as GM/r2, where G is points where they cross. For the two equa- the gravitational constant, M the mass of tions above, the graphs cross at two points: the object at the center of the field, and r x = –2, y = 4 and x = 5, y = 25. the distance between the object and the There are various types of graph. Some, point in question. The standard value of such as the HISTOGRAM and the PIE CHART, the acceleration of free fall at the Earth’s are used to display numerical information surface is 9.8 m s–2, but it varies with alti- in a form that is simple and quickly under- tude (i.e. with r2). stood. Some, such as CONVERSION GRAPHS, are used as part of a calculation. Others, gravitational mass The MASS of a body as such as SCATTER DIAGRAMS, may be used in measured by the force of attraction be- analysing the results of a scientific experi- tween masses. The value is given by New- ment. See also bar chart. ton’s law of universal gravitation. Inertial 2. (topology) A network of lines and ver- and gravitational masses appear to be tices. See Königsberg bridge problem. equal in a uniform gravitational field. See also inertial mass. graphics display (graphical display unit) See visual display unit. gravity The gravitational pull of the Earth (or other celestial body) on an object. The gravitation The concept originated by force of gravity on an object causes its Isaac Newton around 1666 to account for weight. the apparent motion of the Moon around the Earth, the essence being a force of at- gravity, center of See center of mass. traction, called gravity, between the Moon and the Earth. Newton used this theory of great circle A circle on the surface of a gravitation to give the first satisfactory ex- sphere that has the same radius as the planations of many facts, such as Kepler’s sphere. A great circle is formed by a cross- laws of planetary motion, the ocean tides, section by any plane that passes through and the precession of the equinoxes. See the center of the sphere. Compare small also Newton’s law of universal gravitation. circle.

96 gyroscope greatest upper bound See bound. 2. There is an identity element for the operation – i.e. an element that, combined Green’s function A solution for certain with another, leaves it unchanged. In the types of partial differential equation. The example, the identity element is zero: method of Green’s functions is used exten- adding zero to any member leaves it un- sively in many branches of theoretical changed; 3 + 0 = 3, etc. physics, particularly mechanics, electrody- 3. For each element of the group there is namics, acoustics, the many-body problem another element – its inverse. Combining in quantum mechanics and quantum field an element with its inverse leads to the theory. In these physical applications the identity element. In the example, the num- Green’s function can be considered to be a ber +3 has an inverse –3 (and vice versa); response to an impulse, with the Green’s thus +3 + (–3) = 0. function method for finding the solution to 4. The associative law holds for the mem- a partial differential equation being the bers of the group. In the example: sum or integral of the responses to im- 2 + (3 + 5) = (2 + 3) + 5 pulses. An example of a Green’s function is Any set of elements obeying the above rules the electric potential due to a unit point forms a group. Note that the binary opera- source of charge. tion need not be addition. Group theory is important in many branches of mathematics – for instance, in the theory of roots of Green’s theorem A result in VECTOR CAL- equations. It is also very useful in diverse CULUS that is a corollary of the DIVERGENCE branches of science. In chemistry, group THEOREM (Gauss theorem). If u and v are scalar functions, S indicates a surface inte- theory is used in describing symmetries of atoms and molecules to determine their en- gral and V a volume integral, Green’s the- ergy levels and explain their spectra. In orem states that physics, certain elementary particles can be ∫ (U∇.∇v – v∇.∇u)dV = v classified into mathematical groups on the ∫S(U∇v–v∇u).dS. basis of their quantum numbers (this led to An alternative form of Green’s theorem is the discovery of the omega-minus particle ∫ u∇v.dS = ∫ u∇.∇v + ∫ ∇u.∇vdV. S v V as a missing member of a group). Group Green’s theorem is used extensively in theory has also been applied to other sub- physical applications of vector calculus jects, such as linguistics. such as problems in electrodynamics. See also Abelian group; cyclic group. gross 1. Denoting a weight of goods in- group speed If a wave motion has a phase cluding the weight of the container or speed that depends on wavelength, the dis- packing. turbance of a progressive wave travels with 2. Denoting a profit calculated before de- a different speed than the phase speed. This ducting overhead costs, expenses, and is called the group speed. It is the speed (usually) taxes. with which the group of waves travels, and Compare net. is given by: U = c – λdc/dλ group A set having certain additional where c is the phase speed. The group properties: 1. In a group there is a binary speed is the one that is usually obtained by operation for which the elements of the set measurement. If there is no dispersion of can be related in pairs, giving results that the wave motion, as for electromagnetic are also members of the group (the prop- radiation in free space, the group and erty of closure). For example, the set of all phase speeds are equal. positive and negative numbers and zero form a group under the operation of addi- Guldinus theorem See Pappus’ theo- tion. Adding any member to any other rems. gives an element that is also a member of the group; e.g. 3 + (–2) = 1, etc. gyroscope /jÿ-rŏ-skohp/ A rotating object

97 gyroscope that tends to maintain a fixed orientation clist are stable when moving at speed be- in space. For example, the axis of the ro- cause of the gyroscopic effect. Practical tating Earth always points in the same di- applications are the navigational gyro- rection toward the Pole Star (except for a compass and automatic stabilizers in ships small PRECESSION). A spinning top or a cy- and aircraft.

98 H

half-angle formulae See addition formu- different, with the important exception lae. that v0 = vl. The Hamiltonian graph is of interest in graph theory when the problem half-plane A set of points that lie on one of going around the graph along edges so side of a straight line. If the line l is denoted as to visit each vertex only once is consid- by ax + by + c = 0, where a, b, and c are ered. constants, then the open half-plane on one side of the line is given by the set of points ham-sandwich theorem 1. A ham sand- (x,y) that satisfy the inequality ax + by + c wich can be cut with one stroke of a knife > 0. The open half-plane on the other side so that the ham and each slice of bread are of the plane is given by the set of points exactly cut in half. More formally, if A, B, (x,y) that satisfy the inequality ax + by + c and C are bounded connected sets in space, < 0. The inequalities ax + by + c > 0 and ax then there is a plane that cuts each set into + by + c ≤ 0 define closed half planes on two sets with equal volume. opposite sides of the plane. Half planes 2. If f(x) ≤ g(x) ≤ h(x) for all x and the func- (both open and closed) are used extensively tions f and h have the same limit, then g in linear programming. also has this limit.

Hamiltonian /ham-ăl-toh-nee-ăn/ In clas- hard copy In computer science, a docu- sical mechanics, a function of the coordi- ment that can be read, such as a computer nates qi, i = 1, 2, … n, and momenta, pi, i = printout in plain language. See printer. 1, 2, … n, generally denoted by H and defined by hard disk A rigid magnetic disk that H = ∑piqi – L stores programs and data in a computer. where qi denotes the derivative with re- They are usually fixed in the machine and spect to time and L is the Lagrangian func- cannot be removed. See also disk; floppy tion of the system expressed as a function disk. of the coordinates momenta and time. If the Lagrangian function does not depend hardware The physical embodiment of a explicitly on time, the system is said to be computer system, i.e. its electronic cir- conservative and H is the total energy of cuitry, disk and magnetic tape units, line the system. The Hamiltonian function is printers, cabinets, etc. Compare software. named for the Irish mathematician Sir William Rowan Hamilton (1805–65). harmonic analysis The use of trigonometric series to study mathematical func- Hamiltonian graph A graph that has a tions. See Fourier series. Hamiltonian cycle, i.e. a cycle contains each vertex, where a cycle of a graph is a harmonic mean See mean. sequence of alternating vertices and edges which can be written as v0, l1, v1, l2, v2 ..., harmonic motion A regularly repeated ll, vl, where li is an edge which connects the sequence that can be expressed as the sum vertices vi–1 and vi. In this definition all the of a set of sine waves. Each component sine edges are different and all the vertices are wave represents a possible simple har-

99 harmonic progression monic motion. The complex vibration of of T: ∂T/∂t = c∇2T, where c is a constant. sound sources (with fundamental and In the case of one space dimension with co- overtones), for instance, is a harmonic mo- ordinate x this equation becomes ∂T/∂t = tion, as is the sound wave produced. See c∂2T/∂x2. The heat equation is different also simple harmonic motion. from the wave equation since the heat equation has a first derivative with respect harmonic progression See harmonic to time whereas the wave equation has a sequence. second derivative with respect to time. Physically, this corresponds to heat con- harmonic sequence (harmonic progres- duction being an irreversible process, like sion) An ordered set of numbers, the reci- friction, whereas wave motion is re- procals of which have a constant difference versible. The heat equation has the same ½ ⅓ ¼ between them; for example, {1, , , , … form as the general equation for diffusion. 1/n}. In this example {1, 2, 3, 4, … n} have a constant difference – i.e. they form an hectare /hek-tair/ Symbol ha. A unit of ARITHMETIC SEQUENCE. The reciprocals of area equal to 10 000 square metres. tHE- the terms in a harmonic sequence form an hectare is most often used as a convenient arithmetic sequence, and vice versa. unit of land. harmonic series The sum of the terms in hecto- Symbol: h A prefix denoting 102. a harmonic sequence; for example: 1 + ½ + 2 ⅓ ¼ For example, 1 hectometer (hm) = 10 + + … The harmonic series is a diver- meters (m). gent series. height A vertical distance, usually up- HCF Highest common factor. See com- ward, from a base line or plane. For exam- mon factor. ple, the perpendicular distance from the base of a triangle to the vertex opposite, head An input/output device in a com- and the distance between the uppermost puter that can read, write, or erase signals and the base planes of a cuboid, are both onto or from magnetic tape or disk. See known as the height of the figure. erase head; write head. hee-liks heat equation An equation which de- helix / / A spiral-shaped space scribes the flow of heat. In three spatial di- curve. A cylindrical helix lies on a cylinder. mensions it states that the rate of change of A conical helix lies on a cone. For example, the absolute temperature T with respect to the shape of the thread on a screw is a time t is proportional to the LAPLACIAN ∇2 helix. In a straight screw, it is a cylindrical helix and in a conically tapered screw it is a conical helix.

Helmholtz’s theorem /helm-holts/ A theorem in VECTOR CALCULUS that states that if a vector V satisfies certain general mathematical conditions then V can be written as the sum of an IRROTATIONAL VECTOR and a SOLENOIDAL VECTOR. This theorem is important in electrodynamics. It is named for the German physicist Hermann von Helm- holtz (1821–94).

cylindrical conical helix helix hemisphere The surface bounded by half of a SPHERE and a plane through the center Types of helix of the sphere.

100 Hilbert’s problems henry /hen-ree/ Symbol: H The SI unit of hexadecimal number is made up with six- inductance, equal to the inductance of a teen different digits instead of the ten in the closed circuit that has a magnetic flux of decimal system. Normally these are 0, 1, 2, one weber per ampere of a current in the 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. For ex- circuit. 1 H = 1 Wb A–1. The unit is named ample, 16 is written as 10, 21 is written as for the American physicist Joseph Henry 15 (16 + 5), 59 is written as 3B [(3 × 16) + (1797–1878). 11]. Hexadecimal numbers are sometimes used in computer systems because they are heptagon /hep-tă-gon/ A plane figure with much shorter than the long strings of bi- seven straight sides. A regular heptagon nary digits that the machine normally uses. has seven equal sides and seven equal an- Binary numbers are easily converted into gles. hexadecimal numbers by grouping the digits in fours. Compare binary; decimal; Hermitian matrix /her-mee-shăn, -mish- duodecimal; octal. ăn, air-mee-shăn/ The Hermitian conjugate of a matrix is the transpose of the complex hexagon /heks-ă-gon/ A plane figure with conjugate of the matrix, where the com- six straight sides. A regular hexagon is one plex conjugate of a matrix is the matrix with all six sides and all six angles equal, whose elements are the complex conju- the angles all being 120°. Congruent regu- gates of the corresponding elements of the lar hexagons can be fitted together to cover given matrix (see conjugate complex num- completely a plane surface. Apart from bers). A Hermitian matrix is a matrix that squares and equilateral triangles, they are is its own Hermitian conjugate; i.e. a the only regular polygons with this prop- square matrix such that aij is the complex erty. conjugate of aij for all i and j where aij is the element in the ith row and jth column. The hexahedron /heks-ă- hee-drŏn/ A POLYHE- matrix is named for the French mathemati- DRON that has six faces. For example, the cian Charles Hermite (1822–1901). cube, the cuboid, and the rhombohedron are all hexahedrons. The cube is a regular Hero’s formula A formula for the area of hexahedron; all six faces are congruent a triangle with sides a, b, and c: squares. A = √[s(s – a)(s – b)(s – c)] where s is half the perimeter; i.e. ½(a + b + higher derivative The n-th derivative of c). The formula is named for the Greek a function f of x, where n is greater than or mathematician and inventor Hero of equal to two. If one writes y = f(x) the Alexandria (fl. AD 62). higher derivatives are denoted by dny/dxn or y(n). The second derivative of y with re- hertz /herts/ Symbol: Hz The SI unit of fre- spect to x is denoted by d2y/dx2 or y″. It is quency, defined as one cycle per second found by differentiating dy/dx with respect (s–1). Note that the hertz is used for regu- to x. Similarly, the third derivative of y larly repeated processes, such as vibration with respect to x is denoted by d3y/dx3 or or wave motion. An irregular process, such y′″. For example, if y = x4 + x3 + 1, dy/dx = as radioactive decay, would have units ex- 4x3 + 3x2, d2y/dx2 = 12x2 + 6x, d3y/dx3 = pressed as s–1 (per second). The unit is 24x + 6. named for the German physicist Heinrich Rudolf Hertz (1857–94). highest common factor See common factor. heuristic /hyû-ris-tik/ Based on trial and error, as for example some techniques in it- high-level language See program. erative calculations. See also iteration. Hilbert’s problems /hil-berts/ A set of 23 hexadecimal /heks-ă-dess-ă-măl/ Denot- important mathematical problems posed ing or based on the number sixteen. A by the German mathematician David

101 histogram

Hilbert (1862–1941) at the International Hooke’s law For an elastic material Congress of Mathematics in 1900. This set below its elastic limit, the extension result- of problems has stimulated a great deal of ing from the application of a load (force) is important work in mathematics since proportional to the load. The law is named 1900. Some of Hilbert’s problems have for the English physicist Robert Hooke been solved but, at the time of writing, (1635–1703). some of them remain unsolved. horizontal Describing a line that is paral- histogram /his-tŏ-gram/ A statistical lel with the horizon. It is at right angles to graph that represents, by the length of a the vertical. rectangular column, the number of times that each class of result occurs in a sample horsepower Symbol: HP A unit of power or experiment. See also frequency polygon. equal to 550 foot-pounds per second. It is equivalent to 746 W. holomorphic /hol-ŏ-mor-fik, hoh-lŏ-/ See analytic. hour A unit of time equal to 60 minutes or 1/24 of a day. homeomorphism /hoh-mee-ŏ-mor-fiz- ăm, hom-ee-/ A one-one transformation hundredweight Symbol: cwt In the UK, a between two topological spaces that is con- unit of mass equal to 112 pounds. It is tinuous in both directions. What this equivalent to 50.802 3 kg. In the USA a means is that if two figures are homeomor- hundredweight is equal to 100 pounds, but phic one can be continuously deformed this unit is rarely used. into the other without tearing. For exam- Huygens’ principle /hÿ-gĕnz/ (also ple, any two spheres of any size are home- known as HUYGENS’ CONSTRUCTION) A re- omorphic. But a sphere and a torus are not. sult in the theory of waves which states that each point of a wave-front which is homogeneous /hoh-mŏ-jee-nee-ŭs, hom- propagating serves as the source of sec- ŏ-/ 1. (in a function) Having all the terms ondary waves, with the secondary waves to the same degree in the variables. For a having the same frequency and speed as the homogeneous function f(x,y,z,…) of de- original wave. Huygens’ principle is used gree n extensively in optics to analyze light waves. F(kx,ky,kz,…) = knf(x,y,z) 2 The modification of Huygens’ principle to for all values of k. For example, x + xy + incorporate diffraction and interference is 2 y is a homogeneous function of degree 2 sometimes called the Huygens–Fresnel and principle. Both Huygens’ principle and the 2 2 (kx) + kx.ky + (ky) = Huygens–Fresnel principle can be derived 2 2 2 k (x + xy + y ) as mathematical consequence of the WAVE 2. Describing a substance or object in EQUATION. The principle is named for the which the properties do not vary with po- Dutch astronomer and physicist Christiaan sition; in particular, the density is constant Huygens (1629–95). throughout. hybrid computer A computer system homomorphism /hoh-mŏ-mor-fiz-ăm, containing both analog and digital devices hom-ŏ-/ If S and T are sets on which binary so that the properties or each can be used relations * and • are defined respectively, a to the greatest advantage. For instance, a mapping h from S and T is a homomor- digital and an analog computer can be in- phism if it satisfies the condition h(x*y) = terconnected so that data can be trans- h(x)•h(y) for all x and y in S, i.e. it pre- ferred between them. This is achieved by serves structure. If the mapping is one-to- means of a hybrid interface. Hybrid com- one it is called an isomorphism and the sets puters are designed for specific tasks and S and T are isomorphic. have a variety of uses, mainly in scientific

102 hyperbola

asymptote

V2 V1 F F x 2 O 1

Hyperbola: F1 and F2 aere foci. V1 and V2 are the vertices with coordinates (a,0) and (–a,0) respectively. Y1 and Y2 have coordinates (b,0) and (–b,0). and technical fields. See also analog com- hyperbola /hÿ-per-bŏ-lă/ (pl. hyperbolas puter; computer. or hyperbolae) A CONIC with an eccentricity greater than 1. The hyperbola has two hydraulic press A MACHINE in which branches and two axes of symmetry. An forces are transferred by way of pressure in axis through the foci cuts the hyperbola at a fluid. In a hydraulic press the effort F1 is two vertices. The line segment joining these vertices is the transverse axis of the hyper- applied over a small area A1 and the load bola. The conjugate axis is a line at right F2 exerted over a larger area A2. Since the angles to the transverse axis through the pressure is the same, F1/A1 = F2/A2. The center of the hyperbola. A chord through a force ratio for the machine, F2/F1, is A1/A2. Thus, in this case (and in the related hy- focus perpendicular to the transverse axis is a latus rectum. draulic braking system and hydraulic jack) In Cartesian coordinates the equation: the force exerted by the user is less than the x2/a2 – y2/b2 = 1 force applied; the force ratio is greater than represents a hyperbola with its center at the 1. If the distance moved by the effort is s1 origin and the transverse axis along the x- and that moved by the load is s2 then, since axis. 2a is the length of the transverse axis. the same volume is transmitted through the 2b is the length of the conjugate axis. This system, s1A1 = s2A2; i.e. the distance ratio is the distance between the vertices of a dif- is A2/A1. In practical terms, the device is ferent hyperbola (the conjugate hyperbola) not very efficient since frictional effects are with the same asymptotes as the given one. large. The foci of the hyperbola are at the points (ae,0) and (–ae,0), where e is the eccentric- hydrostatics /hÿ-drŏ-stat-iks/ The study ity. The asymptotes have the equations: of fluids (liquids and gases) in equilibrium. x/a – y/b = 0

103 hyperbolic functions

y y = cosh x

3 – y = sinh x

2 –

1 – y = tanh x

0 x – – – – – – – -3 -2 -1 1 2 3

y = tanh x -1 –

-2 –

-3 – y = sinh x

The graphs of the hyperbolic functions cosh x, sinh x, and tanh x.

x/a + y/b = 0 The hyperbolic sine (sinh) of an angle a The equation of the conjugate hyper- is defined as: bola is sinha = ½(ea – e–a) x2/a2 – y2/b2 = –1 The hyperbolic cosine (cosh) of an The length of the latus rectum is 2b2/ae. angle a is defined as: A hyperbola for which a and b are equal is cosha = ½(ea + e–a) a rectangular hyperbola: The hyperbolic tangent (tanh) of an 2 2 2 x – y = a angle a is defined as: If a rectangular hyperbola is rotated so tanha = sinha/cosha = that the x- and y-axes are asymptotes, then (ea – e–a)/(ea + e–a) its equation is Hyperbolic secant (sech), hyperbolic xy = k cosecant (cosech), and hyperbolic cotan- where k is a constant. gent (coth) are defined as the reciprocals of hyperbolic functions /hÿ-per-bol-ik/ A cosh, sinh, and tanh respectively. Some of set of functions that have properties similar the fundamental relationships between hy- in some ways to the trigonometric func- perbolic functions are: tions, called the hyperbolic sine, hyperbolic sinh(–a) = –sinha cosine, etc. They are related to the hyper- cosh(–a) = +cosha bola in the way that the trigonometric cosh2a – sinh2a = 1 functions (circular functions) are related to sech2a + tanh2a = 1 the circle. coth2a – cosech2a = 1

104 hypothesis test hyperbolic paraboloid A CONICOID that inside of a larger fixed circle. See also cusp; is described by the equation x2/a2 – y2/b2 = epicycloid. 2z/c, where a, b, and c are constants and the coordinate system is such that the hypotenuse /hÿ-pot-ĕ-news/ The side origin is a SADDLE POINT. Cross-sections opposite the right angle in a right-angled through the z-axis cut this surface in triangle. The ratios of the hypotenuse parabolas, with the origin being the vertex length to the lengths of the other sides are of all these parabolas. Cross-sections of the used in trigonometry to define the sine and surface formed by planes parallel to the xy- cosine functions of angles. planes are hyperbolas. The cross-section with the xy-plane is a pair of straight lines. hypothesis /hÿ-poth-ĕ-sis/ (pl. hypoth- Cross-sections with planes which are par- eses) A statement, theory, or formula that allel to the xz- and yz-planes are parabolas. has yet to be proved but is assumed to be The xz- and yz-planes are symmetry true for the purposes of the argument. planes. hypothesis test (significance test) A rule for deciding whether an assumption (hy- hyperboloid /hÿ-per-bŏ-loid/ A surface pothesis) about the distribution of a ran- generated by rotating a hyperbola about dom variable should be accepted or one of its axes of symmetry. Rotation rejected, using a sample from the distribu- about the conjugate axis gives a hyper- tion. The assumption is called the null hy- boloid of one sheet. Rotation about the pothesis, written H0, and it is tested against transverse axis gives a hyperboloid of two some alternative hypothesis, H . For ex- sheets 1 ample, when a coin is tossed H0 can be P(heads) = ½ and H1 that P(heads) >½. A hypertext /hÿ-per-text/ A method of cod- statistic is computed from the sample data. ing and displaying text on a computer If it falls in the critical region, where the screen in such a way that key words or value of the statistic is significantly differ- phrases in the document can act as direct ent from that expected under H0, H0 is re- links to other documents or to other parts jected in favor of H1. Otherwise H0 is of the document. It is extensively used on accepted. A type I error occurs if H0 is re- the World Wide Web. An extension of hy- jected when it should be accepted. A type II pertext, known as hypermedia, allows error occurs if it is accepted when it should linkage to sounds, images, and video clips. be rejected. The significance level of the test, α, is the maximum probability with hypocycloid /hÿ-pŏ-sÿ-kloid/ A cusped which a type I error can be risked. For ex- curve that is the locus of a point on the cir- ample, α = 1% means H0 is wrongly re- cumference of a circle that rolls around the jected in one case out of 100.

105 I

icosahedron /ÿ-kos-ă- hee-drŏn/ (pl. such as x2 + 2 = 0, for which the solutions icosahedrons or icosahedra) A POLYHE- are x = +i√2 and x = –i√2. See complex DRON that has twenty triangular faces. A number. regular icosahedron has twenty congruent faces, each one an equilateral triangle. imaginary part Symbol Imz. The part iy of a complex number z that can be written identity, law of See laws of thought. as z = x + iy, where x and y are both real numbers. identity element An element of a set that, combined with another element, leaves it impact A collision between two bodies. If unchanged. See group. the initial masses and velocities of the bodies are known then the velocities following identity matrix See unit matrix. the impact can be calculated in terms of the initial velocities and the COEFFICIENT OF identity set A set consisting of the same RESTITUTION by using the principle of the elements as another. For example, the set conservation of momentum. The instance of natural numbers greater than 2 and the of a direct (head-on) impact is easiest to set of integers greater than 2 are identity analyze theoretically but the motions of sets. bodies which are not initially moving along the same line can also be analyzed by a if and only if (iff) See biconditional. modification of the method used for bodies that are initially moving along the same if…then… See implication. line. image The result of a geometrical trans- Imperial units The system of measure- formation or a mapping. For example, in ment based on the yard and the pound. The geometry, when a set of points are trans- f.p.s. system was a scientific system based formed into another set by reflection in a on Imperial units. line, the reflected figure is called the image. Similarly, the result of a rotation or a pro- implication 1. (material implication; con- jection is called the image. The algebraic ditional) Symbol: → or ⊃ In logic, the rela- equivalent of this occurs when a function tionship if…then… between two f(x) acts on a set A of values of x to pro- propositions or statements. Strictly, impli- duce an image set B. See also domain; cation reflects its ordinary language inter- range; transformation. imaginary axis The axis on the complex P Q P → Q plane that purely imaginary numbers lie on. This axis is usually drawn as the y-axis. T T T T F F F T T imaginary number A multiple of i, the F F T square root of minus one. The use of imaginary numbers is needed to solve equations Implication

106 incircle

∞ pretation (if…then…) much less than con- then the improper integral ∫–∞ f(x)dx exists junction, disjunction, and negation do and has the value l1 + l2. theirs. Formally, P → Q is equivalent to ‘ei- If the function f(x) which is being inte- ther not P or Q’ (∼ P ∨ Q), hence P → Q is grated becomes infinite at some point over false only when P is true and Q is false. the interval of integration it is possible to Thus, logically speaking, if ‘pigs can fly’ is examine whether the improper integral ex- substituted for P and ‘grass is green’ for Q ists. This is the case whether the point at then ‘if pigs can fly then grass is green’ is which the function becomes infinite is one true. The truth-table definition of implica- of the limits of integration or is within the tion is given in the illustration. interval of integration. The case when the ⇒ 2. In algebra, the symbol is used be- point is one of the limits is easier to ana- tween two equations when the first implies lyze. If the point is the lower limit a then the second. For example: one can consider the integral with the ⇒ 2 2 x = y x = y lower limit a+δ, where δ is a small number, See also condition; truth table. perform the integration and take the limit as δ→0. If such a limit exists then the im- implicit Denoting a function that con- proper integral exists and has a value given tains two or more variables that are not by this limit. The case when the function independent of each other. An implicit becomes infinite at the upper limit b can be function of x and y is one of the form f(x,y) investigated in a similar way. = 0, for example, If the function becomes infinite at some x2 + y2 – 4 = 0 point c which is between a and b the inte- Sometimes an explicit function, that is, gral is split into two integrals, with c being one expressed in terms of an independent the upper limit of one integral and the variable, can be derived from an implicit lower limit of the other integral. If both function. For example, y + x2 – 1 = 0 these improper integrals exist then the can be written as original improper integral exists, with its y = 1 – x2 value being equal to the sum of the values where y is an explicit function of x. of the two integrals into which it was split. improper fraction See fraction. impulse (impulsive force) A force acting for a very short time, as in a collision. If the δ δ improper integral An integral in which force (F) is constant the impulse is F t, t either the interval of integration is infinite being the time period. If the force is vari- or the value of the function f(x) which is able the impulse is the integral of this over being integrated becomes infinite at some the short time period. An impulse is equal point in the interval of integration. to the change of momentum that it pro- An improper integral of the type duces. ∞ ∫a f(x)dx is said to exist if the value of the integral of the function f(x) with respect to impulsive force See impulse. x in which the interval of integration runs from a to b tends to a finite limit l as b→∞, incenter /in-sen-ter/ See incircle. ∞ b i.e. ∫a f(x)dx exists if lim∫af(x)dx = l. An ex- inch Symbol: in or ″ A unit of length equal ample of an∞ improper integral of this type 2 is given by ∫1 (1/x )dx = 1. to one twelfth of a foot. It is equivalent to It is also possible to define an improper 0.025 4 m. integral if the interval of integration is from –∞ to a. This means that if the integral incircle /in-ser-kăl/ (inscribed circle) A cir- ∞ a ∫ ∞ ∫ ∞ cle drawn inside a figure, touching all the – f(x∞)dx is written as the sum – f(x)dx and ∫ a f(x)dx and if these two integrals sides. The center of the circle is the incen- exist and have values l1 and l2 respectively ter of the figure. Compare circumcircle. 107 inclined plane

increasing function A real function f of x for which f(x1) ≤ f(x2) for any x1 and x2 in an interval I for which x1 < x2. If f(x1) < f(x2) when x1 < x2 then f is said to be a strictly increasing function.

increment A small difference in a variable. For example, x might change by an increment ∆x from the value x1 to the value x2; ∆x = x2 – x1. In calculus, infinitely small increments are used. See also differentiation; integration. Incircle: an incircle touches all the sides of its surrounding figure. indefinite integral The general integration of a function f(x), of a single variable, x, without specifying the interval of x to inclined plane A type of MACHINE. Effec- which it applies. For example, if f(x) = x2, tively a plane at an angle, it can be used to the indefinite integral raise a weight vertically by movement up ∫f(x)dx = ∫x2dx = (x3/3) + C an incline. Both distance ratio and force where C is an unknown constant (the con- ratio depend on the angle of inclination (θ) stant of integration) that depends on the in- and equal 1/sinθ. The efficiency can be terval. Compare definite integral. See also fairly high if friction is low. The screw and integration. the wedge are both examples of inclined planes. independence See probability. inclusion See subset. independent See dependent. inclusive disjunction (inclusive or) See independent variable See variable. disjunction. indeterminate equation An equation inconsistent Describing a set of equations that has an infinite number of solutions. for which the solution set is empty. Geo- For example, metrically, a set of equations is inconsistent x + 2y = 3 if there is no point common to all the lines is indeterminate because an infinite num- or curves represented by the equations. In ber of values of x and y will satisfy it. An the case of two linear equations in two indeterminate equation in which the variables can take only integer values is called variables x and y the two equations can be a Diophantine equation and it has an infi- represented by straight lines L and L . The 1 2 nite but denumerable set of solutions. Dio- two equations are inconsistent if L and L 1 2 phantine equations are named for the are parallel. In the case of three linear Greek mathematician Diophantus of equations in three variables x, y, and z the Alexandria (fl. AD 250). three equations can be represented by planes P1, P2, and P3. There are three ways indeterminate form An expression that in which the equations can be inconsistent. can have no quantitative meaning; for ex- The first way is that the three planes are all ample 0/0. parallel and distinct. The second way is that two of the planes are parallel and dis- index (pl. indexes or indices) A number tinct. The third way is that one plane is that indicates a characteristic or function parallel to the line of intersection of the in a mathematical expression. For exam- other two planes. ple, in y4, the exponent, 4, is also known as

108 inertial system

3 the index. Similarly in √27 and log10x, the while denying the conclusion in an induc- numbers 3 and 10 respectively are called tion does not lead to a CONTRADICTION. indices (or indexes). The conclusion is not guaranteed to be true if the premisses are. Compare deduction. indirect proof (reductio ad absurdum) A logical argument in which a proposition or inelastic collision /in-i-las-tik/ A collision statement is proved by showing that its for which the restitution coefficient is less negation or denial leads to a CONTRADIC- than one. In effect, the relative velocity TION. Compare direct proof. after the collision is less than that before; the kinetic energy of the bodies is not con- induction /in-duk-shŏn/ 1. (mathematical served in the collision, even though the sys- induction) A method of proving mathe- tem may be closed. Some of the kinetic matical theorems, used particularly for se- energy is converted into internal energy. ries sums. For instance, it is possible to See also restitution, coefficient of. show that the series 1 + 2 + 3 + 4 + … has a sum to n terms of n(n + 1)/2. First we inequality A relationship between two show that if it is true for n terms it must expressions that are not equal, often writ- also be true for (n + 1) terms. According to ten in the form of an equation but with the the formula symbols > or < meaning ‘is greater than’ Sn = n(n + 1)/2 and ‘is less than’. For example, if x < 4 then if the formula is correct, the sum to (n + 1) x2 < 16. If y2 > 25, then y > 5 or y < –5. If terms is obtained by adding (n + 1) to this the end values are included, the symbols ≥ Sn+1 = n(n + 1)/2 + (n + 1) (is greater than or equal to) and ≤ (is less Sn+1 = (n + 1)(n + 2)/2 than or equal to) are used. When one quan- This agrees with the result obtained by tity is very much smaller or greater than replacing n in the general formula by (n + another, it is shown by << or >>. For ex- 1), i.e.: ample, if x is a large number x >> 1/x or 1/x Sn+1 = (n + 1)(n + 1 + 1)/2 << x. See also equality. Sn+1 = (n + 1)(n + 2)/2 Thus, the formula is true for (n + 1) inequation /in-i-kway-zhŏnz/ Another terms if it is true for n terms. Therefore, if word for inequality. it is true for the sum to one term (n = 1), it must be true for the sum to two terms (n + inertia /i-ner-shă/ An inherent property of 1). Similarly, if true for two terms, it must matter implied by Newton’s first law of be true for three terms, and so on through motion: inertia is the tendency of a body to all values of n. It is easy to show that it is resist change in its motion. See also inertial true for one term: mass; Newton’s laws of motion. Sn = 1(1 + 1)/2 Sn = 1 inertial mass /i-ner-shăl/ The mass of an which is the first term in the series. Hence object as measured by the property of iner- the theorem is true for all integer values of tia. It is equal to the ratio force/accelera- n. tion when the object is accelerated by a 2. In logic, a form of reasoning from indi- constant force. In a uniform gravitational vidual cases to general ones, or from ob- field, it appears to be equal to GRAVITA- served instances to unobserved ones. TIONAL MASS – all objects have the same Inductive arguments can be of the form: F1 gravitational acceleration at the same is A, F2 is A … Fn is A, therefore all Fs are place. A (‘this swan has wings, that swan has wings … therefore all swans have wings’); inertial system A frame of reference in or: all Fs observed so far are A, therefore which an observer sees an object that is free all Fs are A (‘all swans observed so far are of all external forces to be moving at con- white, therefore all swans are white’). Un- stant velocity. The observer is called an in- like deduction, asserting the premisses ertial observer. Any FRAME OF REFERENCE

109 inf that moves with constant velocity and quantity by +∞. If x is positive, y = –1/x without rotation relative to an inertial tends to –∞ as x tends to 0. frame is also an inertial frame. NEWTON’S LAWS OF MOTION are valid in any inertial inflection See point of inflection. frame (but not in an accelerated frame), and the laws are therefore independent of information theory The branch of prob- the velocity of an inertial observer. ability theory that deals with uncertainty, accuracy, and information content in the inf See infimum. transmission of messages. It can be applied to any system of communication, including inference /in-fĕ-rĕns/ 1. The process of electrical signals and human speech. Ran- reaching a conclusion from a set of pre- dom signals (noise) are often added to a misses in a logical argument. An inference message during the transmission process, may be deductive or inductive. See also de- altering the signal received from that sent. duction; induction. Information theory is used to work out the 2. See sampling. probability that a particular signal received is the same as the signal sent. Redundancy, infimum (inf) The greatest lower BOUND for example simply repeating a message, is of a set. needed to overcome the limitations of the system. Redundancy can also take the form infinite number The smallest infinite of a more complex checking process. In transmitting a sequence of numbers, their number is '0 (aleph zero). This is the number of members in the set of integers. A sum might also be transmitted so that the receiver will know that there is an error whole hierarchy of increasingly large infi- when the sum does not correspond to the nite numbers can be defined on this basis. rest of the message. The sum itself gives no ' , the next largest, is the number of sub- 1 extra information since, if the other num- sets of the set of integers. See also aleph; bers are correctly received, the sum can continuum; countable. easily be calculated. The statistics of choosing a message out of all possible messages See sequence. infinite sequence (letters in the alphabet or binary digits for example) determines the amount of infor- infinite series See series. mation contained in it. Information is measured in bits (binary digits). If one out of infinite set A set in which the number of two possible signals are sent then the infor- elements is infinite. For example, the set of mation content is one bit. A choice of one ‘positive integers’, z = {1, 2, 3, 4, …}, is in- out of four possible signals contains more finite but the set of ‘positive integers less information, although the signal itself than 20’ is a finite set. Another example of might be the same. an infinite set is the number of circles in a particular plane. Compare finite set. inner product Consider a vector space V over a scalar field F. An inner product on V infinitesimal /in-fi-nă-tess-ă-măl/ Infi- is a mapping of ordered pairs of vectors in nitely small, but not equal to zero. Infini- V into F; i.e. with every pair of vectors x tesimal changes or differences are made use and y there is associated a scalar, which is of in CALCULUS (infinitesimal calculus). written 〈x,y〉 and called the inner product of x and y, such that for all vectors x, y, z infinity Symbol: ∞ The value of a quantity and scalars α that increases without limit. For example, (i) 〈x+y,z〉 = 〈x,z〉 + 〈y,z〉 if y = 1/x, then y becomes infinitely large, (ii) 〈x,y〉 = α〈x,y〉 or approaches infinity, as x approaches 0. (iii) 〈x,y〉 = 〈y,x〉, where 〈a,b〉 is the com- An infinitely large negative quantity is de- plex conjugate of 〈a,b〉 noted by –∞ and an infinitely large positive (iv) 〈x,x〉≥0, 〈x,x〉 = 0 if and only if x = 0.

110 integral transform

An inner product on V defines a norm on x < n + 1. The integer n is said to be the in- V given by ||x|| = √〈x,x〉. See norm. teger part of x. For example, [13/3] = 4 and [e] = 2. Care needs to be taken in the input 1. The signal or other form of in- case of negative numbers. For example, formation that is applied (fed in) to an elec- [–13/3] = –5. The difference between posi- trical device, machine, etc. The input to a tive numbers can be significant if one has a computer is the data and programmed in- computer program that converts a real structions that a user communicates to the number into an integer by truncation. In machine. An input device accepts com- the case of positive numbers and zero trun- puter input in some appropriate form and cation and taking the integer part of the converts the information into a code of real number would give the same integer electrical pulses. The pulses are then trans- but this is not the case for negative num- mitted to the central processor of the com- bers. puter. 2. The process or means by which input is integer variable See variable. applied. 3. To feed information into an electrical integral /in-tĕ-grăl, in-teg-răl/ The result device or machine. of integrating a function. See integration. See also input/output; output. integral equation An equation in which input/output (I/O) The equipment and an unknown function appears under an in- operations used to communicate with a tegral sign. Many problems in physical sci- computer, and the information passed in ence and engineering can be expressed as or out during the communication. Input/ integral equations. It is sometimes easier to output devices include those used only for solve problems which can be expressed as INPUT or for OUTPUT of information and ordinary or partial differential equations those, such as visual display units, used for by expressing them in terms of integral both input and output. equations. In particular, solving initial value problems and boundary-value prob- inscribed Describing a geometric figure lems of differential equations is frequently that is drawn inside another geometrical performed by converting the differential figure. Compare circumscribed. equation into an integral equation. Just as there is no general technique for perform- inscribed circle See incircle. ing integration so there is no general technique for solving all integral equations. instantaneous value /in-stăn-tay-nee-ŭs/ The value of a varying quantity (e.g. veloc- integral transform A transform in which ity, acceleration, force, etc.) at a particular a function f(x) of a variable x is trans- instant in time. formed into a function g(y) of a variable y by the integral equation: integer /in-tĕ-jer/ Symbol: z Any of the set g(y) = ∫K(x, y)f(x)dx, of whole numbers, {…, –2, –1, 0, 1, 2, …} where K(x,y) is a part of the integral equa- used for counting. The integers include tion known as its kernel. For example, zero and the negative whole numbers. integral transforms can map a function of time into another function in frequency integer part Symbol [x]. An integer n when wave phenomena are being analyzed. which is related to a real number x by n ≤ See Laplace transform. – – – – – – – – -3 -2 -1 0 1 2 3 4

Integers: a number line showing positive and negative integers.

111 integrand y

f(x )

x2 Ύ f(x )dx x1

0 x x1 x2

The integration of a function y = f(x) as the area between the curve and the x-axis. integrand /in-tĕ-grand/ A function that is integral of velocity is distance. For exam- to be integrated. For example, in the inte- ple, a car traveling with a velocity V in a gral of f(x).dx, f(x) is the integrand. See time interval t1 to t2 goes a distance s given also integration. by t2 ∫ vdt integrating factor A multiplier used to t1 simplify and solve DIFFERENTIAL EQUA- Integrals of this type, between definite lim- TIONS, usually given the symbol ξ. For ex- its, are known as definite integrals. An in- ample, xdy – ydx = 2x3dx may be definite integral is one without limits. The multiplied by ξ(x) = 1/x2 to give the stan- result of an indefinite integral contains a dard form: constant – the constant of integration. For d(y/x) = (x.dy – y.dx)/x2 = 2xdx example, which has the solution y/x = x2 + C, where ∫xdx = x2/2 + C C is a constant. where C is the constant of integration. A table of integrals is given in the Appendix. integration The continuous summing of change in a function, f(x), over an interval integration by parts A method of inte- of the variable x. It is the inverse process of grating a function of a variable by express- DIFFERENTIATION in calculus, and its result ing it in terms of two parts, both of which is known as the integral of f(x) with respect are differentiable functions of the same to x. An integral: variable. A function f(x) is written as the x 2 product of u(x) and the derivative dv/dx. ∫ vdt x1 The formula for the differential of a product is: can be regarded as the area between the d(u.v)/dx = u.dv/dx + v.du/dx curve and the x-axis, between the values x1 Integrating both sides over x and rearrang- and x2. It can be considered as the sum of ing the equation gives a number of column areas of width ∆x and ∫u.(dv/dx)dx = uv – ∫v(du/dx)dx heights given by f(x). As ∆x approaches which can be used to evaluate the integral zero, the number of columns increases infi- of a product. For example, to integrate nitely and the sum of the column areas ap- xsinx, dv/dx is taken to be sinx, so v = proaches the area under the curve. The –cosx + C (C is a constant of integration).

112 interpolation u is taken to be x, so du/dx = 1. The inte- interface A shared boundary. It is the gral is then given by area(s) or place(s) at which two devices or ∫xsinxdx = two systems meet and interact. For exam- x(–cosx + C) – ∫(–cosx + C) = ple, there is a simple interface between an –xcosx + sinx + k electric plug and socket. A far more com- where k is a constant of integration. Usu- plicated interface of electronic circuits pro- ally a trigonometric or exponential func- vides the connection between the central tion is chosen for dv/dx. processor of a computer and each of its peripheral units. Human-machine interface integration by substitution A method refers to the interaction between people of integrating a function of one variable by and machines, including computers. For expressing it as a simpler or more easily in- good, i.e. efficient, interaction, devices tegrated function of another variable. For such as visual display units and easily un- example, to integrate √(1 – x2) with respect derstandable programming languages have to x, we can make x = sinu, so that √(1 – x2) been introduced. = √(1 – sin2u) = √(cos2u) = cosu, and dx = (dx/du).du = cosudu. Therefore: interior angle An angle formed on the in- b d side of a plane figure by two of its straight ͐ ͙(1 – x2)dx = ͐ cos2u.du sides. For example, there are three interior a c angles in a triangle, which add up to 180°. 1 d = [u/2 – 2sinucosu]c Compare exterior angle.

Note that for a definite integral the limits intermediate value theorem A theorem must also be changed from values of x to concerning continuous functions which corresponding values of u. states that if a real function f is continuous on an interval I bounded by a and b for integro-differential equation /in-teg- which f(a) ≠ f(b) then if n is some real num- roh-dif-ĕ-ren-shăl/ A differential equation ber between f(a) and f(b) there must be that also has an integral as part of it. Such some number c in the interval, i.e. a < c < b equations occur in the quantitative descrip- for which f(c) = n. tion of transport processes. General meth- The intermediate value theorem can be ods for the solution of integro-differential used to find the solutions of an equation. equations do not exist and approximation For example, if f(a) > 0 and f(b) < 0 then methods have to be used. f(x) = 0 between a and b. interaction Any mutual action between internal store See central processor; particles, systems, etc. Examples of interac- store. tions include the mutual forces of attraction between masses (gravitational interpolation /in-terp-ŏ-lay-shŏn/ The interaction) and the attractive or repulsive process of estimating the value of a func- forces between electric charges (electro- tion from known values on either side of it. magnetic interaction). For example, if the speed of an engine, controlled by a lever, increases from 40 to 50 intercept A part of a line or plane cut off revolutions per second when the lever is by another line or plane. pulled down by four centimeters, one can interpolate from this information and as- interest The amount of money paid each sume that moving it two centimeters will year at a stated rate on a borrowed capital, give 45 revolutions per second. This is the or the amount received each year at a simplest method of interpolation, called stated rate on loaned or invested capital. linear interpolation. If known values of The interest rate is usually expressed as a one variable y are plotted against the other percentage per annum. See compound in- variable x, an estimate of an unknown terest; simple interest. value of y can be made by drawing a

113 interquartile range

A ∩ B

The shaded area in the Venn diagram is the intersection of set A and set B. straight line between the two nearest interval A set of numbers, or points in a known values. coordinate system, defined as all the values The mathematical formula for linear in- between two end points. See also closed interpolation is: terval; open interval. y3 = y1 + (x3 – x1)(y2 – y1)/(x2 – x1) y3 is the unknown value of y (at x3) and y2 into A mapping from one set to another is and y1 (at x2 and x1) are the nearest known said to be into if the range of the mapping values, between which the interpolation is is a proper subset of the second set; i.e. if made. If the graph of y against x is a there are members of the second set which are not the image of any element of the first smooth curve, and the interval between y1 set under the mapping. Compare onto. and y2 is small, linear interpolation can give a good approximation to the true intransitive /in-tran-să-tiv/ Describing a value, but if (y2 – y1) is large, it is less likely that y will fit sufficiently well to a straight relation that is not transitive; i.e. when xRy line between y and y . A possible source and yRz then it is not true that xRz. An ex- 1 2 ample of an intransitive relation is being of error occurs when y is known at regular the square of. If x = y2 and y = z2 it does not intervals, but oscillates with a period follow that x = z2. shorter than this interval. Compare extrapolation. intuitionism /in-too-ish-ŏ-niz-ăm/ An approach to the foundations of mathematics interquartile range /in-ter-kwor-tÿl/ A that was developed in the late nineteenth measure of dispersion given by (P – P ) 75 25 century and the early decades of the twen- where P75 is the upper QUARTILE and P25 tieth century. Intuitionism tried to build up the lower quartile. The semi-interquartile mathematics based on intuitive concepts ½ range is (P75 – P25). rather than from a rigid series of axioms. Using this approach the intuitionists were intersection 1. The point at which two or able to build up various parts of algebra more lines cross each other, or a set of and geometry and to formulate calculus, points that two or more geometrical fig- albeit in a complicated way. As with other ures have in common. attempts such as FORMALISM to provide 2. In set theory, the set formed by the ele- foundations for mathematics, intuitionism ments common to two or more sets. For was not accepted universally as providing a example, if set A is {black four-legged ani- foundation for mathematics. mals} and set B is {sheep} then the intersec- The main difficulties for all attempts to tion of A and B, written A∩B, is {black provide foundations for mathematics, in- sheep}. This can be represented on a VENN cluding formalism and intuitionism involve DIAGRAM by the intersection of two circles, dealing with infinite sets and infinite one representing A and the other B. processes.

114 inverse matrix invariant /in-vair-ee-ănt/ Describing a inverse mapping A mapping, denoted property of an equation, function, or geo- f–1, from one set B to another set A, where metrical figure that is unaltered after the f is a one-to-one and onto mapping from A application of any member of some given to B which satisfies the definition in the fol- family of transformations. For example, lowing sentence. The inverse mapping f–1 the area of a polygon is invariant under the from B to A exists if, for an element b of B, group of rotations. there is a unique element a of A given by f–1(b) = a which satisfies f(a) = b. Like the inverse element An element of a set that, mapping f, the inverse mapping f–1 is a one- combined with another element, gives the to-one and onto mapping. identity element. See group. inverse matrix The unique n × n matrix, inverse function The reverse mapping of denoted A–1 corresponding to the n × n ma- the set B into the set A when the FUNCTION trix A that satisfies AA–1 = A–1A = I, where that maps A into B has already been de- I is the UNIT MATRIX of order n. If a matrix fined. is not a square matrix it cannot have an inverse matrix. It is not necessarily the case inverse hyperbolic functions The in- that a square matrix has an inverse. If a verse functions of hyperbolic sine, hyper- matrix does have an inverse then the ma- bolic cosine, hyperbolic tangent, etc., trix is invertible or non-singular. defined in an analogous way to the inverse There are several techniques for finding trigonometric functions. For instance, the inverse matrices. In general, the inverse of inverse hyperbolic sine of a variable x, A is given by (1/detA) adj A, where adjA IS written arc sinhx or sinh–1x, is the angle (or the ADJOINT of A, provided that detA ≠ 0. If number) of which x is the hyperbolic sine. detA = 0 then the matrix is said to be SIN- Similarly, the other inverse hyperbolic GULAR and an inverse does not exist. In the functions are: case of a 2 × 2 matrix A of the inverse hyperbolic cosine of x (written arc form coshx or cosh–1x) a b A = inverse hyperbolic tangent of x (written arc c d tanhx or tanh–1x) The inverse A–1 exists if ad–bc ≠ 0 and is inverse hyperbolic cotangent of x (written given by –1 arc cothx or coth x) −1 1 d −b A = inverse hyperbolic secant of x (written arc ad − bc −c a sechx or sech–1x) inverse hyperbolic cosecant of x (written In the case of large matrices it is not arc cosechx or cosech–1x). convenient to find the inverse of a matrix

x + 3y = 5 1 3 x 5 <=> x = 2x + 4y = 6 2 4 y 6

-2 3 1 3 1 0 2 x = 1 -½ 2 4 0 1

3 -2 2 5 -1 => x = x = y 1 -½ 6 2

The solution of simultaneous equations by finding the inverse of a matrix. The equations are written as the equivalent matrix equation, both sides of which are then multiplied by the inverse of the coefficient matrix.

115 inverse of a complex number by using the general formula involving the I/O See input/output. determinant. It is more convenient to use a technique similar to GAUSSIAN ELIMINATION, irrational number /i-rash-ŏ-năl/ A num- which is used to solve sets of linear equa- ber that cannot be expressed as a ratio of tions. two integers. The irrational numbers are precisely those infinite decimals that are inverse of a complex number A com- not repeating. Irrational numbers are of plex number, denoted by 1/z or z–1, which two types: is the inverse of the complex number z = x (i) Algebraic irrational numbers are roots + i y, i.e. it is the number z–1 for which zz–1 of polynomial equations with rational √ = 1. It is given by z = x/(x2 + y2) – i y/(x2 + numbers as coefficients. For example, 3 = 1.732 050 8… is a root of the equation x2 y2). If z is written in terms of polar coordi- = 3. This equation does not have a rational nates z = r (cos θ + i sin θ) = r exp (iθ) then –1 θ solution since such a solution could be z is given by (1/r) exp (–i ) = (1/r) (cos 2 2 θ θ ≠ written x = m/n with m = 3n , but this is – i sin ), which exists if r 0. impossible since 3 divides the left-hand side an even number of times and the right- inverse ratio A reciprocal ratio. For ex- hand side an odd number of times. ample, the inverse ratio of x to y is the ratio (ii) Transcendental numbers are irrational of 1/x to 1/y. numbers that are not algebraic, e.g. e and π. inverse square law A physical law in which an effect varies inversely as the irrational term A term in which at least square of the distance from the source pro- one of the indices is an irrational number. ducing the effect. An example is Newton’s For example, xy√2 and 2xπ are irrational law of universal gravitation. terms. inverse trigonometric functions The in- irrotational vector A vector V for which verse functions of sine, cosine, tangent, etc. the curl of V vanishes, i.e. ∇×V = 0. Ex- For example, the inverse sine of a variable amples of irrotational vectors include the is called the arc sine of x; it is written arc gravitational force described by Newton’s sinx or sin–1x and is the angle (or number) law of gravity and the electrostatic force of which the sine is x. Similarly, the other governed by Coulomb’s law. If u is the dis- inverse trigonometric functions are: placement of a plane wave in an elastic inverse cosine of x (arc cosine, written arc medium (or a spherical wave a long way cosx or cos–1x) from the source of the wave) then the con- inverse tangent of x (arc tangent, written dition that u is an irrotational vector means that the wave is a longitudinal wave. arc tanx or tan–1x) When this result is combined with the inverse cotangent of x (arc cotangent, writ- concept of a SOLENOIDAL VECTOR and ten arc cotx or cot–1x) HELMHOLTZ’S THEOREM it is useful in ana- inverse cosecant of x (arc cosecant, written –1 lyzing waves in seismology. arc cosecx or cosec x) If a vector is an irrotational vector then inverse secant of x (arc secant, written arc it can be written as –1 times the gradient of –1 secx or sec x). a scalar function. In physical applications the scalar function is usually called the inverter gate See logic gate. scalar potential. involute /in-vŏ-loot/ The involute of a isolated point A point that satisfies the curve is a second curve that would be ob- equation of a curve but is not on the main tained by unwinding a taut string wrapped arc of the curve. For example, the equation around the first curve. The involute is the y2(x2 – 4) = x4 has a solution at x = 0 and y curve traced out by the end of the string. = 0, but there is no real solution at any

116 iteration point near the origin, so the origin is an iso- integral) A succession of integrations per- lated point. See also double point; multiple formed on the same function. For example, point. a DOUBLE INTEGRAL or a TRIPLE INTEGRAL. isolated system See closed system. iteration /it-ĕ-ray-shŏn/ A method of solving a problem by successive approxima- isometric paper /ÿ-sŏ-met-rik/ Paper on tions, each using the result of the preceding which there is a grid of equilaterial trian- approximation as a starting point to obtain gles printed. This type of paper is useful for a more accurate estimate. For example, the drawing shapes such as cubes and cuboids, square root of 3 can be calculated by writ- with each face being a parallelogram in the ing the equation x2 = 3 in the form 2x2 = x2 drawing. Using this type of paper it is pos- + 3, or x = ½(x + 3/x). To obtain a solution sible to draw all the dimensions of the fig- for x by iteration, we might start with a ures correctly and to measure them by first estimate, x1 = 1.5. Substituting this in using the regular grid pattern. the equation gives the second estimate, x2 = ½(1.5 + 2) = 1.750 00. Continuing in this isometry /ÿ-som-ĕ-tree/ A transformation way, we obtain: in which the distances between the points x3 = ½(1.75 + 3/1.75) = 1.732 14 remain constant. x4 = ½(1.732 14 + 3/1.732 14) = 1.732 05 isomorphic /ÿ-sŏ-mor-fik/ See homomor- and so on, to any required accuracy. The phism. difficulty in solving equations by iteration is in finding a formula for iteration (algo- isomorphism /ÿ-sŏ-mor-fiz-ăm/ See ho- rithm) that gives convergent results. In this momorphism. case, for example, the algorithm xn+1 = 3/xn does not give convergent results. isosceles /ÿ-soss-ĕ-leez/ Having two equal There are several standard techniques, sides. See triangle. such as NEWTON’S METHOD, for obtaining convergent algorithms. Iterative calcula- issue price See nominal value. tions, although often tedious for manual computation, are widely used in comput- iterated integral /it-ĕ-ray-tid/ (multiple ers.

117 J

j An alternative to i for the square root complex number. One takes the expression of –1. The use of j rather than i in complex z2 + c, where z and c are complex numbers, numbers is particularly common among and calculates it for a given value of z and electrical engineers. takes the result as a new starting value of z. This process can be repeated indefinitely job A unit of work submitted to a com- and three possibilities occur, depending on puter. It usually includes several programs. the initial value for z. One is that the value The information necessary to run a job is tends to zero with successive iterations. input in the form of a short program writ- Another is that the value diverges to infin- ten in the job-control language (JCL) of the ity. There is, however, a set of initial values computer. The JCL is interpreted by the of z for which successive iterations give val- OPERATING SYSTEM and is used to identify ues that stay in the set. This set of values is the job and describe its requirements to the a Julia set and it can be represented by operating system. See also batch process- points in the complex plane. The Julia set is ing. the boundary between values that have an joule /jool/ Symbol: J The SI unit of energy attractor at zero and values that have an and work, equal to the work done where ATTRACTOR at infinity. The actual form of the point of application of a force of one the Julia set depends on the value of the newton moves one meter in the direction of constant complex number c. Thus, if c = 0, → 2 action of the force. 1 J = 1 N m. The joule the iteration is z z . In this case the Julia is the unit of all forms of energy. The unit set is a circle with radius 1. There is an in- is named for the British physicist James finite number of Julia sets showing a wide Prescott Joule (1818–89). range of complex patterns. The set is named for the French mathematician Gas- Julia set /joo-lee-ă/ A set of points in the ton Julia (1893–1978). See also Mandel- complex plane defined by iteration of a brot set.

118 K

kelvin Symbol: K The SI base unit of ther- (3) The square of the period of each planet modynamic temperature. It is defined as is proportional to the cube of the semi- the fraction 1/273.16 of the thermody- major axis of the ellipse. namic temperature of the triple point of Application of the third law to the orbit water. Zero kelvin (0 K) is absolute zero. of the Moon about the Earth gave support One kelvin is the same as one degree on the to Newton’s theory of gravitation. Celsius scale of temperature. The unit is named for the British theoretical and ex- keyboard A computer input device that a perimental physicist Baron William Thom- human user can operate to type in data in son Kelvin (1824–1907). the form of alphanumeric characters. It has a standard QWERTY key layout with some Kendall’s method A method of measur- additional characters and function keys. ing the degree of association between two See alphanumeric; input device. different ways of ranking n objects, using 3 two variables (x and y), which give data kilo- Symbol: k A prefix denoting 10 . For example, 1 kilometer (km) = 103 meters (x1,y1),…,(xn,yn). The objects are ranked using first the xs and then the ys. For each (m). of the 2n(n – 1)/2 pairs of objects a score is kilogram /kil-ŏ-gram/ (kilogramme) Sym- assigned. If the RANK of the jth object is bol: kg The SI base unit of mass, equal to greater (or less) than that of the kth, re- the mass of the international prototype of gardless of whether the xs or ys are used, the kilogram, which is a piece of plat- the score is plus one. If the rank of the jth inum–iridium kept at Sèvres in France. is less than that of the kth using one variable but greater using the other, the score is kilogramme /kil-ŏ-gram/ An alternative minus one. Kendall’s coefficient of rank spelling of kilogram. correlation τ = (sum of scores)/½n(n – 1). τ The closer is to one, the greater the degree kilometer /kil-ŏ-mee-ter, kă-lom-ĕ-ter/ of association between the rankings. The (km) A unit of distance equal to 1000 me- method is named for the British statistician tres, approximating to 0.62 mile. Maurice Kendall (1907–83). See also Spearman’s method. kilowatt-hour /kil-ŏ-wot/ Symbol: kwh A unit of energy, usually electrical, equal to Kepler’s laws /kep-lerz/ Laws of plane- the energy transferred by one kilowatt of tary motion deduced in about 1610 by the power in one hour. It is the same as the German astronomer Johannes Kepler Board of Trade unit and has a value of 3.6 (1571–1630) using astronomical observa- × 106 joules. tions made by the Danish astronomer Tycho Brahe (1546–1601): kinematics /kin-ĕ-mat-iks/ The study of (1) Each planet moves in an elliptical orbit the motion of objects without considera- with the Sun at one focus of the ellipse. tion of its cause. See also mechanics. (2) The line between the planet and the Sun sweeps out equal areas in equal times. kinetic energy /ki-net-ik/ Symbol: T The

119 kinetic friction work that an object can do because of its knot 1. A curve formed by looping and in- motion. For an object of mass m moving terlacing a string and then joining the ends. with velocity v, the kinetic energy is mv2/2. The mathematical theory of knots is a This gives the work the object would do in branch of TOPOLOGY. coming to rest. The rotational kinetic en- 2. A unit of velocity equal to one nautical ergy of an object of moment of inertia I mile per hour. It is equal to 0.414 m s–1. and angular velocity ω is given by Iω2/2. See also energy. Koch curve /kokh/ See fractal. kinetic friction See friction. Königsberg bridge problem /koh-nigz- berg/ A classical problem in topology. The kite A plane figure with four sides, and river in the Prussian city of Königsberg di- two pairs of adjacent sides equal. Two of vided into two branches and was crossed the angles in a kite are opposite and equal. by seven bridges in a certain arrangement. Its diagonals cross perpendicularly, one of The problem was to show that it is impos- them (the shorter one) being bisected by sible to walk in a continuous path across the other. The area of a kite is equal to half all the bridges and cross each one only the product of its diagonal lengths. In the once. The problem was solved by Euler in special case in which the two diagonals the eighteenth century, by replacing the have equal lengths, the kite is a rhombus. arrangement by an equivalent one of lines Klein bottle /klÿn/ A curved surface with and vertices. He showed that a network the unique topological property that it has like this (called a graph) can be traversed in only one surface, no edges, and no inside a single path if and only if there are fewer and outside. It can be thought of as formed than three vertices at which an odd number by taking a length of flexible stretchy tub- of line segments meet. In this case there are ing, cutting a hole in the side through four. which one end can fit exactly, passing an end of the tube through this hole, and then Kronecker delta /kroh-nek-er/ Symbol δ δ joining it to the other end from the inside. ij. A quantity defined by ij = 1, if i = j, and Starting at any point on the surface a con- δij = 0 if i ≠ j. It is named for the German tinuous line can be drawn along it to any mathematician Leopold Kronecker (1823– other point without crossing an edge. The 91). bottle is named for the German mathematician (Christian) Felix Klein (1849– Kuratowski–Zorn lemma See Zorn 1925). See also topology. lemma.

X X a b c c ab

island d l l Z d Z e gf g e Y f Y map graph

Königsberg bridge problem

120 L

lambda calculus /lam-dă/ A branch of Langlands program /lang-lănds/ An at- mathematical logic that is used to express tempt to unify different, apparently uncon- the idea of COMPUTABILITY. Any mathemat- nected, branches of mathematics that was ical process that can be performed by using initiated by the American mathematician an ALGORITHM can be achieved using the Robert Langlands in the 1960s. This pro- lambda calculus. The ideas of the lambda gram started off as a series of conjectures calculus are very closely related to the relating different branches of mathematics. foundations of the theory of how comput- A major boost to the Langlands program ers operate, particularly the concept of a was supplied in 1995 when FERMAT’S LAST TURING MACHINE. THEOREM was proved by showing that a conjecture relating two different branches lamina /lam-ă-nă/ (pl. laminae) An ideal- of mathematics is true. If the other conjec- ization of a thin object that is taken to have tures in the Langlands program are true it uniform density and has the dimensions of would mean that problems in one branch area but zero thickness. A sheet of metal or of mathematics could be related to prob- paper or a card can be represented by a lems in another branch of mathematics, lamina. with the hope that problems that are very difficult to solve in one branch of mathe- Lami’s theorem /lam-ee, lah-mee/ A rela- matics might be easier to solve in another tion between three forces A, B, and C that branch. In addition, the program would, if are in equilibrium and the angles α, β, and realized, give a great deal of unity to math- γ between the forces shown in the figure. ematics. The theorem states that A/sinα = B/sinβ = C/sinγ. Lami’s theorem is derived by apply- Laplace’s equation /lah-plahs/ See par- ing the SINE RULE to the triangle of forces tial differential equation. The equation is associated with A, B, and C and using the named for the French mathematician result that sin(180°– θ) – sin θ. It is used to Pierre Simon, Marquis de Laplace (1749– solve problems involving three forces. 1827).

Laplace transform An integral trans-

form f(x) of a function∞ F(t) defined by: f(x) = ∫ 0 exp(–xt)F(t)dt A It is possible for this integral not to exist but it is possible to specify under what conditions it does exist. β The Laplace transform of a function γ F(t) is sometimes denoted by L{F(t)}. For α example, if F(t) = 1, L{F(t)} = 1/x. If F(t) = sin at, L{F(t)} = a/(x2 + a2). If F(t) = B C exp(–at), L{F(t)} = 1/(x + a). Tables of Laplace transforms are available and they can be used in the solution of ordinary and Lami’s theorem partial differential equations.

121 Laplacian

all points on this circle have latitude N

P • •O equator

The latitude θ of a point P on the Earth’s surface.

Laplacian /lă-plass-ee-ăn/ (Laplace opera- latin square An n × n square array of n tor) Symbol ∇2. The operator defined in different symbols with the property that three dimensions by each symbol appears once and only once in ∇2 = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2. each row and each column. Such an The Laplacian occurs in many PARTIAL DIF- arrangement is possible for every n. For ex- FERENTIAL EQUATIONS, including equations ample, for n = 4 and the letters A, B, C, D: of physical interest such as the LAPLACE A B C D EQUATION and the POISSON EQUATION. C D A B D C B A laptop /lap-top/ A type of small light- B A D C weight portable computer with a flip-up Such arrays are used in statistics to analyze screen and powered by rechargeable bat- experiments with three factors influencing teries. the outcome; for example the experimenter, the method, and the material laser printer A type of computer printer under test. The significance of each factor in which the image is formed by scanning a in the experiment may be tested by using a charged plate with a laser. Powdered ink latin square. For example, denoting the adheres to the charged areas and is trans- four methods by A, B, C, D we may use the ferred to paper as in a photocopier. Laser latin square printers can produce text almost as good as material typesetting machines. A B C D experimenter C D A B lateral /lat-ĕ-răl/ Denoting the side of a D C B A solid geometrical figure, as opposed to the B A D C base. For instance, a lateral edge of a pyra- The latin-square array is used since, in mid is one of the edges from the vertex studying the effects of one factor, the influ- (apex). A lateral face of a pyramid or prism ences of the other factors occur to the same is a face that is not a base. The lateral sur- extent. face (or area) of a cylinder or cone is the curved surface (or area), excluding the latitude The distance of a point on the plane base. Earth’s surface from the equator, measured

122 laws of thought as the angle in degrees between a plane large number and N(E)/n converges to p as through the equator (the equatorial plane) n tends to infinity. and the line from the point to the center of the Earth. A point on the equator has a lat- law of moments See moment. itude of 0° and the North Pole has a latitude of 90° N. See also longitude. law of the mean (mean value theorem) The rule in differential calculus that, if f(x) lattice A partially ordered set such that is continuous in the interval a ≤ x ≤ b, and each pair of elements a and b has both: the derivative f′(x) exists everywhere in this 1. A greatest lower bound c; i.e. an element interval, then there is at least one value of c such that c ≤ a and c ≤ b and if c′≤a and x (x0) between a and b for which: c′≤b then c′≤c. [f(b) – f(a)]/(b – a) = f′(x0) 2. A least upper bound d; i.e. an element d Geometrically this means that if a straight such that d ≥ a and d ≥ b and if d′≥a and line is drawn between two points, (a,f(a)) d′ ≥ b then d′≥d. and (b,f(b)), on a continuous curve, then The elements c and d are called the meet there is at least one point between these and join respectively of a and b and are de- where the tangent to the curve is parallel to noted by c = a∩b and d = a∪b. An example the line. This law is derived from ROLLE’S of a lattice is the set of all subsets of a given THEOREM. set, where A ≤ B means that each element of A is also an element of B. In this exam- laws of conservation See conservation ple, A∩B is the intersection of the sets A law. and B and A∪B is their union. laws of friction See friction. latus rectum /lay-tŭs rek-tŭm/ (pl. latera recta) See ellipse; hyperbola; parabola. laws of thought Three laws of logic that are traditionally considered – as are other law of flotation See flotation; law of. logic rules – to exemplify something fundamental about the way we think; that is, it is law of large numbers A theorem in not arbitrary that we say certain forms of probability stating that if an event E has reasoning are correct. On the contrary, it probability p, and if N(E) represents the would be impossible to think otherwise. number of occurrences of the event in n tri- 1. The law of contradiction (law of non- als, then N(E)/n is very close to p if n is a contradiction). It is not the case that some-

tangent at C parallel to AB B C • • y = f(x )

•A

0 x axo b

The law of the mean for a function f(x) that is continuous between x = a and x = b.

123 LCD thing can both be true and not be true: Leibniz theorem /lÿb-nits/ A formula for symbolically finding the nth derivative of a product of ∼(p ∧∼p) two functions. The nth derivative with re- 2. The law of excluded middle. Something spect to x of a function f(x) = u(x)v(x), must either be true or not be true: symbol- written Dn(uv) = dn(uv)/dxn, is equal to the ically series: ∨∼ n n–1 2 n–2 p p uD v + nC1DuD v + nC2D uD v + n–1 n 3. The law of identity. If something is true, … + nCn–1D uDv + vD u then it is true: symbolically where nCr = n!/[(n – r)!r!]. p → p The formula holds for all positive integer values of n. LCD Lowest common denominator. See For n = 1, D(uv) = uDv + vDu common denominator. For n = 2, D2(uv) = uD2v + 2DuDv + vD2u For n = 3, D3(uv) = uD3v + 3DuD2v + LCM Lowest common multiple. See com- 3D2uDv + D3u mon multiple. Note the similarity between the differential coefficients and the binomial expansion leading diagonal See square matrix. coefficients. The theorem is named for the German mathematician, philosopher, his- least common denominator See com- torian, and physicist Gottfried Wilhelm mon denominator. Leibniz (1646–1716). least common multiple See common lemma /lem-ă/ (pl. lemmas or lemmata) A multiple. THEOREM proved for use in the proof of another theorem. least squares method A method of fit- ting a REGRESSION LINE to a set of data. If length The distance along a line, plane fig- the data are points (x ,y ),…(x y ), the 1 1 n n ure, or solid. In a rectangle, the greater of corresponding points (x ,y ′),…(x ,y ′) are 1 1 n n the two dimensions is usually called the found using the linear equation y = ax + b. The least squares line minimizes length and the smaller the breadth. 2 2 [(y1 – y1′) + (y2 – y2′) + … + (yn – 2 lever A class of MACHINE; a rigid object yn′) ] It is found by solving the normal equations able to turn around some point (pivot or ∑y = an + b∑x fulcrum). The force ratio and the distance ∑xy = a∑x + b∑x2 ratio depend on the relative positions of the for a and b. The technique is extended for fulcrum, the point where the user exerts the regression quadratics, cubics, etc. effort, and the point where the lever applies force to the load. There are three types (or- least upper bound See bound. ders) of lever. First order, in which the fulcrum is be- left-handed system See right-handed sys- tween the load and the effort. An example tem. is a crowbar. Second order, in which the load is be- Legendre polynomial /lă-zhahn-drĕ/ A tween the effort and the fulcrum. An ex- series of functions that arises as solutions ample is a wheelbarrow. to Laplace’s equation in spherical polar co- Third order, in which the effort is be- ordinates. They form an infinite series. The tween the load and the fulcrum. An exam- polynomial is named for the French math- ple is a pair of sugar tongs. ematician Adrien-Marie Legendre (1752– Levers can have high efficiency; the main 1833). See also partial differential equa- energy losses are by friction at the pivot, tion. and by bending of the lever itself.

124 linear programming library programs Collections of com- line A join between two points in space or puter programs that have been purchased, on a surface. A line has length but no contributed by users, or supplied by the breadth, that is, it has only one dimension. computer manufacturers for the use of the A straight line is the shortest distance be- computing community. tween two points on a flat surface. light pen A computer input device that linear Relating to a straight line. Two enables a user to write or draw on the variables x and y have a linear relationship screen of a VISUAL DISPLAY UNIT (VDU). if it can be represented graphically by a straight line, i.e. by an equation of the form light-year Symbol: ly A unit of distance y = mx + c (where m is the slope of the line used in astronomy, defined as the distance when plotted and c is a constant). The that light travels through space in one year. equation is known as a LINEAR EQUATION. It is approximately equal to 9.460 5 × 1015 meters. linear dependence See dependent. limit In general, the value approached by linear equation An EQUATION in which a function as the independent variable ap- the highest power of an unknown variable proaches some value. The idea of a limit is is one. The general form of a linear equa- the basis of the branch of mathematics tion is known as analysis. There are several exam- mx + c = 0 ples of the use of limits. where m and c are constants. On a Carte- 1. The limit of a function is the value it ap- sian coordinate graph y = mx + c proaches as the independent variable tends is a straight line that has a gradient m and to some value or to infinity. For instance, crosses the y-axis at y = c. The equation the function x/(x + 3) for positive values of x + 4y2 = 4 x is less than 1. As x increases it ap- is linear in x but not in y. See also equation. proaches 1 – the value approached as x becomes infinite. This is written linear extrapolation See extrapolation. Lim x/(x + 3) = 1 x →∞ linear independence See dependent. stated as ‘the limit of x/(x + 3) as x approaches (or tends to) infinity is 1’. 1 is the linear interpolation See interpolation. limiting value of the function. 2. The limit of a CONVERGENT SEQUENCE is linearly ordered A linearly ordered set is the limit of the nth term as n approaches a partially ordered set that satisfies the tri- infinity. chotomy principle: for any two elements x 3. The limit of a CONVERGENT SERIES is the and y exactly one of x > y, x < y, is true. For limit of the sum of n terms as n approaches example, the set of positive integers with infinity. their natural order is a linearly ordered set. 4. A DERIVATIVE of a function f(x) is the limit of [f(x + δx) – f(x)]/δx as δx ap- linear momentum See momentum. proaches zero. 5. A definite INTEGRAL is the limit of a finite linear momentum, conservation of See sum of terms yδx as δx approaches zero. constant linear momentum; law of. limit cycle A closed curve in phase space linear programming The process of to which a system evolves. A limit cycle is finding maximum or minimum values of a a type of ATTRACTOR characteristic of oscil- linear function under limiting conditions lating systems. or constraints. For example, the function x – 3y might be minimized subject to the con- limiting friction See friction. straints that x + y ≤ 10, x ≤ y – 2, 8 ≥ x ≥ 0

125 linear transformation

8 x + y = 10

7 x = y − 2

5 x = 8

− 1 x 3y = k

O x −1 1234567 89101112

−2

−3

Linear programing: possible values lie in the shaded area. The minimum value is at (8,0). and y ≥ 0. The constraints can be shown as and constraints are maximized or mini- the area on a Cartesian coordinate graph mized by computer techniques that are bounded by the lines x + y = 10; x = y – 2, similar in principle to this graphical tech- x = 8, and y = 0. The minimum value for x nique for two variables. – 3y is chosen from points within this area. A series of parallel lines x – 3y = k are linear transformation 1. In one dimen- drawn for different values of k. The line k sion, the general linear transformation is = –9 just reaches the constraint area at the given by: point (10,0). Lower values are outside it, x′ = (ax + b)/(dx + c) and so x = 10, y = 0 gives the minimum where a, b, c, and d are constants. In two value of x – 3y within the constraints. Lin- dimensions, the general linear transforma- ear programming is used to find the best tion is given by: possible combination of two or more vari- x′ = (a1x + b1y + c1)/(d1x + e1y + f2) able quantities that determine the value of and another quantity. In most applications, for y′ = (a2x + b2y + c2)/(d2x + e2y + f2) example, finding the best combination of General linear transformations in more quantities of each product from a factory than two dimensions are defined similarly. to give the maximum profit, there are 2. In an n-dimensional vector space, a lin- many variables and constraints. Linear ear transformation has the form y = Ax, functions with large numbers of variables where x and y are column vectors and A is

126 local maximum

P2(x2, y2, z2) force at any • point P in space C F

P2 x2 y2 z2 r+dr P F3 • ͐ F,dr = ͐ F dx + ͐ F dy + ͐ F dz •P1(x1, y1, z1) x y z Fx P x y z 0 r Fy 1 1 1 1 y

The line integral of a force vector F along a path C from a point P1 to a point P2. a matrix. A linear transformation takes ax and are in phase the motion is a straight + by into ax′ + by′ for all a and b if it takes line. If they are out of phase by 90°, it is a x and y into x′ and y′. circle. Other phase differences produce ellipses. If the frequencies of the components line integral (contour integral; curvilinear differ, more complex patterns are formed. integral) The integration of a function Lissajous figures can be demonstrated with along a particular path, C, which may be a an oscilloscope by deflecting the spot with segment of straight line, a portion of space one oscillating signal along one axis and curve, or connected segments of several with another signal along the other axis. curves. The function is integrated with re- The patterns are named for the French spect to the position vector r = ix + jy + kz, physicist Jules Antoine Lissajous (1822– which denotes the position of each point 80). P(x,y,z) on a curve C. For example, the direction and magni- liter /lee-ter/ Symbol: l A unit of volume tude of a force vector F acting on a particle defined as 10–3 meter3. The name is not may depend on the particle’s position in a recommended for precise measurements. gravitational field or a magnetic field. The Formerly, the liter was defined as the vol- work done by the force in moving the par- ume of one kilogram of pure water at 4°C ticle over a distance dr is F.dr. The total and standard pressure. On this definition, work done in moving the particle along a 1 l = 1000.028 cm3. particular path from point P1 to point P2 is the line integral shown in the diagram. literal equation An equation in which the line printer An output device of a com- numbers are, like the unknowns, repre- puter system that prints characters (letters, sented by letters. For example, ax + b = c is numbers, punctuation marks, etc.) on a literal equation in x. To solve it, it is nec- paper a complete line at a time and can essary to find x in terms of a, b, and c, i.e. therefore operate very rapidly. x = (c – b)/a. Finding the solution of a literal equation can be regarded as changing Lissajous figures /lee-sa-zhoo/ Patterns the subject of a formula. obtained by combining two simple harmonic motions in different directions. For load The force generated by a machine. example, an object moving in a plane so See machine. that two components of the motion at right angles are simple harmonic motions, traces local maximum (relative maximum) A out a Lissajous figure. If the components value of a function f(x) that is greater than have the same frequency and amplitude for the adjacent values of x, but is not the

127 local meridian greatest of all values of x. See maximum the form n + log10x, where x is between 1 point. and 10 and n is an integer. For example, log1015 = log10(10×1.5) = log1010 + local meridian See longitude. log101.5 = 1 + 0.1761 log10150 = log10(100×1.5) = 2.1761 local minimum (relative minimum) A × log100.15 = log10 (0.1 1.5) = –1_ + 0.1761. value of a function f(x) that is less than for This is written in the notation 1.1761. the adjacent values of x but is not the low- The integer part of the logarithm is est of all values of x. See minimum point. called the characteristic and the decimal fraction is the mantissa. Natural loga- location See store. rithms (Napierian logarithms) use the base e = 2.718 28…, log x is often written as locus of points A set of points, often de- e lnx. fined by an equation relating coordinates. For example, in rectangular Cartesian co- logarithmic function /lôg-ă-rith-mik/ ordinates, the locus of points on a line through the origin at 45° to the x-axis and The function logax, where a is a constant. the y-axis is defined by the equation x = y; It is defined for positive values of x. the line is said to be the locus of the equation. A circle is the locus of all points that logarithmic scale 1. A line in which the lie a fixed distance from a given point. distance, x, from a reference point is proportional to the logarithm of a number. logarithm /lôg-ă-rith-m/ A number ex- For example, one unit of length along the pressed as the exponent of another num- line might represent 10, two units 100, ber. Any number x can be written in the three units 1000, and so on. In this case the form x = ay. y is then the logarithm to the distance x along the logarithmic scale is base a of x. For example, the logarithm to given by the equation x = log10a. Logarith- the base ten of 100 (log10100) is two, since mic scales form the basis of the slide rule 100 = 102. Logarithms that have a base of since two numbers may be multiplied by ten are known as common logarithms (or adding lengths on a logarithmic scale Briggsian logarithms), named for the Eng- log(a × b) = log a + log b. lish mathematician Henry Briggs (1561– The graph of the curve y = xn, when 1630). They are used to carry out multipli- plotted on graph paper with logarithmic cation and division calculations because coordinate scales on both axes (log-log numbers can be multiplied by adding their graph paper), is a straight line since logy = × logarithms. In general p q can be written nlogx. This method can be used to estab- c × d (c+d) c d as a a = a , p = a and q = a . Both lish the equation of a non-linear curve. logarithms and antilogarithms (the inverse Known values of x and y are plotted on function) are available in the form of log-log graph paper and the gradient n of printed tables, one used for calculations. the resulting line is measured, enabling the For example, 4.91 × 5.12 would be calcu- equation to be found. See also log-linear lated as follows: log 4.91 is 0.6911 (from 10 graph. tables) and log105.12 is 0.7093 (from tables). Therefore, 4.91 × 5.12 is given by an- 2. Any scale of measurement that varies tilog (0.6911 + 0.7093), being 25.14 (from logarithmically with the quantity mea- antilog tables). Similarly, division can be sured. For instance, pH in chemistry is a carried out by subtraction of logarithms measure of acidity or alkalinity – i.e. of hy- and the nth root of a number (x) is the an- drogen-ion concentration. It is defined as + tilogarithm of (logx)/n. log10(1/[H ]). An increase in pH from 5 to For numbers between 0 and 1 the com- 6 represents a decrease in [H+] from 10–5 to mon logarithm is negative. For example, 10–6, i.e. a factor of 10. An example of a log100.01 = –2. The common logarithm of logarithmic scale in physics is the decibel any positive real number can be written in scale used for noise level.

128 logic gate

| | | | –1 0 1 2

| | | | | | | | | | 0.1 0.2 0.5 1 2 5 10 20 50 100

Logarithmic scale logarithmic series The infinite power se- The incorrectness of the argument shows ries that is the expansion of the function up clearly when, after making reasonable loge(1+x), namely: substitutions for A, B, and C, we get true x – x2/2 + x3/3 – x4/4 + … premisses but a false conclusion: This series is convergent for all values of x all dogs are mammals greater than –1 and less than or equal to 1. all cats are mammals therefore all dogs are cats. log graph See logarithmic scale. Such an argument is called a fallacy. Logic puts forward and examines rules logic The study of the methods and prin- that will insure that – given true premisses ciples used in distinguishing correct or – a true conclusion can automatically be valid arguments and reasoning from incor- reached. It is not concerned with examin- rect or invalid arguments and reasoning. ing or assessing the truth of the premisses; The main concern in logic is not whether a it is concerned with the form and structure conclusion is in fact accurate, but whether of arguments, not their content. See deduc- the process by which it is derived from a set tion; induction; symbolic logic; truth- of initial assumptions (premisses) is cor- value; validity. rect. Thus, for example, the following form of argument is valid: logic circuit An electronic switching cir- all A is B cuit that performs a logical operation, such all B is C as ‘and’ and ‘implies’ on its input signals. therefore all A is C, There are two possible levels for the input and thus the conclusion and output signals, high and low, some- all fish have wings times indicated by the binary digits 1 and can be derived, correctly, from the pre- 0, which can be combined like the values misses ‘true’ and ‘false’ in a truth table. For exam- all fish are mammals ple, a circuit with two inputs and one out- and put might have a high output only when all mammals have wings the inputs are different. The output there- even though the premisses and the conclu- fore is the logical function ‘either…or…’ of sion are untrue. Similarly, true premisses the two inputs (the exclusive disjunction). and true conclusions are no guarantee of See truth table. a valid argument. Therefore, the true conclusion logic gate An electronic circuit that car- all cats are mammals ries out logical operations. Examples of does not follow, logically, from the true such operations are ‘and’, ‘either – or’, premisses: all cats are warm-blooded and input 1 input 2 output all mammals are warm-blooded because it is an example of the invalid ar- high high low gument form high low high all A is B low high high all C is B low low low therefore, all A is C. Table for a logic circuit

129 log-linear graph

100 –

10 –

1 –

– – – x 0.1 1 2 3

In this log-linear graph, the function y = 4.9 e1 5x is shown as a straight line with a gradient of 1.5.

‘not’, ‘neither – nor’, etc. Logic gates oper- ues of x are plotted on the linear scale and ate on high or low input and output volt- values of y on the LOGARITHMIC SCALE. ages. Binary logic circuits, those that switch between two voltage levels (high log–log graph A graph on which both and low), are widely used in digital com- axes have LOGARITHMIC SCALES. puters. The inverter gate or NOT gate simply changes a high input to a low output long division A method used to divide and vise versa. In its simplest form, the one number by another number or one al- AND gate has two inputs and one output. gebraic expression by another algebraic ex- The output is high if and only if both in- pression. In both cases it is customary to puts are high. The NAND gate (not and) is work from the left. In the case of numbers, similar, but has the opposite effect; that is, the first figure of the number to be divided a low output if and only if both inputs are is divided by the dividing number. This high. The OR gate has a high output if one gives an integer (which may be zero) and a or more of the inputs are high. The exclu- remainder. The remainder is added to the sive OR gate has a high input only if one of next figure of the number to be divided and the inputs, but not more than one, is high. then divided by the dividing number. This The NOR gate has a high output only if all process is repeated until all the figures in the inputs are low. Logic gates are con- the number to be divided have been divided structed using transistors, but in a circuit by the dividing number, with the answer diagram they are often shown by symbols being an integer and a remainder. The cal- that denote only their logical functions. culation of numbers using long division These functions are, in effect, those rela- has largely been replaced by the use of elec- tionships that can hold between proposi- tronic calculators. Nevertheless, the same tions in symbolic logic, with combinations sort of procedure is used to divide one that can be represented in a TRUTH TABLE. algebraic expression by another algebraic See also conjunction; disjunction; nega- expression. tion. longitude The east-west position of a log-linear graph (semilogarithmic graph) point on the Earth’s surface measured as A graph on which one axis has a logarith- the angle in degrees from a standard merid- mic scale and the other has a linear scale. ian (taken as the Greenwich meridian). A On a log-linear graph, an exponential meridian is a great circle that passes function (one of the form y = keax where k through the North and South poles. The and a are constants) is a straight line. Val- local meridian of a point is a great circle

130 Lorentz force

all points on this circle have longitude φ N

zero P φ longitude

P’ φ equator

The longitude φ of a point P on the Earth’s surface.

passing through that point and the two Lorentz–Fitzgerald contraction /lo- poles. See also latitude. rents fits-je-răld/ A reduction in the length of a body moving with a speed v relative to longitudinal wave A wave motion in an observer, as compared with the length which the vibrations in the medium are in of an identical object at rest relative to the the same direction as the direction of en- observer. The object is supposed to con- ergy transfer. Sound waves, transmitted by tract by a factor √(1 – v2/c2), c being the alternate compression and rarefaction of speed of light in free space. The contraction the medium, are an example of longitudi- was postulated to account for the negative nal waves. Compare transverse wave. result of the Michelson–Morley experiment using the ideas of classical physics. long multiplication A technique for the The idea behind it was that the electro- multiplication of two numbers or algebraic magnetic forces holding atoms together expressions. It is customary to write the were modified by motion through the two numbers to be multiplied above each ether. The idea was made superfluous other. The top number is multiplied by the (along with the concept of the ether) by the figures of the bottom number separately, theory of relativity, which supplied an al- with a separate line for each product in the ternative explanation of the Michelson– appropriate place underneath the initial Morley experiment. The contraction was numbers, with a final line at the bottom named for the Dutch theoretical physicist giving the final product. The calculation of Hendrik Antoon Lorentz (1853–1928) and numbers by long multiplication has largely the Irish physicist George Francis Fitzger- been replaced by the use of electronic cal- ald (1851–1901), who arrived at the solu- culators. Long multiplication of algebraic tion independently. expressions can be performed by a similar procedure. Lorentz force /lo-rents/ The force F exerted on an electric charge q that is moving loop A sequence of instructions in a com- in a magnetic field of field strength B with puter program that is performed either a a velocity v. This force is given by: F = q v specified number of times or is performed × b. The force is perpendicular to both the repeatedly until some condition is satisfied. direction of the magnetic field and the ve- See also branch. locity. This means that the Lorentz force

131 Lorentz transformation deflects the electric charge but does not low-level language See program. change its speed. There are many physical applications of the Lorentz force. lumen /loo-mĕn/ Symbol: lm (pl. lumens or lumina) The SI unit of luminous flux, Lorentz transformation A set of equa- equal to the luminous flux emitted by a tions that correlate space and time coordi- point source of one candela in a solid angle of one steradian. 1 lm = 1 cd sr. nates in two frames of reference. See also relativity, theory of. lune /loon/ A portion of the area of a sphere bounded by two great semicircles lower bound See bound. that have common end points. lowest common denominator See com- lux /luks/ Symbol: lx (pl. lux) The SI unit mon denominator. of illumination, equal to the illumination produced by a luminous flux of one lumen lowest common multiple See common falling on a surface of one square meter. multiple. 1 lx = 1 lm m–2.

132 M

machine A device for transmitting force relationship is measured by the force ratio or energy between one place and another. (or mechanical advantage) – the force ap- The user applies a force (the effort) to the plied by the machine (load, F2) divided by machine; the machine applies a force (the the force applied by the user (effort, F1). load) to something. These two forces need The work done by the machine cannot not be the same; in fact the purpose of the exceed the work done to the machine. machine is often to overcome a large load Therefore, for a 100% efficient machine: with a small effort. For any machine this if F2 > F1 then s2 < s1 and if F2 s1. Here s2 and s1 are the distances moved by x y F2 and F1 in a given time. (a) The relationship between s1 and s2 in a L E given case is measured by the distance ratio E (or velocity ratio) – the distance moved by the effort (s1) divided by the distance (b) moved by the load (s2). α L Neither distance ratio nor force ratio has a unit; neither has a standard symbol. See also hydraulic press; inclined plane; lever; pulley; screw; wheel and axle.

machine code See program.

Maclaurin series /mă-klor-in/ See Taylor E series.

L magic square A square array of numbers

A1 whose columns, rows and diagonals add to A 2 the same total. For the magic square: L (d) E 6 7 2 1 5 9 8 3 4 the total is 15. radius R radius r magnetic disk See disk. (e) magnetic tape A long strip of flexible plastic with a magnetic coating on which EL information can be stored. Its use in the Machine: (a) lever; (b) inclined plane; (c) pul- recording and reproduction of sound is ley; (d) hydraulic press; (e) wheel and axle. well-known. It is also widely used in com-

133 magnificent seven problems puting to store information. The data is z2 + c, where z and c are complex numbers. stored on the tape in the form of small This process can produce an infinite num- closely packed magnetic spots arranged in ber of patterns depending on the value cho- rows across the tape. The spots are magne- sen for the constant complex number c (see tized in one of two directions so that the Julia set). These patterns are of two types: data is in binary form. The magnetization they are either connected, so that there is a pattern of a row of spots represents a letter, single area within a boundary, or they are digit (0–9), or some other character.from disconnected and broken into distinct the central processor. It is widely used as a parts. The Mandelbrot set is the set of all backing store. Julia sets that are connected. It can be represented by points on the complex plane magnificent seven problems A set of and has a characteristic shape. seven mathematical problems put forward It is not necessary to generate Julia sets by a committee of mathematicians in 2000 for the Mandelbrot set to be produced. It as the most outstanding problems at that can be shown that a given value of c is in time. The POINCARÉ CONJECTURE was one of the Mandelbrot set if the starting value z = the seven problems. It is hoped that at- 0 is bounded for the iteration. Thus, if the tempts to solve these problems will stimu- sequence late progress in mathematics in the c, c2 + c, (c2 + c)2 + c, … twenty-first century just as attempts to remains bounded, then the point c is in the solve HILBERT’S PROBLEMS stimulated pro- Mandelbrot set. gress in mathematics in the twentieth cen- The set has been the subject of much in- tury. A reward of one million dollars is terest. The figure is a FRACTAL and can be being offered for the solution of each of examined on the computer screen in high these problems. In order to claim this re- ‘magnification’ (i.e. by calculating the set ward the solution has to be published in a over a short range of values of c). It shows mathematics journal and be scrutinized for an amazingly complex self-similar struc- two years. At the time of writing, no re- ture characterized by the presence of wards have yet been given. smaller and smaller copies of the set at in- magnitude 1. The absolute value of a creasing levels of detail. The set is named number (without regard to sign). for the American mathematician Benoit Mandelbrot (1924– ). 2. The non-directional part of a VECTOR, corresponding to the length of the line representing it. manifold A space that is locally Euclidean but is not necessarily Euclidean globally. mainframe See computer. Examples of manifolds that are not Euclid- ean globally include the circle and the sur- main store See store; central processor. face of a sphere. It is convenient to discuss many aspects of geometry and topology in major arc See arc. terms of manifolds. major axis See ellipse. mantissa /man-tiss-ă/ See logarithm. major sector See sector. many-valued function A function for which one value of x gives more than one major segment The segment of a CIRCLE value of y. If one value of x gives n values divided by a chord that contains the larger of y then the function is said to be an n- part of the circle. valued function of x. If n = 2 then the function is called a two-valued function of x. Mandelbrot set /man-dĕl-brot/ A set of For example, y2 = x is a two-valued func- points in the complex plane generated by tion of x since every real value of x gives considering the iterations of the form z → two values of y.

134 matrix

Mandelbrot set: the shape of the set with a detail showing the fine structure (top left). map See function. energy + kinetic energy + potential energy) of a mass m, c being the speed of light in mapping See function. free space. The equation is a consequence of Einstein’s special theory of relativity. It market price See nominal value; yield. is a quantitative expression of the idea that mass is a form of energy and energy also Markov chain /mar-koff/ A sequence of has mass. Conversion of rest-mass energy discrete random events or variables that into kinetic energy is the source of power in have probabilities depending on previous radioactive substances and the basis of nu- events in the chain. The sequence is named clear-power generation. for the Russian mathematician Andrey An- dreyevich Markov (1856–1922). material implication See implication. mass Symbol: m A measure of the quan- mathematical induction See induction. tity of matter in an object. The SI unit of mass is the kilogram. Mass is determined in mathematical logic See symbolic logic. two ways: the inertial mass of a body determines its tendency to resist change in matrix /may-triks, mat-riks/ (pl. matrices motion; the GRAVITATIONAL MASS deter- or matrixes) A set of quantities arranged in mines its gravitational attraction for other rows and columns to form a rectangular masses. See also inertial mass; weight. array. The common notation is to enclose these in parentheses. Matrices do not have mass, center of See center of mass. a numerical value, like DETERMINANTS. They are used to represent relations be- mass-energy equation The equation E = tween the quantities. For example, a plane mc2, where E is the total energy (rest mass vector can be represented by a single col-

135 matrix of coefficients

2 6 10 1 2 3 +=3 8 13 4 5 6 4 8 12 8 13 18

Matrix addition.

1 2 3 3 6 9 3 x = 4 5 6 12 15 18

Multiplication of a matrix by a number.

1 2 3 6 7 (6 + 16 + 30) (7 + 18 + 33) X= 4 5 6 8 9 10 11 (24 + 40 + 60) (28 + 45 + 66)

Matrix multiplication

Matrix: addition and multiplication of matrices. umn matrix with two numbers, a 2 × 1 ma- by a number or constant k, the result is an- trix, in which the upper number represents other m × n matrix. If the element in the ith its component parallel to the x-axis and the row and jth column is aij then the corre- lower number represents the component sponding element in the product is kaij. parallel to the y-axis. Matrices can also be This operation is distributive over matrix used to represent, and solve, simultaneous addition and subtraction, that is, for two equations. In general, an m × n matrix – matrices A and B, one with m rows and n columns – is writ- k(A + B) = kA + kB ten with the first row: Also, kA = Ak, as for multiplication of a11a12 … a1n numbers. In the multiplication of two ma- The second row is: trices, the matrices A and B can only be a21a22 … a2n multiplied together to form the product AB and so on, the mth row being: if the number of columns in A is the same am1am2 … amn as the number of rows in B. In this case The individual quantities a11, a21, etc., are they are called conformable matrices. If A called elements of the matrix. The number is an m × p matrix with elements aij and B of rows and columns, m × n, is the order or is a p × n matrix with elements bij, then dimensions of the matrix. Two matrices their product AB = C is an m × n matrix are equal only if they are of the same order with elements cij, such that cij is the sum of and if all their corresponding elements are the products equal. Matrices, like numbers, can be aijbij + ai2b2j + ai3b3j + … + aipbpj added, subtracted, multiplied, and treated Matrix multiplication is not commutative, algebraically according to certain rules. that is, AB ≠ BA. However, the commutative, associative, See also square matrix. and distributive laws of ordinary arithmetic do not apply. Matrix addition con- matrix of coefficients An m × n matrix, sists of adding corresponding elements with general entry aij associated with the together to obtain another matrix of the coefficients in the set of m linear equations same order. Only matrices of the same for n unknown variables x1, x2, ... xn: order can be added. Similarly, the result of a11x1 + a12x2 + ... a1nxn = b1, subtracting two matrices is the matrix a21x1 + a22x2 + ... a2nxn = b2, formed by the differences between corre- am1x1 + am2x2 + ... amnxn = bm, sponding elements. Expressing a set of linear equations in Matrix multiplication also has certain terms of the matrix of coefficients can be a rules. In multiplication of an m × n matrix step towards finding the set of solutions of

136 measure theory

A || ||| C’ B’

P ||| ||

B C | A’ |

Median: the three medians intersect at P, the centroid of the triangle.

the equations. The matrix formed by in- When data is classified, as for example in a cluding the constants b1, b2, etc., is the frequency table, x1 is replaced by the class augmented matrix. mark. The weighted mean W = maximum likelihood A method of esti- (w1x1 + w2x2 + … + wnxn)/ mating the most likely value of a parame- (w1 + w2 + … + wn) ter. If a series of observations x1,x2,…,xn where weight wi is associated with xi. are made, the likelihood function, L(x), is The harmonic mean H = the joint probability of observing these val- n/[(1/x1) + (1/x2) + … + (1/xn)] ues. The likelihood function is maximized The geometric mean G = 1/n when [dlogL(x)]/dp = 0. In many cases, an (x1.x2 … xn) intuitive estimate, such as the mean, is also The mean of a random variable is its ex- the maximum likelihood estimate. pected value. maximum point A point on the graph of mean center See centroid. a function at which it has the highest value within an interval. If the function is a mean deviation A measure of the disper- smooth continuous curve, the maximum is sion of a set of numbers. It is equal to the a TURNING POINT, that is, the slope of the average, for a set of numbers, of the differ- tangent to the curve changes continuously ences between each number and the set’s from positive to negative by passing mean value. If x is a random variable with through zero. If there is a higher value of mean value µ, then the mean deviation is the function outside the immediate neigh- the average, or expected value, of |x – µ|, borhood of the maximum, it is a local max- written imum (or relative maximum). If it is higher ∑|xi – µ|/n than all other values of the function it is an i absolute maximum. See also stationary mean value theorem See law of the point. mean. mean A representative or expected value measurements of central tendency The for a set of numbers. The arithmetic mean general name given for the types of average or average (called mean) of x1, x2, …, xn is used in statistics, i.e. the MEAN, the MEDIAN, given by: and the MODE. (x1 + x2 + x3 + … + xn)/n If x1, x2, …, xk occur with frequencies f1, measure theory A branch of mathemat- f2, …, fk then the arithmetic mean is ics concerned with constructing the theory (f1x1 + f2x2 + … + fkxk)/ of integration in a mathematically rigorous (f1 + f2 + … + fk) way, starting from set theory. 137 mechanical advantage mechanical advantage See force ratio; become straight horizontal lines. Areas fur- machine. ther from the equator are more stretched out in the horizontal direction. For a par- mechanics The study of forces and their ticular point on the surface of the sphere at effect on objects. If the forces on an object angle θ latitude and angle φ longitude, the or in a system cause no change of momen- corresponding Cartesian coordinates on tum the object or system is in equilibrium. the map are: The study of such cases is statics. If the x = kθ forces acting do change momentum the y = k log tan(φ/2) study is of dynamics. The ideas of dynam- Mercator’s projection is an example of a ics relate the forces to the momentum conformal mapping, in which the angles changes produced. Kinematics is the study between lines are preserved (except at the of motion without consideration of its poles). This method of mapping is named cause. for the Dutch cartographer and geographer median /mee-dee-ăn/ 1. The middle num- Gerardus Mercator (1512–94). ber of a set of numbers arranged in order. See also projection. When there is an even number of numbers the median is the average of the middle meridian /mĕ-rid-ee-ăn/ See longitude. two. For example, the median of 1,3,5,11,11 is 5, and of 1,3,5,11,11,14 is Mersenne prime /mair-sen/ a prime p (5 + 11)/2 = 8. The median of a large pop- number that can be written as 2 –1, where ulation is the 50th percentile (P50). Com- p is a prime number. Mersenne primes are pare mean. See also percentile; quartile. discovered using computers, with the num- 2. In geometry, a straight line joining the ber being discovered increasing as the vertex of a triangle to the mid-point of the power of computers increases. For exam- opposite side. The medians of a triangle in- ple, the Mersenne prime 219937–1 was dis- tersect at a single point, which is the cen- covered in this way. Mersenne primes are troid of the triangle. related to even PERFECT NUMBERS since the only even perfect numbers are numbers of mediator A line that bisects another line the form 2(p–1)(2p–1). The prime is named at right angles. for the French mathematician Marin Mersenne (1588–1648). mega- Symbol: M A prefix denoting 106. 6 For example, 1 megahertz (MHz) = 10 meter /mee-ter/ Symbol: m The SI base hertz (Hz). unit of length, defined as the distance traveled by light in a vacuum in 1/299 792 458 See element. member of a second. memory See store. metric /met-rik/ An expression for the distance between two points in some geo- mensuration /men-shŭ-ray-shŏn/ The metrical space that is not necessarily Eu- study of measurements, especially of the dimensions of geometric figures in order to clidean. The expressions for the metric in calculate their areas and volumes. non-Euclidean geometrics are generaliza- tions of the distance between two points in Mercator’s projection /mer-kay-terz/ A Euclidean geometry which is given by ap- method of mapping points from the sur- plying PYTHAGORAS’ THEOREM to Cartesian face of a sphere onto a plane surface. Mer- coordinates. The concept of the metric is cator’s projection is used to make maps of very important in the geometries that de- the World. The lines of longitude on the scribe flat space–time in special relativity sphere become straight vertical lines on the theory and curved space–time in general plane. The lines of latitude on the sphere relativity theory.

138 micron metric prefix Any of various numerical m.g.f. See moment generating function. prefixes used in the metric system. Michelson–Morley experiment /mÿ- metric space Any set of points, such as a kĕl-sŏn mor-lee/ A famous experiment con- plane or a volume in geometrical space, in ducted in 1887 in an attempt to detect the which a pair of points a and b with a dis- ether, the medium that was supposed to be tance d(a,b) between them, satisfy the con- necessary for the transmission of electro- ditions that d(a,b) ≥ 0 and d(a,b) = 0 if and magnetic waves in free space. In the exper- only if a and b are the same point. Another iment, two light beams were combined to property of a metric space is that d(a,b) + produce interference patterns after travel- ≥ d(b,c) d(a,c). The set of all the functions ing for short equal distances perpendicular of x that are continuous in the interval x = to each other. The apparatus was then a to x = b is also a metric space. If f(x) and turned through 90° and the two interfer- g(x) are in the space, ence patterns were compared to see if there b had been a shift of the fringes. If light has a ͐ [f(x) – g(x)]dx a velocity relative to the ether and there is an ether ‘wind’ as the Earth moves through is defined for all values of x between a and space, then the times of travel of the two b, and the integral is zero if and only if f(x) beams would change, resulting in a fringe = g(x) for all values of x between a and b. shift. No such shift was detected, not even metric system A system of units based on when the experiment was repeated six the meter and the kilogram and using mul- months later when the ether wind would tiples and submultiples of 10. SI units, have reversed direction. The experiment is c.g.s. units, and m.k.s. units are all scien- named for its instigators, the American tific metric systems of units. physicist Albert Abraham Michelson (1852–1931) and the American chemist metric ton See tonne. and physicist Edward William Morley (1838–1923). See also relativity; theory of. metrology /mĕ-trol-ŏ-jee/ The study of units of measurements and the methods of micro- Symbol: µ A prefix denoting 10–6. making precise measurements. Every phys- For example, 1 micrometer (µm) = 10–6 ical quantity that can be quantified is ex- meter (m). pressed by a relationship of the type q = nu, where q is the physical quantity, n is a microcomputer /mÿ-kroh-kŏm-pyoo-ter/ number, and u is a unit of measurement. See computer. One of the prime concerns of metrologists is to select and define units for all physical micron /mÿ-kron/ (micrometer) Symbol: quantities. µm A unit of length equal to 10–6 meter.

METRIC PREFIXES Prefix Symbol Multiple Prefix Symbol Multiple atto- a × 10–18 deca- da × 10 femto f × 10–15 hecto- h × 102 pico- p × 10–12 kilo- k × 103 nano- n × 10–9 mega- M × 106 micro- µ×10–6 giga- G × 109 milli- m × 10–3 tera- T × 1012 centi- c × 10–2 peta- P × 1015 deci- d × 10–1 exa- E × 1018

139 microprocessor

A minimum point A point on the graph of a function at which it has the lowest value within an interval. If the function is a smooth continuous curve, the minimum is a turning point, that is, the slope of the tan- MNgent to the curve changes continuously from negative to positive by passing through zero. If there is a lower value of the function outside the immediate neighborhood of the minimum, it is a local min- B C imum (or relative minimum). If it is lower Midpoint theorem. than all other values of the function it is an absolute minimum. See also stationary point; turning point. microprocessor /mÿ-kroh-pross-ess-er/ See central processor. minor arc See arc. mid-point theorem A result in geometry minor axis See ellipse. that states that the line that joins the mid- points of two sides of a triangle has half the minor sector See sector. length of the third side of the triangle and is parallel to the third side. This result can minor segment The segment of a CIRCLE be proved very easily using vectors in the divided by a chord that contains the following way. smaller part of the circle. Denote AB by b and AC by c. The directed line BC is given by BC = BA + AC = –b + c minuend /min-yoo-end/ The term from = c–b. One also has AM = (1/2) AB = (1/2) which another term is subtracted in a dif- b and AN = (1/2)AC = (1/2) c. Thus, MN = ference. In 5 – 4 = 1, 5 is the minuend (4 is MA + AN = –(1/2) b + (1/2) c = (1/2) c the subtrahend). –(1/2) b = (1/2) (c – b). The last result can be written as MN = (1/2)BC. This means minute (of arc) A unit of plane angle that MN has half the magnitude of BC and equal to one sixtieth of a degree. is parallel to it. mirror line A line that is the axis of sym- mil 1. A unit of length equal to one thou- metry for reflection symmetry. If part of an sandth of an inch. It is commonly called a object is in contact with the mirror line ‘thou’ and is equivalent to 2.54 × 10–5 m. then the image of the object is also in con- 2. A unit of area, usually called a circular tact with the mirror line. If part of an ob- mil, equal to the area of a circle having a ject is a certain distance from the mirror diameter of 1 mil. line then the image of that part of the object is the same distance from the mirror mile A unit of length equal to 1760 yards. line. If the mirror line passes through an It is equivalent to 1.6093 km. object then it also passes through the image of the object. milli- Symbol: m A prefix denoting 10–3. For example, 1 millimeter (mm) = 10–3 mixed number See fraction. meter (m). m.k.s. system A system of units based on million A number equal to 1 000 000 the meter, the kilogram, and the second. It (106). formed the basis for SI units. minicomputer /min-ee-kŏm-pyoo-ter/ See mmHg (millimeter of mercury) A former computer. unit of pressure defined as the pressure that

140 modulus

Möbius strip, which has one surface and one edge. will support a column of mercury one mil- modulation /moj-ŭ-lay-shŏn/ A process limeter high under specified conditions. It in which the characteristics of one wave are is equal to 133.322 4 Pa. It is also called altered by some other wave. Modulation is the torr. used extensively in radio transmission. The wave which is altered is called the carrier Möbius strip /moh-bee-ŭs/ (Möbius band) wave and the wave responsible for the A continuous flat loop with one twist. It is change is called the modulating wave. In- formed by taking a flat rectangular strip, formation is transmitted in this way. twisting it in the middle so that each end There are several ways in which modu- turns through 180° with respect to the lation can be performed. In amplitude other, and then joining the ends together. modulation the amplitude of the carrier Because of the twist, a continuous line can wave rises and falls as the amplitude of the be traced along the surface between any modulating wave rises and falls. two points, without crossing an edge. The If the carrier wave is regarded as a sine unique topological property of the Möbius wave then amplitude modulation alters the strip is that it has one surface and one edge. amplitude of the sine wave but does not If a Möbius strip is cut along a line parallel change the angular aspect of the carrier to the edge it is transformed into a doubly wave, either as regards phase angle or fre- twisted band that has two edges and two quency. Modulation which involves sides. It is named for the German mathe- change of the angle is known as angle mod- matician August Ferdinand Möbius ulation. There are two types of angle mod- (1790–1868). See also topology. ulation: frequency modulation and phase modulation. In frequency modulation the modal class /moh-d’l/ The CLASS that oc- frequency of the carrier wave rises and falls curs with the greatest frequency, for exam- as the amplitude of the modulating wave ple in a frequency table. See also mode. rises and falls. In phase modulation the relative phase of the carrier wave changes by mode The number that occurs most fre- an amount proportional to the amplitude quently in a set of numbers. For example, of the modulating wave. In angle modula- the mode (modal value) of {5, 6, 2, 3, 2, 1, tion the amplitude of the carrier wave is 2} is 2. If a continuous random variable has not altered. probability density function f(x), the mode There exist several other types of mod- is the value of x for which f(x) is a maxi- ulation, including some which make use of mum. If such a variable has a frequency pulses (pulse modulation). curve that is approximately symmetrical and has only one mode, then modulus /moj-ŭ-lŭs/ (pl. moduli) The ab- (mean – mode) = 3(mean – median). solute value of a quantity, not considering its sign or direction. For example, the mod- modem /moh-dem/ A device for sending ulus of minus five, written |–5|, is 5. The computer data over long distances using modulus of a vector quantity corresponds telephone lines. It is short for to the length or magnitude of the vector. modulator/demodulator. The modulus of a COMPLEX NUMBER x + iy

141 mole is √(x2 + y2). If the number is written in the moment of area For a given surface, the form r(cosθ + isinθ), the modulus is r. See moment of area is the moment of mass that also argument. the surface would have if it had unit mass per unit area. mole Symbol: mol The SI base unit of amount of substance, defined as the moment of inertia Symbol: I The rota- amount of substance that contains as many tional analog of mass. The moment of in- elementary entities as there are atoms in ertia of an object rotating about an axis is 0.012 kilogram of carbon-12. The elemen- given by 2 tary entities may be atoms, molecules, ions, I = mr electrons, photons, etc., and they must be where m is the mass of an element a distance r from the axis. See also radius of gy- specified. One mole contains 6.022 52 × ration; theorem of parallel axes. 1023 entities. One mole of an element with relative atomic mass A has a mass of A moment of mass The moment of mass of grams (this was formerly called one gram- a point mass about a point, line, or plane is atom). One mole of a compound with rel- the product of the mass and the distance ative molecular mass M has a mass of M from the point or of the mass and the per- grams (this was formerly called one gram- pendicular distance from the line or plane. molecule). For a system of point masses, the moment of mass is the sum of the mass-distance moment (of a force) The turning effect products for the individual masses. For an produced by a force about a point. If the object the integral must be used over the point lies on the line of action of the force volume of the object. the moment of the force is zero. Otherwise it is the product of the force and the per- momentum, conservation of /mŏ-men- pendicular distance from the point to the tŭm/ See constant linear momentum; law line of action of the force. If a number of of. forces are acting on a body, the resultant moment is the algebraic sum of all the in- momentum, linear Symbol: p The prod- dividual moments. For a body in equilib- uct of an object’s mass and its velocity: p = rium, the sum of the clockwise moments is mv. The object’s momentum cannot equal to the sum of the anticlockwise mo- change unless a net outside force acts. This ments (this law is sometimes called the law relates to Newton’s laws and to the defini- of moments). See also couple; torque. tion of force. It also relates to the principle of constant momentum. See also angular moment generating function (m.g.f.) A momentum. function that is used to calculate the statis- /mon-ŏ-tonn-ik/ Always tical properties of a random variable, x. It monotonic changing in the same direction. A monoto- is defined in terms of a second variable, t, nic increasing function of a variable x in- such that the m.g.f., M(t), is the expecta- creases or stays constant as x increases, but tion value of etx, E(etx). For a discrete ran- never decreases. A monotonic decreasing dom variable function of x decreases or stays constant as ∑ tx M(t) = e p x increases, but never increases. Each term and for a continuous random variable in a monotonic series is either greater than ∫ tx M(t) = e f(x)dx or equal to the one before it (monotonic in- Two distributions are the same if their creasing) or less than or equal to the one m.g.f.s are the same. The mean and vari- before it (monotonic decreasing). Compare ance of a distribution can be found by dif- alternating series. ferentiating the m.g.f. The mean E(x) = M′(0) and the variance, Var(x) = M″(0) – mouse A computer INPUT DEVICE that is (M′(0))2. held under the palm of the hand and

142 myria- moved on a flat surface to control the ple, 2 × (4 × 5) = (2 × 4) × 5. This is the as- movements of a cursor (pointer) on the sociative law for arithmetic multiplication. computer screen. Instructions can be sent Multiplication of vector quantities and to the computer by pressing (‘clicking on’) matrices do not follow the same rules. one or more buttons on the mouse. multiplication of complex numbers multiple A number or expression that has The rules of multiplication which emerge a given number or expression as a factor. from the definition of a complex number. For example, 26 is a multiple of 13. If z1 = a + ib and z2 = c + id, then the product z1z2 is given by: z1z2 = (ac – bd) + i(ad multiple integral See iterated integral. + bc). The product of two complex numbers multiple point A point on the curve of a can be written neatly in terms of their polar function at which several arcs intersect, or forms: θ θ θ which forms an isolated point, and where a z1 = r1 exp (i 1) = r1 (cos 1 + i sin 1) and θ θ θ simple derivative of the function does not z2 = r2 exp (i 2) = r2 (cos 2 + i sin 2). exist. If the equation of the curve is written This gives θ θ θ in the form: z1z2 = r1r2 exp [i( 1 + 2)] = r1r2 [cos ( 1 + 2 2 θ2) + i sin (θ1 + θ2)]. In addition to being (a1x + b1y) + (a2x + b2xy + c2y ) 3 very convenient for many purposes, the ex- + (a3x + …) + … = 0 in which the multiple point is at the origin pression for the product of two complex numbers in terms of polar forms has a sim- of a Cartesian-coordinate system, the val- ple geometrical interpretation that the ues of the coefficients of x and y indicate modulus of the product is the product of the type of multiple point. If a and b are 1 1 the moduli of the two complex numbers zero, that is, if all the first degree terms are being multiplied together and the argu- zero, then the origin is a singular point. If ment of the product is the sum of the argu- the terms a , b , and c are also zero it is a 2 2 2 ments of the two complex numbers. double point. If, in addition, the terms a3, b3, etc., of the third degree terms are zero, multiplication of fractions See frac- it is a triple point, and so on. See also dou- tions. ble point; isolated point. multiplication of matrices See matrix. multiplicand /mul-ti-plă-kand/ A number or term that is multiplied by another (the multiplication of vectors See scalar multiplier) in a multiplication. product; vector product. × multiplication Symbol: The operation multiplier See multiplicand. of finding the product of two or more quantities. In arithmetic, multiplication of mutually exclusive events Two events one number, a, by another, b, consists of that cannot occur together. If the two adding a to itself b times. This kind of mul- events are denoted by M1 and M2 respec- tiplication is commutative, that is, a × b = tively then the two events being mutually × b a. The identity element for arithmetic exclusive is denoted by M1 ∩ M2 = ∅ (the multiplication is 1, i.e. multiplication by 1 null event). produces no change. In a series of multipli- cations, the order in which they are carried myria- Symbol: my A prefix used in out does not change the result. For exam- France to denote 104.

143 N

NAND gate See logic gate. natural numbers Symbol: N The set of numbers {1,2,3, …} used for counting sep- nano- Symbol: n A prefix denoting 10–9. arate objects. For example, 1 nanometer (nm) = 10–9 meter (m). nautical mile A unit of length equal to 6080 feet (about 1.852 km), equivalent to Napierian logarithm /nă-peer-ee-ăn/ See 1 minute of arc along a great circle on the logarithm. Earth’s surface. A speed of 1 nautical mile per hour is a knot. Napier’s formulae /nay-pee-erz/ A set of equations used in spherical trigonometry necessary condition See condition. to calculate the sides and angles in a spherical triangle. In a spherical triangle with negation Symbol: ∼ or ¬ In logic, the op- sides a, b, and c, and angles opposite these eration of putting not or it is not the case of α, β, and γ respectively: that in front of a proposition or statement, sin½(a – b)/sin½(a + b) = thus reversing its truth value. The negation tan½(α – β)/tan½γ of a proposition p is false if p is true and cos½(a – b)/cos½(a + b) = vice versa. The truth-table definition for tan½(α + β)/tan½γ negation is shown in the illustration. See sin½(α – β)/sin½(α + β) = also truth table. tan½(a – b)/cot½c cos½(α – β)/cos½(α + β) = tan½(a + b)/cot½c P ~P The formulae are named for the Scot- F T T F tish mathematician John Napier (1550– 1617). See also spherical triangle. Negation nappe /nap/ One of the two parts of a conical surface that lie either side of the vertex. See cone. negative Denoting a number of quantity that is less than zero. Negative numbers are natural frequency The frequency at also used to denote quantities that are which an object or system will vibrate below some specified reference point. For freely. A free vibration occurs when there is example, in the Celsius temperature scale a no external periodic force and little resis- temperature of –24°C is 24° below the tance. The amplitude of free vibrations freezing point of water. Compare positive. must not be too great. For instance, a pendulum swinging with small swings under negative binomial distribution See Pas- its own weight moves at its natural fre- cal’s distribution. quency. Normally, an object’s natural frequency is its fundamental frequency. neighborhood See topology. natural logarithm See logarithm. nested intervals A sequence of intervals

144 Newton’s method such that each interval contains the previ- kg m s–2. The unit is named for the English ous one. The nested interval theorem states physicist and mathematician Sir Isaac that for any sequence of bounded and Newton (1642–1727). closed nested intervals there is at least one point that belongs to all the intervals. If the Newtonian mechanics /new-toh-nee-ăn/ lengths of the intervals tend to zero as one Mechanics based on Newton’s laws of mo- goes through the sequence then there is ex- tion; i.e. relativistic or quantum effects are actly one such point. not taken into account. nesting The embedding of a computer Newton’s law of universal gravitation subroutine or a loop of instructions within The force of gravitational attraction be- another subroutine or loop, which in turn tween two point masses (m1 and m2) is may lie within yet another, and so on. proportional to each mass and inversely proportional to the square of the distance net 1. Denoting a weight of goods exclud- (r) between them. The law is often given in ing the weight of the containers or packing. the form 2. Denoting a profit calculated after de- 2 F = Gm1m2/r ducting all overhead costs, expenses, and where G is a constant of proportionality taxes. Compare gross. called the gravitational constant. The law 3. A surface that can be folded to form a can also be applied to bodies; for example, solid. spherical objects can be assumed to have 4. A network. their mass acting at their center. See also relativity; theory of. network A graph consisting of VERTICES that are joined by arcs (directed edges), Newton’s laws of motion Three laws of with each arc having an arrow on it to in- mechanics formulated by Sir Isaac Newton dicate its direction, and each arc is associ- in 1687. They can be stated as: ated with a non-negative number called its 1. An object continues in a state of rest or weight. There are many physical applica- constant velocity unless acted on by an ex- tions of networks including electrical cir- ternal force. cuits and representations of streets in 2. The resultant force acting on an object towns, with the physical significance of the weight depending on the physical problem. is proportional to the rate of change of mo- For example, in some applications there is mentum of the object, the change of mo- physical flow (transport) of something be- mentum being in the same direction as the tween the vertices, such as electrical cur- force. rent in an electrical circuit or cars moving 3. If one object exerts a force on another along a street, with the weight being the ca- then there is an equal and opposite force pacity of the arc. Another type of example (REACTION) on the first object exerted by is the one in which a network represents the second. some process, with the vertices represent- The first law was discovered by Galileo, ing the steps in the process and the weight and is both a description of inertia and a of an arc joining two vertices represents the definition of zero force. The second law time between the two steps. provides a definition of force based on the inertial property of mass. The third law is neutral equilibrium Equilibrium such equivalent to the law of conservation of that if the system is disturbed a little, there linear momentum. is no tendency for it to move further nor to return. See stability. Newton’s method A technique for obtaining successive approximations (itera- newton Symbol: N The SI unit of force, tions) to the solution of an equation, each equal to the force needed to accelerate one more accurate than the preceding one. The kilogram by one meter per second. 1 N = 1 equation in a variable x is written in the

145 node form f(x) = 0, and the general formula or i.e. the price paid by the first buyers of the algorithm: shares, may not be the same as the nominal xn+1 = xn – f(xn)/f′(xn) value, although it is likely to be close to it. is applied, where xn is the nth approxima- A share with a nominal value of 50¢ may tion. Newton’s method can be thought of be offered at an issue price of 55¢; it is then as repeated estimates of the position on a said to be offered at a premium of 5¢. If of- graph of f(x) against x at which the curve fered at an issue price of 45¢ it is said to be crosses the x-axis, by extrapolation of the offered at a discount of 5¢. Once estab- tangent to the curve. The slope of the tan- lished as a marketable share on a stock ex- gent at (x1,f(x1)) is df/dx at x = x1, that is change, the nominal value has little f′(x1) = f(x1)/(x2 – x1) importance and it is the market price at x2 = x1 – f(x1)/f′(x1) is therefore the point which it is bought and sold. However, the where the tangent crosses the x-axis, and is dividend is always expressed as a percent- a closer approximation to x at f(x) = 0 than age of the nominal value. x1 is. Similarly, x3 = x2 – f(x2)/f′(x2) nomogram /nom-ŏ-gram/ A graph that is a better approximation still. For exam- consists of three parallel lines, each one a ple, if f(x) = x2 – 3 = 0, then f′(x) = 2x and scale for one of three related variables. A we obtain the algorithm straight line drawn between two points, 2 xn+1 = xn – (x n – 3)/2xn = ½(xn + 3/xn) representing known values of two of the See also iteration. variables, crosses the third line at the corresponding value of the third variable. For node A point of minimum vibration in a example, the lines might show the temper- stationary wave pattern, as near the closed ature, volume, and pressure of a known end of a resonating pipe. Compare antin- mass of gas. If the volume and pressure are ode. See also stationary wave. known, the temperature can be read off the nomogram. Noether’s theorem /noh-terz/ A fundamental result in physics which relates sym- nonagon /non-ă-gon/ A plane figure with metry to conservation laws. It states that nine straight sides. A regular nonagon has for each continuous symmetry under nine equal sides and nine equal angles. which a physical system is invariant there is a conserved quantity. For example, invari- non-Cartesian coordinates Coordinates ance of a system under rotational symme- which are not Cartesian coordinates. try is associated with the conservation of POLAR COORDINATES and SPHERICAL POLAR angular momentum. It does not follow that COORDINATES are examples of non-Carte- if a certain symmetry and conservation law sian coordinates. associated with Noether’s theorem exists In many physical problems which are in a system described by classical physics specified in terms of a partial differential then the symmetry and conservation law equation it is frequently the case that the necessarily have to exist in the correspond- equation can only be solved exactly using a ing system described by QUANTUM ME- particular set of non-Cartesian coordi- CHANICS. nates, with the coordinate system being used depending on the geometry or sym- nominal value (per value) The value metry of the problem. For example, spher- given to a stock or share by the government ical polar coordinates are convenient for or corporation that offers it for sale. Stocks problems with spherical symmetry. have a nominal value of $100. Shares, however, may have any nominal value. For non-contradiction, law of See laws of example, a corporation wishing to raise thought. $100 000 by an issue of shares may issue 100 000 $1 shares or 200 000 50¢ shares, non-Euclidean geometry Any system of or any other combination. The issue price, geometry in which the parallel postulate of

146 normal

Euclid does not hold. This postulate can be solved exactly but, in general, non-linear stated in the form that, if a point lies out- oscillations require approximate methods. side a straight line, only one line parallel to Associated with this, it is possible for the straight line can be drawn through the chaotic motion to occur for non-linear os- point. In the early nineteenth century it was cillators. shown that it is possible to have a whole self-consistent formal system of geometry non-linear waves Waves that are associ- without using the parallel postulate at all. ated with NON-LINEAR OSCILLATIONS. The There are two types of non-Euclidean form of a non-linear wave is more compli- geometry. In one (called elliptic geometry) cated than a sine wave and only approxi- there are no parallel lines through the mates to a sine wave in the case of small point. An example of this is a system de- oscillations. It has been suggested that very scribing the properties of lines, figures, an- large ‘freak waves’ in oceans are non-linear gles, etc., on the surface of a sphere in waves. Other physical examples exist. A which all lines are parts of great circles (i.e. SOLITON is a particular type of non-linear circles that have the same center as the cen- wave. ter of the sphere). Since all great circles intersect, no parallel can be drawn through non-uniform motion Motion in which the point. Note also that the angles of a tri- the velocity is not constant. This can mean angle on such a sphere do not add up to that there is acceleration (including the 180°. The other type of non-Euclidean case of negative acceleration, i.e. decelera- geometry is called a hyperbolic geometry – tion) and/or the direction of the motion is here an infinite number of parallels can be not constant. For example, a body moving drawn through the point. in a circle is an example of non-uniform Note that a type of geometry is not in it- motion since the direction of the body is al- self based on ‘experiment’ – i.e. of mea- ways changing, which means that there is surements of distance, angles, etc. It is a always an acceleration, even if the speed of purely abstract system based on certain as- the body is constant. If the speed of a body sumptions (such as Euclid’s axioms). moving in a circle is not uniform then there Mathematicians study such systems for is said to be non-uniform circular motion. their own sake – without necessarily look- The analysis of non-uniform circular mo- ing for practical applications. The practical tion requires a slight generalization of the applications come in when a particular analysis of uniform circular motion. In the mathematical system gives an accurate de- case of non-uniform circular motion there scription of physical properties – i.e. the is a component of the acceleration directed properties of the ‘real world’. In practical along the tangent to the circle as well as a uses (in architecture, surveying, engineer- component directed towards the center of ing, etc.) it is assumed that Euclidean the circle. geometry applies. However, it is found that this is only an approximation, and that the NOR gate See logic gate. space-time continuum of relativity theory is non-Euclidean in its properties. norm A generalization of the concept of magnitude to any vector space. The norm non-isomorphism See isomorphism. of a vector x is usually written ||x||. It is a real number associated with the vector and non-linear oscillations Oscillations in is positive or zero (for the zero vector). If a which the force is not proportional to the is a real number, then displacement from the equilibrium posi- ||ax|| = a||x|| tion; i.e. the oscillations are not simple har- and monic motion. The general motion of a ||x + y|| ≤ ||x|| + ||y|| pendulum is an example of a non-linear oscillation. The motion of a pendulum is a normal Denoting a line or plane that is case of a non-linear oscillation that can be perpendicular to another line or plane. A

147 normal chord

can be found. The alternative term ‘Gauss- N ian distribution’ is named for the German S mathematician Karl Friedrich Gauss (1777–1855).

P normal form See canonical form. T normalize To multiply a quantity (e.g. a vector or matrix) by a suitable constant so that its norm is equal to one.

normal subgroup A subgroup H of a group G is normal if and only if for any el- Normal: NP is the normal to the curve ement h in H, h–1 gh is in H for all elements g of G. line or plane is said to be normal to a curve NOT gate See logic gate. if it is perpendicular to the tangent to the curve at the point at which the line and the NP-problem A type of problem where curve meet. A radius of a circle, for exam- the size of the problem is characterized by ple, is normal to the circumference. A plane some number n and the number of steps passing through the center of a sphere is which an algorithm would need to solve normal to the surface at all points at which this problem is N, and the dependence of N they meet. on n is such that N increases with n more rapidly than any polynomial of n. Such normal chord A straight line joining two problems are called NP-problems because points on a curve that is also a normal to the time it takes to solve them increases the curve at one or both of the two points. more rapidly than any polynomial of n. The diameter of a circle can be regarded as Examples of NP-problems include the a normal chord. It is also possible to con- TRAVELING SALESMAN PROBLEM and the struct normal chords of a hyperbola that factorization of large integers. NP-prob- connect the two branches of the hyperbola. lems are more difficult to solve than P-PROBLEMS. It may be the case that some normal distribution (Gaussian distribu- NP-problems, such as the factorization of tion) The type of statistical distribution large integers, could be solved much more followed by, for example, the same mea- rapidly using a quantum computer. surement taken several times, where the variation of a quantity (x) about its mean n-th root of unity A complex number z value (µ) is entirely random. A normal dis- that satisfies the relation zn = 1. There are tribution has the probability density func- n such n-th roots of unity. They are given tion by z = exp (i2πl/n), where l takes all the in- f(x) = exp[–(x – µ)2/2σ2]/σ√2π teger values from 0 to n–1. The complex where σ is known as the standard devia- numbers that are the n-th root of unity can tion. The distribution is written N(µ,σ2). be represented in an ARGAND DIAGRAM for The graph of f(x) is bell-shaped and sym- the complex plane as vertices of a regular metrical about x = µ. The standard normal polygon with n sides. These points lie on a distribution has µ = 0 and σ2 = 1. x can be circle with its center at the origin of the standardized by letting z = (x – µ)/σ. The complex plane and a radius of one unit. If values zα, for which the area under the n is an even integer then the roots must in- curve from – ∞ to zα is α, are tabulated; i.e. clude both 1 and –1. If n is an odd integer z is such that P(z ≤ zα) = α. Hence the roots that are not real numbers have to P(a

148 numerical integration roots of unity can also be written as: z = notes a number. Examples include 0, 1, 2, cos(2πl/n) + i sin(2πl/n), with l taking all in- 3, 4, 5, 6, 7, 8, 9 of Arabic numerals and I, teger values from 0 to n–1. V, X, L, C, D, M of ROMAN NUMERALS.

NTP Normal temperature and pressure. numerator /new-mĕ-ray-ter/ The top part See STP. of a fraction. For example, in the fraction ¾, 3 is the numerator and 4 is the denomi- null hypothesis See hypothesis test. nator. The numerator is the dividend. null matrix (zero matrix) A MATRIX in numerical analysis /new-me-ră-kăl/ The which all the elements are equal to zero. study of methods of calculation that involve approximations, for example, itera- null set See empty set. tive methods. See also iteration. number line A straight horizontal line on numerical integration A procedure for which each point represents a real number. calculating approximate values of inte- Integers are points marked at unit distance grals. Sometimes a function is known only apart. as a set of values for corresponding values of a variable and not as a general formula numbers Symbols used for counting and that can be integrated. Also, many func- measuring. The numbers now in general tions cannot be integrated in terms of use are based on the Hindu-Arabic system, which was introduced to Europe in the known standard integrals. In these cases, 14th and 15th centuries. The Roman nu- numerical integration methods, such as the merals used before this made simple arith- TRAPEZIUM RULE and SIMPSON’S RULE, can be metic very difficult, and most calculations used to calculate the area under a graph needed an abacus. Hindu-Arabic numerals corresponding to the integral. The area is (0, 1, 2, … 9) enabled calculations to be divided into vertical columns of equal performed with far greater efficiency be- width, the width of each column represent- cause they are grouped systematically in ing an interval between two values of x for units, tens, hundreds, and so on. See also which f(x) is known. Usually a calculation integers; irrational numbers; natural num- is first carried out with a few columns; bers; rational numbers; real numbers; these are further subdivided until the de- whole numbers. sired accuracy is attained, i.e. when further subdivision makes no significant difference numeral /new-mĕ-răl/ A symbol that de- to the result.

–10123456 7

Number line: this shows an open interval of the real numbers between –1 and +2 and a closed interval from 4 to 6 (including 4 and 6)

149 O

object The set of points that undergoes a each character and translates it into a series geometrical transformation or mapping. of electrical pulses. See also projection. octagon /ok-tă-gon/ A plane figure with oblate /ob-layt, ŏ-blayt/ Denoting a spher- eight straight sides. A regular octagon has oid that has a polar diameter that is smaller eight equal sides and eight equal angles. than the equatorial diameter. The Earth, for example, is not a perfect sphere but is octahedron /ok-tă-hee-drŏn/ (pl. octahe- an oblate spheroid. Compare prolate. See drons or octahedra) A POLYHEDRON that also ellipsoid. has eight faces. A regular octahedron has eight faces, each one an equilateral trian- oblique /ŏ-bleek/ Forming an angle that is gle. not a right angle. oblique coordinates See Cartesian coordinates. oblique solid A solid geometrical figure that is ‘slanted’; for example, a cone, cylinder, pyramid, or prism with an axis that is not at right angles to its base. Compare right solid. oblique spherical triangle See spherical triangle. oblique triangle A triangle that does not Octahedron: a regular octahedron contain a right angle. oblong /ob-long/ An imprecise term for a RECTANGLE. octal /ok-tăl/ Denoting or based on the number eight. An octal number system has obtuse /ŏb-tewss/ Denoting an angle that eight different digits instead of the ten in is greater than 90° but less than 180°. the decimal system. Eight is written as 10, Compare acute; reflex. nine as 11, and so on. Compare binary; decimal; duodecimal; hexadecimal. OCR (optical character recognition) A system used to input information to a com- octant /ok-tănt/ 1. One of eight regions puter. The information, usually in the form into which space is divided by the three of letters and numbers, is printed, typed, or axes of a three-dimensional Cartesian co- sometimes hand-written. The characters ordinate system. The first octant is the one used can be read and identified optically by in which x, y, and z are all positive. The an OCR reader. This machine interprets second, third, and fourth octants are num-

150 open set

O B

Octant: ABCO is an octant of the sphere. bered anticlockwise around the positive z- ond, and vice versa. See also function; ho- axis. The fifth octant is underneath the momorphism. first, the sixth under the second, etc. 2. A unit of plane angle equal to 45 degrees one-to-one mapping A mapping f from (π/4 radians). a set M to a set N such that if m1, and m2 are different elements of M then their im- odd Not divisible by two. The set of odd ages f(m1) and f(m2) are different elements numbers is {1, 3, 5, 7, …}. Compare even. of N. This definition means that if f is a one-to-one mapping then the result f(m1) = odd function A function f(x) of a variable f(m2) means that m1 = m2. x for which f(–x) = –f(x). For example, sinx and x3 are odd functions of x. Compare onto A mapping from one set S to another even function. set T is onto if every member of T is the image of some member of S under the map- odds When bets are placed on some event ping. Compare into. the odds are the probability of it happen- ing. open curve A curve in which the ends do not meet, for example, a parabola or a hy- oersted Symbol Oe A unit of magnetic perbola. Compare closed curve. field strength in the c.g.s. system. It is equal to 103/4π amperes per meter (103/4π open interval A set consisting of the Am–1). The unit is named for the Danish numbers between two given numbers (end physicist Hans Christian Oersted (1777– points), not including the end points, for 1851). example, all the real numbers greater than 1 and less than 4.5 constitute an open in- ohm /ohm/ Symbol: Ω The SI unit of elec- terval. The open interval between two real trical resistance, equal to a resistance numbers a and b is written (a,b). Here, the through which a current of one ampere round brackets indicate that the points a flows when there is an electric potential and b are not included in the INTERVAL. On difference of one volt across it. 1 Ω = 1 a number line, the end points of an open in- VA–1. The unit is named for the German terval are circled. Compare closed interval. physicist Georg Simon Ohm (1787–1854). open sentence In formal logic, a sentence one-to-one correspondence A function that contains one or more free variables. or mapping between two sets of things or numbers, such that each element in the first open set A set defined by limits that are set maps into only one element in the sec- not included in the set itself. The set of all

151 operating system rational numbers greater than 0 and less orbit that is a conic section; i.e. a parabola, than ten, written {x:0

152 orthogonal vectors

B C Y

Orthocenter of a triangle ABC: the pedal triangle is XYZ. ordinate The vertical coordinate (y-coor- square of the distance between two orthog- dinate) in a two-dimensional rectangular onal circles is equal to the sum of the Cartesian coordinate system. See Cartesian squares of their two radii. coordinates. orthogonal group See orthogonal ma- OR gate See logic gate. trix. origin The fixed reference point in a co- orthogonal matrix A matrix for which ordinate system, at which the values of all its transpose is also its inverse. Thus, if A is ~ the COORDINATES are zero and at which the a matrix and its transpose is denoted by A axes meet. then A is an orthogonal matrix if AÃ = I, where I is the unit matrix. The set of all n × orthocenter /or-thoh-sen-ter/ A point in a n orthogonal matrices forms a group called triangle that is the point of intersection of the orthogonal group. The set of all n × n lines from each vertex perpendicular to the orthogonal matrices with determinants opposite sides. The triangle formed by that have the value 1 is a subgroup of the joining the feet of these vertices is the pedal orthogonal group called the special orthog- triangle. onal group. The orthogonal group is denoted O(n) and the special orthogonal orthogonal circles /or-thog-ŏ-năl/ Two group is denoted SO(n). The groups O(n) circles that intersect at right angles to each and SO(n) have several important physical other. Circles which intersect in this way applications, including the description of are said to cut orthogonally. If A and B are rotations. the centers of two circles that cut orthogonally at the points P and Q then the tangent orthogonal projection /or-thog-ŏ-năl/ A at P to the circle with center A is perpen- geometrical transformation that produces dicular to the radius AP. In a similar way an image on a line or plane by perpendicu- the tangent at P to the circle with center B lar lines crossing the plane. If a line of is perpendicular to the radius BP. Analo- length l is projected orthogonally from a gous results hold for the tangents at Q. plane at angle θ to the image plane, its Since the circles are orthogonal the tan- image length is lcosθ. The image of a circle gents at P are perpendicular, as are the tan- is an ellipse. See also projection. gents at Q. This means that each of the tangents at both P and Q goes through the orthogonal vectors Vectors which are center of the other circle. This means that perpendicular. This means that if a and b AP2 + BP2 = AQ2 + BQ2 = AB2, i.e. the are non-zero vectors then they are orthog-

153 orthonormal onal vectors if and only if their SCALAR it is one twentieth of a UK pint, equivalent PRODUCT a.b = 0. If there is a set of mutu- to 2.841 3 × 10–5 m3. 1 UK fluid ounce is ally orthogonal unit vectors then there is equal to 0.960 8 US fluid ounce. said to be an orthonormal set of vectors. In three-dimensional space the UNIT VECTORS output 1. The signal or other form of in- i, j, and k along the x, y, and z axes respec- formation obtained from an electrical de- tively form a set of orthonormal vectors. vice, machine, etc. The output of a computer is the information or results de- orthonormal /or-thŏ-nor-măl/ See or- rived from the data and programmed in- thogonal vectors. structions fed into it. This information is transferred as a series of electrical pulses OS See operating system. from the central processor of the computer to a selected output device. Some of these Osborn’s rule /oz-born/ A rule that can output units convert the pulses into a read- be used to convert relations between able or pictorial form; examples include trigonometric identities and the analogous the printer, plotter, and visual display unit relations between hyperbolic functions. It (which can also be used as an input device). states that a cos2 term becomes a cosh2 Other output devices translate the pulses term without change of sign but that a sin2 into a form that can be fed back into the term (or more generally a product of two computer at a later stage; the magnetic tape sine terms) becomes a –sinh2 term, or, unit is an example. more generally, the sign of the term is 2. The process or means by which output is changed. For example, cos2x + sin2x = 1 obtained. becomes cosh2x – sinh2x = 1 and cos2x – 3. To deliver as output. sin2x= cos2x becomes cosh2x + sinh2x = See also input; input/output. cosh2x. Osborn’s rule has to be used care- fully since there are some trigonometric overdamping See damping. functions, such as tan2x, that can be expressed in terms of sines and cosines in sev- overflow The situation arising in com- eral ways. Although Osborn’s rule can be puting when a number, such as the result of used to find an identity for hyperbolic an arithmetical operation, has a greater functions, finding such an identity in this magnitude than can be represented in the way does not constitute a proof for the space allocated to it in a register or a loca- identity. tion in store. oscillating series /os-ă-lay-ting/ A special overlay A technique used in computing type of nonconvergent series for which the when the total storage requirements for a sum does not approach a limit but contin- lengthy program exceeds the space avail- ually fluctuates. Oscillating series can able in the main store. The program is split either fluctuate between bounds, for ex- into sections so that only the section or sec- ample the series 1 – 1 + 1 – 1 + …, or it can tions required at any one time will be trans- be unbounded, for example 1 – 2 + 3 – 4 + ferred. These program segments (or 5 – … . overlays) will all occupy the same area of the main store. oscillation /os-ă-lay-shŏn/ A regularly repeated motion or change. See vibration. overtones The wave patterns which are present in sound and music in which the ounce 1. A unit of mass equal to one six- frequency is greater than the fundamental teenth of a pound. It is equivalent to frequency. It is possible to analyze the 0.0283 49 kg. overtones of a wave pattern using FOURIER 2. A unit of capacity, often called a fluid SERIES. The overtones which are heard ounce, equal to one sixteenth of a pint. It is along with the main note of a specific fre- equivalent to 2.057 3 × 10–5 m3. In the UK, quency f vary in different musical instru-

154 overtones ments. This means that the waveform asso- a note deviates from being a sine wave, is ciated with each note varies in different therefore dependent on the presence or ab- musical instruments. The quality or timbre sence of overtones and hence depends on of a note, i.e. how much the waveform for the instrument.

155 P

palindrome A number that is the same mathematician and astronomer Paul sequence of integers forward and back- Guldin (1577–1643). ward. An example of a palindrome is the number 35753. A conjecture concerning parabola /pă-rab-ŏ-lă/ A conic with an ec- palindromes is that starting from any num- centricity of 1. The curve is symmetrical ber, if the integers of that number are re- about an axis through the focus at right an- versed and the resulting number is added to gles to the directrix. This axis intercepts the the original number, then if this process is parabola at the vertex. A chord through repeated long enough a palindrome will re- the focus perpendicular to the axis is the sult. However, this conjecture has not been latus rectum of the parabola. proved. In Cartesian coordinates a parabola can be represented by an equation: pandigital number /pan-dij-i-tăl/ A num- y2 = 4ax ber which contains all of the numbers 0, 1, In this form the vertex is at the origin and 2, 3, 4, 5, 6, 7, 8, 9 only once. the x-axis is the axis of symmetry. The focus is at the point (0,a) and the directrix paper tape A long strip of paper, or some- is the line x = –a (parallel to the y-axis). times thin flexible plastic, on which infor- The latus rectum is 4a. mation can be recorded as a pattern of If a point is taken on a parabola and round holes punched in rows across the two lines drawn from it – one parallel to tape, once used extensively in data process- the axis and the other from the point to the ing. focus – then these lines make equal angles with the tangent at that point. This is Pappus’ theorems Two theorems con- known as the reflection property of the cerning the rotation of a curve or a plane parabola, and is utilized in parabolic re- shape about a line that lies in the same flectors and antennas. See paraboloid. plane. The first theorem states that the sur- The parabola is the curve traced out by face area generated by a curve revolving a projectile falling freely under gravity. For about a line that does not cross it, is equal example, a tennis ball projected horizon- to the length of the curve times the circum- tally with a velocity v has, after time t, trav- ference of the circle traced out by its cen- eled a distance d = vt horizontally, and has troid. The second theorem states that the also fallen vertically by h = gt2/2 because of volume of a solid of revolution generated the acceleration of free fall g. These two by a plane area that rotates about a line not equations are parametric equations of the crossing it, is equal to the area times the parabola. Their standard form, corre- circumference of the circle traced out by sponding to y2 = 4ax, is the centroid of the area. (Note that the x = at2 plane area and the line both lie in the same y = 2at plane.) The theorems are named for the where x represents h, the constant a is g/2, Greek mathematician Pappus of Alexan- and y represents d. See illustration over- dria (fl. AD 320). The second Pappus theo- leaf. See also conic. rem is sometimes known as the Guldinus theorem as it was rediscovered by the Swiss paraboloid /pă-rab-ŏ-loid/ A curved sur-

156 paradox

axis of rotation

curve length l

circle circumference c

centroid of curve

Pappus’ theorem: the curved surface area A = l x c.

centroid of axis of rotation plane area

plane area A circle circumference c

Pappus’ theorem: the volume enclosed by the curved surface V = A x c.

Pappus’ theorems

face in which the cross-sections in any Sections parallel to the other two planes (x plane passing through a central axis is a = 0 or y = 0) are parabolas. See illustration parabola. A paraboloid of revolution is overleaf. See also conicoid. formed when a parabola is rotated around its axis of symmetry. Parabolic surfaces are paradox /pa-ră-doks/ (antinomy) A used in telescope mirrors, searchlights, ra- proposition or statement that leads to a diant heaters, and radio antennas on ac- contradiction if it is asserted and if it is de- count of the focusing property of the nied. parabola. A famous example of a paradox is Rus- Another type of paraboloid is the hy- sell’s paradox in set theory. A set is a col- perbolic paraboloid. This is a surface with lection of things. It is possible to think of the equation: sets as belonging to two groups: sets that x2/a2 – y2/b2 = 2cz contain themselves and sets that do not where c is a positive constant. Sections par- contain themselves. A set that contains it- allel to the xy plane (z = 0) are hyperbolas. self is itself a member of the set. For in-

157 parallax

D y y T P P (x,y) Q

F S

a a N V F (0,a) x F x

Parabola: F is the focus and DE is the directrix. FG is the latus rectum. If PQ is parallel to the x- axis, angles SPF and TPQ are equal (ST is a tangent at P).

y y

O O x x

Paraboloid of revolution stance, the set of all sets is itself a set, so it it doesn’t contain itself. On the other hand, contains itself. Other examples are the ‘set if it does not contain itself it must be one of of all things that one can think about’ and all the things that do not contain them- ‘the set of all abstract ideas’. On the other selves, so it must belong to the set. The hand the set of all things with four legs short answer to the question is, ‘If it does, does not itself have four legs – it is an ex- it doesn’t; if it doesn’t, it does!’ – hence the ample of a set that does not contain itself. paradox. Paradoxes like this can be used in Other examples of sets that do not contain investigating fundamental questions in set themselves (or are not members of them- theory. selves) are ‘the set of all oranges’ and ‘the set of all things that are green’. parallax /pa-ră-laks/ The angle between Consider now the set of all sets that do the direction of an object, for example a not contain themselves. The paradox arises star or a planet, from a point on the surface from the question, ‘Does the set of all sets of the Earth and the direction of the same that do not contain themselves contain it- object from the center of the Earth. The self or not?’ If one asserts that it does con- measurement of parallax is used to find the tain itself then it cannot be one of the distance of an object from the Earth. The things that do not contain themselves. This object is viewed from two widely separated means that it does not belong to the set, so points on the Earth and the distance be-

158 parameter

area = hb h

Parallelepiped Parallelogram

tween these two points and the direction of length of one side and the perpendicular the object as seen from each is measured. distance from that side to the side opposite. The parallax at the observation points and In the special case in which the angles are the distance of the object from the observer all right angles the parallelogram is a rec- can be found by simple trigonometry. tangle; when all the sides are equal it is a rhombus. parallel Extending in the same direction and remaining the same distance apart. parallelogram (law) of forces See par- Compare antiparallel. allelogram of vectors. parallel axes, theorem of See theorem of parallelogram (law) of velocities See parallel axes. parallelogram of vectors. parallelepiped /pa-ră-lel-ă-pÿ-ped/ A parallelogram of vectors A method for solid figure with six faces that are parallel- finding the resultant of two vectors acting ograms. In a rectangular parallelepiped the at a point. The two vectors are shown as faces are rectangles. If the faces are two adjacent sides of a parallelogram: the squares, the parallelepiped is a cube. If the resultant is the diagonal of the parallelo- lateral edges are not perpendicular to the gram through the starting point. The tech- base, it is an oblique parallelepiped. nique can be used either with careful scale drawing or with trigonometry. The parallel forces When the forces on an ob- trigonometrical relations give: √ 2 2 θ ject pass through one point, their resultant F = (F1 + F2 + 2F1F2cos ) α –1 θ can be found by using the parallelogram of = sin [(F2/F)sin ] θ vectors. If the forces are parallel the resul- where is the angle between F1 and F2 and α tant is found by addition, taking sign into is the angle between F and F1. See vector. account. There may also be a turning effect in such cases, which can be found by the parallel postulate See Euclidean geome- principle of moments. try. parallelogram /pa-ră-lel-ŏ-gram/ A plane parameter /pă-ram-ĕ-ter/ A quantity that, figure with four straight sides, and with op- when varied, affects the value of another. posite sides parallel and of equal length. For example, if a variable z is a function of The opposite angles of a parallelogram are variables x and y, that is z = f(x,y), then x also equal. Its area is the product of the and y are the parameters that determine z.

159 parametric equations parametric equations Equations that, in tial derivative ∂z/∂x is the rate of change of an implicit function (such as f(x,y) = 0) ex- z with respect to x, with y held constant. Its press x and y separately in terms of a quan- value will depend on the constant value of tity, which is an independent variable or y chosen. In three-dimensional Cartesian parameter. For example, the equation of a coordinates, ∂z/∂x is the gradient of a line circle can be written in the form at a tangent to the curved surface f(x,y) and x2 + y2 = r2 parallel to the x-axis. See also total deriva- or as the parametric equations tive. x = rcosθ y = rsinθ partial differential The infinitesimal where r is the radius. See also parabola. change in a function of two or more variables resulting from changing only one of parametrization /pa-ră-met-ri-zay-shŏn/ the variables while keeping the others con- A method for associating a parameter t stant. The sum of all the partial differen- with a point P, which lies on a curve so that tials is the total differential. See each point on the curve is associated with a differential. value of t, with t lying in some interval of the real numbers. This is frequently done partial differential equation An equa- by expressing the x and y coordinates of P tion that contains partial derivatives of a in terms of t. The resulting equations for x function with respect to a number of vari- and y in terms of t are called the PARAMET- ables. General methods of solution are RIC EQUATIONS for the curve. It is possible available only for certain types of linear to find the expression for dy/dx at any partial differential equation. Many partial point on the curve in terms of the parame- differential equations occur in physical ter t since dy/dx = (dy/dt)/(dx/dt). problems. Laplace’s equation, for exam- Perhaps the simplest parametrization of ple, is a curve is given by a circle of unit radius. ∂2φ/∂x2 + ∂2φ/∂y2 + ∂2φ/∂z2 = 0 The points on the circle are parametrized It occurs in the study of gravitational and by an angle θ, with (in radians) 0 ≤ θ ≤ 2π. electromagnetic fields. The equation is One has x = cosθ and y = sinθ. In the case named for the French mathematician, as- of an ellipse x2/a2 + y2/b2 = 1 the parame- tronomer, and physicist Marquis Pierre trization is given by x = a cosθ and y = b Simon de Laplace (1749–1827). See also sinθ (0 ≤ θ ≤ 2π). In the case of a parabola differential equation. y2 = 4ax the curve is parametrized by t with x = at2 and y = 2at. partial fractions A sum of fractions that is equal to a particular fraction, for exam- paraplanar /pa-ră-play-ner/ Describing a ple, ½ + ¼ = ¾. Writing a ratio in terms of set of vectors that are all parallel to a plane partial fractions can be useful in solving but are not necessarily contained in that equations or calculating integrals. For ex- plane. ample 1/(x2 + 5x + 6) parsec /par-sek/ Symbol: pc A unit of dis- can be written as partial fractions by fac- tance used in astronomy. A star that is one torizing the denominator into (x + 2)(x + parsec away from the earth has a parallax 3), and writing the fraction in the form (apparent shift), due to the Earth’s move- [A/(x + 2)] + [B/(x + 3)] = ment around the Sun, of one second of arc. 1/(x2 + 5x + 6) One parsec is approximately 3.085 61 × This means that 1016 meters. A(x + 3) + B(x + 2) = 1 i.e. partial derivative The rate of change of a Ax + 3A + Bx + 2B = 1 function of several variables as one of the The values of A and B can be found by variables changes and the others are held comparing the coefficients of powers of x, constant. For example, if z = f(x,y) the par- i.e.

160 Pascal’s triangle

3A + 2B = 1 (constant term) then a similar function can be tried. In all A + B = 0 (coefficient of x) these examples, unknown coefficients can Solving these simultaneous equations gives be found by substitution into the original A = 1 and B = –1. So the fraction 1/(x2 + 5x differential equation. + 6) can be expressed as partial fractions 1/(x + 2) – 1/(x + 3). This is a form that can particular solution A solution of a dif- be integrated as a sum of two simpler inte- ferential equation that is given by some grals. particular values of the arbitrary constants that appear in the general solution of the partially ordered A partially ordered set differential equation. A particular solution is a set with a relation x < y defined for can be found if it satisfies both the differ- some elements x and y of the set satisfying ential equation and the BOUNDARY CONDI- the conditions: TIONS that apply. 1. if x < y then y < x is false and x and y are not the same element; partition A partition of a set S is a finite 2. if x < y and y < z then x < z. collection of disjoint sets whose union is S. It need not be the case that x < y or y < x for any two elements x and y. An exam- pascal /pas-kăl/ Symbol: Pa The SI unit of ple of a partially ordered set is the set of pressure, equal to a pressure of one newton subsets of a given set where we define A < per square meter (1 Pa = 1 N m–2). The pas- B for sets A and B to mean that A is a cal is also the unit of stress. It is named for proper subset of B. the French mathematician, physicist and religious philosopher Blaise Pascal (1623– partial product A product of the first n 62). terms a1 a2 ... an, called the n-th partial product, where these n terms are the first n Pascal’s distribution (negative binomial terms of the infinite product. distribution) The distribution of the number of independent Bernoulli trials per- partial sum The sum of a finite number of formed up to and including the rth success. terms from the start of an infinite series. In The probability that the number of trials, a convergent series, the partial sum of the x, is equal to k is given by k–1 r k–r first r terms, Sr, is an approximation to the P(x=k) = Cr–1p q sum to infinity. See series. The mean and variance are r/p and rq/p2 respectively. See also geometric distribu- particle An abstract simplification of a tion. real object – the mass is concentrated at the object’s center of mass; its volume is zero. Pascal’s triangle A triangle array of num- Thus rotational aspects can be ignored. bers in which each row starts and ends with 1 and that is built up by summing two particular integral A name given to the adjacent numbers in a row to obtain the PARTICULAR SOLUTION in the case of a number directly below them in the next second-order linear differential equation of the form ad2y/dx2 + bdy/dx + cy = f(x), where a, b, and c are constants and f is 1 some function of x. A way of finding a par- 1 1 ticular integral which frequently works is to use a function which is similar to f(x). 1 2 1 For example, if f(x) is a polynomial of some degree in x then the particular inte- 1 3 3 1 gral to be tried should also be a polynomial of the same degree. Similarly, if f(x) is a 1 4 6 4 1 trigonometric function such as sinlx or cosmx or an exponential function exp(nx) Pascal’s triangle

161 pedal curve row. Each row in Pascal’s triangle is set of ment of inertia of the body. For small os- binomial coefficients. In the expansion of cillations it is given by the same relation- (x + y)n, the coefficients of x and y are ship as that of the simple pendulum with l given by the (n + 1)th row of Pascal’s trian- replaced by [√(k2 + h2)]/h. Here, k is the RA- gle. DIUS OF GYRATION about an axis through the center of mass and h is the distance pedal curve The pedal curve of a given from the pivot to the center of mass. curve C with respect to a fixed point P is the locus of the foot of the perpendicular Penrose pattern /pen-rohz/ A two-dimen- from P to a variable tangent to the curve C. sional pattern of a tiling that has five-fold The point P is called the pedal point. For rotational axes of symmetry and also long- example, if C is a circle and P is a point on range order, in spite of it being impossible its circumference the pedal curve is a car- to have a two-dimensional crystal in which dioid. five-fold rotational axes of symmetry can occur. It is possible to obtain this type of pedal triangle See orthocenter. pattern by combining two sets of rhombuses, with one of the sets being ‘thin’ pencil A family of geometric objects that rhombuses and the other set being ‘fat’ share a common property. For example, a rhombuses, in specific ways. A three- pencil of circles consists of all the circles in dimensional version of Penrose patterns a given plane that pass through two given occurs in materials that have QUASICRYS- points, a pencil of lines consists of all the TALLINE SYMMETRY. The pattern is named lines in a given plane passing through a for the British mathematician Sir Roger given point, and a pencil of parallel lines Penrose (1931– ). consists of all the lines parallel to a given direction. pentagon /pen-tă-gon/ A plane figure with five straight sides. In a regular pentagon, pendulum A body that oscillates freely one with all five sides and angles equal, the under the influence of gravity. A simple angles are all 108°. A regular pentagon can pendulum consists of a small mass oscillat- be superimposed on itself after rotation ing to and fro at the end of a very light through 72° (2π/5 radians). string. If the amplitude of oscillation is small (less than about 10°), it moves with percentage A number expressed as a frac- simple harmonic motion; the period does tion of one hundred. For example, 5 per- not depend on amplitude. There is a con- cent (or 5%) is equal to 5/100. Any tinuous interchange of potential and ki- fraction or decimal can be expressed as a netic energy through the motion; at the percentage by multiplying it by 100. For ends of the swings the potential energy is a example, 0.63 × 100 = 63% and ¼×100 = maximum and the kinetic energy zero. At 25%. the mid-point the kinetic energy is a maximum and the potential energy is zero. The percentage error The ERROR or uncer- period is given by tainty in a measurement expressed as a per- T = 2π√(l/g) centage. For example, if, in measuring a Here l is the length of the pendulum (from length of 20 meters, a tape can measure to support to center of the mass) and g is the the nearest four centimeters, the measure- acceleration of free fall. If the amplitude of ment is written as 20± 0.04 meters and the oscillation is not small it does not move percentage error is (0.04/20) × 100 = 0.2%. with simple harmonic motion but the problem of its motion can still be solved percentile /per-sen-tÿl, -tăl/ One of the set exactly. of points that divide a set of data arranged A compound pendulum is a rigid body in numerical order into 100 parts. The rth swinging about a point. The period of a percentile, Pr, is the value below and in- compound pendulum depends on the mo- cluding which r% of the data lies and

162 personal computer above which (100 – r)% lies. Pr can be peripheral unit (peripheral) A device found from the cumulative frequency connected to and controlled by the central graph. See also quartile; range. processor of a computer. Peripherals include input devices, output devices, and perfect number A number that is equal backing store. Some examples are visual to the sum of all its factors except itself. 28 display units, printers, magnetic tape units, is a perfect number since its factors are 1, and disk units. See also input; output. 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 = 28. See also Mersenne prime. permutation /per-myŭ-tay-shŏn/ An ordered subset of a given set of objects. For perimeter /pĕ-rim-ĕ-ter/ The distance three objects, A, B, and C, there are six round the edge of a plane figure. For ex- possible permutations: ABC, ACB, BAC, ample, the perimeter of a rectangle is twice BCA, CAB, and CBA. The total number of the length plus twice the breadth. The permutations of n objects is n! perimeter of a circle is its circumference The total number of permutations of r (2πr). objects taken from n objects is given by n!(n – r)!, assuming that each object can be n period Symbol: T The time for one com- selected only once. This is written Pr. For plete cycle of an oscillation, wave motion, example, the possible permutations of two or other regularly repeated process. It is the objects from the set of three objects A, B, reciprocal of the frequency, and is related and C would be AB, BA, AC, CA, BC, CB. to pulsatance, or angular frequency, (ω) by Note that each object is selected only once T = 2π/ω. in this case – if the objects could occur any number of times, the above set of permuta- periodic function A function that repeats tions would include AA, BB, and CC. The number of permutations of r objects se- itself at regular intervals of the variable. lected from n objects when each can occur For example sinx is a periodic function of any number of times is nr. x because sinx = sin (x + 2π) for all values Note also the difference between per- of x. mutations and combinations: permutations are different if the order of selection periodic motion Any kind of regularly is different, so AB and BA are different per- repeated motion, such as the swinging of a mutations but the same combination. The pendulum, the orbiting of a satellite, the vi- number of combinations of r objects from bration of a source of sound, or an electro- n objects is written nC , and nP = nC × r! magnetic wave. If the motion can be r r r represented as a pure sine wave, it is a sim- permutation group The set of permuta- ple harmonic motion. Harmonic motions tions of a number of indistinguishable ob- in general are given by the sum of two or jects. The permutation group is important more pure sine waves. in theoretical physics. It is used to study the energy levels of electrons in atoms period of investment The length of time and molecules and is also used in nuclear for which a fixed amount of capital re- theory. mains invested. In times of historically low interest rates, an investor prepared to com- perpendicular /per-pĕn-dik-yŭ-ler/ At mit his or her money for a long period, right angles. The perpendicular bisector of such as five or ten years, will gain a higher a line crosses it half way along its length rate of interest than can be expected for a and forms a right angle. A vertical surface short-term investment. However, if interest is perpendicular to a horizontal surface. rates are historically high this will not be the case and long-term rates may be lower personal computer (PC) See microcom- than short-term rates. puter.

163 perturbation perturbation /per-ter-bay-shŏn/ A way of showing the complicated behavior that can obtaining approximate solutions of equa- occur in dynamical systems, including the tions that represent the behavior of a sys- presence of ATTRACTORS, chaotic behavior, tem, by making a slight change in some of and LIMIT CYCLES. the basic parameters. It is an important technique in quantum mechanics and celes- phase space A multi-dimensional space tial mechanics. that can be used to define the state of a system. Phase space has coordinates (q1, q2,… phase /fayz/ The stage in a cycle that a p1, p2,…), where q1, q2, etc., are degrees of WAVE (or other periodic system) has freedom of the system and p1, p2, are the reached at a particular time (taken from momenta corresponding to these degrees some reference point). Two waves are in of freedom. For example, a single particle phase if their maxima and minima coin- has three degrees of freedom (correspond- cide. ing to the three coordinates defining its po- For a simple wave represented by the sition). It also has three components of equation momentum corresponding to these degrees y = asin2π(ft – x/λ) of freedom. This means that the state of the The phase of the wave is the expression particle can be defined by six numbers (q1, π λ 2 (ft – x/ ) q2, q3, p1, p2, p3) and it is thus defined by a The phase difference between two point in six-dimensional phase space. If the points distances x1 and x2 from the origin system changes with time (i.e. the particle is changes its position and momentum), then π λ 2 (x1 – x2)/ the point in phase space traces out a path A more general equation for a progres- (known as the trajectory). The system may sive wave is consist of more than one particle. Thus, if y = asin2π[ft – (x/λ) – φ] there are N particles in the system then the Here, φ is the phase constant – the state of the system is specified by a point in phase when t and x are zero. Two waves a phase space of 6N dimensions. The idea that are out of phase have different stages of phase space is useful in chaos theory. See φ at the origin. The phase difference is 1 – also attractor. φ π λ 2. It is equal to 2 x/ , where x is the distance between corresponding points on the phase speed The speed with which the two waves. It is the phase angle between phase in a traveling wave is propagated. It the two waves; the angle between two ro- is equal to λ/T, where T is the period. Com- tating vectors (phasors) representing the pare group speed. waves. phasor /fay-zer/ See simple harmonic mo- phase angle See phase. tion. phase constant See phase. pi /pÿ/ (π) The ratio of the circumference of any circle to its diameter. π is approxi- phase difference See phase. mately equal to 3.14159… and is a transcendental number (its exact value cannot phase portrait A picture which plots the be written down, but it can be stated to any evolution of possible paths in PHASE SPACE degree of accuracy). There are several ex- that a point in that space can have, starting pressions for π in terms of infinite series. from a certain set of initial conditions. In the case of SIMPLE HARMONIC MOTION the pico- /pÿ-koh/ Symbol: p A prefix denot- phase portrait consists of a set of ellipses ing 10–12. For example, 1 picofarad (pF) = which all have the same center. In the case 10–12 farad (F). of NON-LINEAR OSCILLATIONS more complicated patterns appear in the phase portrait. pictogram /pik-tŏ-gram/ (pictograph) A The phase portrait is a convenient way of diagram that represents statistical data in

164 plot a pictorial form. For example, the pro- pixel /piks-ĕl/ See computer graphics. portions of pink, red, yellow, and white flowers that grow from a packet of mixed place value the position of an integer in a seeds can be shown by rows of the appro- number. For example, in the number 375 priate relative numbers of colored flower the 5 represents 5 units, the 7 represents 7 shapes. tens, and the 3 represents 3 hundreds. piecewise A function is piecewise contin- plan An illustration that shows the ap- uous on S if it is defined on S and can be pearance of a solid object as viewed from separated into a finite number of pieces above (vertically downward). See also ele- such that the function is continuous on the vation. interior of each piece. Terms such as piecewise differentiable and piecewise linear are planar /play-ner/ Describing something similarly defined. that occupies a plane or is flat. pie chart A diagram in which proportions plane A flat surface, either real or imagi- are illustrated as sectors of a circle, the rel- nary, in which any two points are joined by ative areas of the sectors representing the a straight line lying entirely on the surface. different proportions. For example, if, out Plane geometry involves the relationships of 100 workers in a factory, 25 travel to between points, lines, and curves lying in work by car, 50 by bus, 10 by train, and the same plane. In Cartesian coordinates, the rest walk, the bus passengers are repre- any point in a plane can be defined by two sented by half of the circle, the car passen- coordinates, x and y. In three-dimensional gers by a quarter, the train users by a 36° coordinates, each value of z corresponds to sector, and so on. a plane parallel to the plane in which the x and y axs lie. For any three points, there exists only one plane containing all three. A particular plane can also be specified by a straight line and a point.

50% plane shape A shape that exists in a bus plane, i.e. a two-dimensional shape that does not have a depth but has a width and height. Examples of plane shapes include triangles, squares, rectangles, kites, rhom- 15% 25% walk buses, and parallelograms. Plane shapes car also include curves such as circles and el- 10% lipses. train Platonic solids /plă-tonn-ik/ A name given to the set of five regular polyhedra. See polyhedron. Pie chart showing how people travel to work. plot To draw on a graph. A series of individual points plotted on a graph may show pint A unit of capacity. The US liquid pint a general relationship between the vari- is equal to one eighth of a US gallon and ables represented by the horizontal and is equivalent to 4.731 8 × 10–4 m3. In the vertical axs. For example, in a scientific ex- UK it is equal to one eighth of a UK gallon periment one quantity can be represented and is equivalent to 5.682 6 × 10–4 m3. The by x and another by y. The values of y at US dry pint is equal to one sixty-fourth of different values of x are then plotted as a a US bushel and is equivalent to 5.506 1 × series of points on a graph. If these fall on 10–4 m3. a line or curve, then the line or curve drawn

165 plotter through the points is said to be a plot of y important application of point groups is to against x. the symmetry of isolated molecules in chemistry. Examples of the symmetry op- plotter An output device of a computer erations in point groups include rotation system that produces a permanent record about a fixed axis, reflection about a plane of the results of some program by drawing of symmetry, inversion through a center of lines on paper. One pen, or maybe two or symmetry, i.e. invariance in the appearance more pens with different colored ink, are of the body when a point in the body with moved over the paper according to instruc- coordinates (x,y,z) is carried to a point tions sent from the computer or from a with coordinates (–x,–y,–z), and rotation backing store. Plotters are used for draw- about some axis followed by reflection in a ing graphs, contour maps, etc. plane perpendicular to that axis. An important set of point groups is the plotting The process of marking points set of CRYSTALLOGRAPHIC POINT GROUPS, i.e. on a system of coordinates, or of drawing the 32 point groups in three dimensions a graph by marking points. that are compatible with the translational symmetry of a crystal lattice (see space Poincaré conjecture /pwank-ka-ray/ A group). In a crystallographic point group conjecture in TOPOLOGY that can be stated only two-fold, three-fold, four-fold, and in the form that there is a HOMEOMORPHISM six-fold rotational symmetries are possible. between any n-dimensional MANIFOLD which is closed, i.e. it is like a loop and has point of contact A single point at which no end points, and is SIMPLY CONNECTED two curves, or two curved surfaces touch. and the sphere of that dimension. The There is only one point of contact between French mathematician Henri Poincaré the circumference of a circle and tangent to (1854–1912) originally made in conjecture the circle. Two spheres also can have only in three dimensions (n = 3). The more gen- one point of contact. eral version of the conjecture is sometimes known as the generalized Poincaré conjec- point of inflection A point on a curved ture. line at which the tangent changes its direc- The generalized Poincaré conjecture is readily proved for n = 1 and n = 2. It has tion of rotation. Approaching from one been proved with more difficulty in the side of the point of inflection, the slope of case of n > 4 in the 1960s. The case of n = the tangent to the curve increases; and 4 was proved, with considerable difficulty moving away from it on the other side, it 3 in the 1980s. Proving the case of n = 3, decreases. For example, the graph of y = x 2 i.e. the original Poincaré conjecture, has – 3x in rectangular Cartesian coordinates, proved to be the hardest of all dimensions has a point of inflection at the point x = 1, 2 2 and a proof of this conjecture is one of the y = –2. The second derivative d y/dx on MAGNIFICENT SEVEN PROBLEMS of mathe- the graph of a function y = f(x) is zero and matics. It has been claimed that the conjec- changes its sign at a point of inflection. ture has been proved in three dimensions Thus, in the example above, d2y/dx2 = 6x – but, at the time of writing, this has not 6, which is equal to zero at the point x = 1. been confirmed. Poisson distribution A probability dis- point A location in space, on a surface, or tribution for a discrete random variable. It in a coordinate system. A point has no di- is defined, for a variable (r) that can take mensions and is defined only by its posi- values in the range 0, 1, 2, …, and has a tion. mean value µ, as P(r) = e–µr/r! point group A set of symmetry opera- It is named for the French mathematician tions on some body in which one of the and mathematical physicist Siméon-Denis points of the body is left fixed in space. An Poisson (1781–1840).

166 polar coordinates

y 4–

3– y = x3 – 3x2 2–

1– 0 – x – – – – – –1 123 –1 –

–2 –

–3 –

–4 –

Point of inflection: in this case the point of inflection is x = 1, y = –2. The derivative dy/dx = –3 at this point.

A binomial distribution with a small traction per unit diameter, denoted c, is frequency of success p in a large number n ∆d/d. If the starting length of the rod is l of trials can be approximated by a Poisson and it is extended by ∆l then the elongation distribution with mean np. per unit length of the rod, denoted s, is given by s = ∆l/l. Poisson’s ratio is defined Poisson equation A generalization of the to be c/s. If the elasticity of a solid is inde- LAPLACE EQUATION in which there is some pendent of direction then the ratio of c/s is function f(x,y,z) on the right-hand side of 0.25. Values of the ratio differ for actual the equation. This means that the Poisson materials. For example, it is found empiri- equation has the form ∇2φ = (x,y,z), where cally that for steels the ratio is about 0.28. ∇2 is the LAPLACIAN. Here, for example, φ In the case of aluminum alloys it is about can be the gravitational potential in a re- 0.33. The value of Poisson’s ratio for a ma- gion in which there is matter, the electro- terial is constant for that material up to its static potential in a region in which there is elastic limit. If the value of Poisson’s ratio electric charge, or the temperature in a re- is less than 0.50 for a material then stretch- gion in which there is a source of heat. In ing that material increases the net volume general, the function f(x,y,z) is called the of the material. If Poisson’s ratio is exactly source density. In the examples given the 0.50 then the volume is constant. source density is respectively proportional to the density of mass, the density of elec- polar coordinates A method of defining tric charge, and the amount of heat gener- the position of a point by its distance and ated per unit time, per unit volume. The direction from a fixed reference point Poisson equation is the main equation of (pole). The direction is given as the angle POTENTIAL THEORY. between the line from the origin to the point, and a fixed line (axis). On a flat sur- Poisson’s ratio The ratio of the contrac- face only one angle, θ, and the radius, r, are tion per unit diameter of a rod that is needed to specify each point. For example, stretched to the elongation per unit length if the axis is horizontal, the point (r,θ) = of the rod. If the starting diameter of the (1,π/2) is the point one unit length away rod is d and the contraction is ∆d the con- from the origin in the perpendicular direc-

167 polar vector

radius vector OP •P(r, )

O = 0

Polar coordinates: coordinates of a point P in two dimensions. tion. Conventionally, angles are taken as notation operators precede their operands. positive in the anticlockwise sense. Thus, a + b is written +ab. The notation In a rectangular Cartesian coordinate was invented by a Polish mathematician, system with the same origin and the x-axis Jan Lukasiewicz (1878–1956). at θ = 0, the x- and y-coordinates of the point (r,θ) are polygon /pol-ee-gon/ A plane figure x = rcosθ bounded by a number of straight sides. In y = rsinθ a regular polygon, all the sides are equal Conversely and all the internal angles are equal. In a r = √(x2 + y2) regular polygon of n sides the exterior and angle is 360°/n. θ = tan–1(y/x) In three dimensions, two forms of polar polyhedron /pol-ee-hee-drŏn/ A solid fig- coordinate systems can be used. See cylin- ure bounded by plane polygonal faces. The drical polar coordinates; spherical polar point at which three or more faces intersect coordinates. See also Cartesian coordi- on a polyhedron is called a vertex, and a nates. line along which two faces intersect is called an edge. In a regular polyhedron polar vector A true vector, i.e. a vector (also known as a Platonic solid), all the which does not change sign in going from faces are congruent regular polygons. a right-handed coordinate system to a left- There are only five regular polyhedrons: handed coordinate system. Compare axial the regular tetrahedron, which has four vector. equilateral triangular faces; the regular hexahedron, or cube, which has six equi- pole 1. One of the points on the Earth’s lateral square faces; the regular octa- surface through which its axis of rotation hedron, which has eight equilateral tri- passes, or the corresponding point on any angular faces; the regular dodecahedron, other sphere. which has twelve regular pentagonal faces; 2. See stereographic projection. and the regular icosahedron, which has 3. See polar coordinates. twenty equilateral triangular faces. These are all convex polyhedrons. That is, all the Polish notation A notation in computer angles between faces and edges are convex science in which parentheses are unneces- and the polyhedron can be laid down flat sary since all formulae can be written un- on any one of the faces. In a concave poly- ambiguously without them. In Polish hedron, there is at least one face in a plane

168 potential theory that cuts through the polyhedron. The tive, and toward it if it is negative. Com- polyhedron cannot be laid down on this pare negative. face. possibility space A two-dimensional polynomial /pol-ee-noh-mee-ăl/ A sum of array that sets out all the possible out- multiples of integer powers of a variable. comes of two events. By arranging all the The general equation for a polynomial in possible outcomes systematically in this the variable x is way there is no chance of missing any of n n–1 n–2 a0x + a1x + a2x +… the possible outcomes. For example if the where a0, a1, etc., are constants and n is the possible outcomes of rolling two dice are highest power of x, called the degree of the listed as a possibility space then it can be polynomial. If n = 1, it is linear expression, seen that there are 36 possible outcomes, for example, f(x) = 2x + 3. If n = 2, it is qua- i.e. the possibility space is a 6 by 6 array. dratic, for example, x2 + 2x + 4. If n = 3, it This enables probabilities of various out- is cubic, for example, x3 + 8x2 + 2x + 2. If comes to be calculated. For example, to n = 4, it is quartic. If n = 5, it is quintic find the probability of finding a specific On a Cartesian coordinate graph on total for two dice it is possible to circle the which (n + 1) individual points are plotted, entries corresponding to that total and cal- there is at least one polynomial curve that culate the probability of that total as the passes through all the points. By choosing ratio of circled entries to the total number suitable values of a0 and a1, the straight of entries in the array. To find the proba- line bility of the total for two dice being 4 the y = a0x + a1 three entries corresponding to a total of 4 can be made to pass through any two are circled, meaning that the probability of points. Similarly a quadratic a total score of 4 is 3/36 = 1/12. 2 y = a0x + a1x + a2 can be made to pass through any three postulate See axiom. points. A polynomial may have more than one potential energy Symbol: V The work an variable: object can do because of its position or 4x2 + 2xy + y2 state. There are many examples. The work is a polynomial of degree 2 (second-degree an object at height can do in falling is its polynomial) of two variables. gravitational potential energy. The ENERGY ‘stored’ in elastic or a spring under tension polytope /pol-ee-tohp/ The analog in n di- or compression is elastic potential energy. mensions of point, line, polygon, and poly- Potential difference in electricity is a simi- hedron in 0, 1, 2, and 3 dimensions lar concept, and so on. In practice the po- respectively. tential energy of a system is the energy involved in bringing it to its current state position vector The vector that repre- from some reference state; i.e. it is the same sents the displacement of a point from a as the work that the system could do in given reference origin. If a point P in polar moving from its current state back to a ref- coordinates has coordinates (r,θ), then r is erence state. the position vector of P – a vector of magnitude r making an angle θ with the axis. potential theory The branch of mathe- See vector. matical physics that analyzes fields, such as gravitational, magnetic and electric fields, positive Denoting a number or quantity in terms of the potentials of these fields. that is greater than zero. Numbers that are The main equations used in potential the- used in counting things and measuring ory are the POISSON EQUATION and the re- sizes are all positive numbers. If a change in lated LAPLACE EQUATION. The development a quantity is positive, it increases, that is it of potential theory had a considerable in- moves away from zero if it is already posi- fluence on the development of the theory of

169 pound differential equations and the CALCULUS OF The precession of Mercury is a movement VARIATIONS. Potential theory is also related of the whole orbit of the planet around an to VECTOR CALCULUS. axis perpendicular to the orbital plane. Relativistic mechanics needs to be used to pound A unit of mass now defined as describe this precession quantitatively. 0.453 592 37 kg. precision The number of figures in a poundal Symbol: pdl The unit of force in number. For example 2.342 is stated to a the f.p.s. system. It is equal to 0.138 255 precision of four significant figures, or newton (0.138 255 N). three decimal places. The precision of a number normally reflects the ACCURACY of power 1. The number of times a quantity the value it represents. is to be multiplied by itself. For example, 24 × × × = 2 2 2 2 = 16 is known as the fourth premiss In LOGIC, an initial proposition or power of two, or two to the power four. statement that is known or assumed to be See also exponent; power series. true and on which a logical argument is 2. Symbol: P The rate of energy transfer (or based. work done) by or to a system. The SI unit of power is the watt – the energy transfer in premium 1. The difference between the joules per second. issue price of a stock or share and its nominal value when the issue price is in excess power series A series in which the terms of the nominal value. Compare discount. contain regularly increasing powers of a 2. The amount of money paid each year to variable. For example, an insurance company to purchase insur- S = 1 + 2x + 3x2 + 4x3 + n ance cover for a specified risk. … + nxn–1 is a power series in the variable x. In gen- pressure Symbol: p The pressure on a sur- eral, a power series has the form face due to forces from another surface or a + a x + a x2 + … + a xn 0 1 2 n from a fluid is the force acting at right an- where a , a , etc. are constants. 0 1 gles to unit area of the surface: pressure = force/area. P-problem A type of problem in which The unit is the pascal (Pa), equal to the the size of the problem is characterized by some number n and the number of steps newton per square meter. that an algorithm would need to solve this Objects are often designed to maximize problem is N, where n and N are related by or minimize pressure applied. To give max- N ≤ cnp, where c and p are constants. Such imum pressure, a small contact area is problems are called P-problems because needed – as with drawing pins and knives. the time it takes to solve them increases as To give minimum pressure, a large contact a polynomial of n. An example of a P-prob- area is needed – as with snowshoes and the lem is the problem of multiplying two large tires of certain vehicles. numbers together. P-problems are easier to Where the pressure on a surface is solve than NP-PROBLEMs. caused by the particles of a fluid (liquid or gas), it is not always easy to find the force precession /pri-sesh-ŏn/ If an object is on unit area. The pressure at a given depth spinning on an axis and a force is applied in a fluid is the product of the depth, the at right angles to this axis, then the axis of average fluid density, and g (the accelera- rotation can itself move around another tion of free fall): axis at an angle to it. The effect is seen in pressure in a fluid = depth × mean tops and gyroscopes, which ‘wobble’ density × g slowly while they spin as a result of the As it is normally possible to measure force of gravity. The Earth also precesses – the mean density of a liquid only, this rela- the axis of rotation slowly describes a cone. tion is usually restricted to liquids.

170 prism

The pressure at a point at a certain principal diagonal See square matrix. depth in a fluid: 1. is the same in all directions; principle of equivalence See relativity; 2. applies force at 90° to any contact sur- theory of. face; 3. does not depend on the shape of the con- principle of moments The principle that tainer. when an object or system is in equilibrium the sum of the moments in any direction pressure of the atmosphere The pres- equals the sum of the moments in the op- sure at a point near the Earth’s surface due posite direction. Because there is no resul- to the weight of air above that point. Its tant turning force, the moments of the value varies around about 100 kPa forces can be measured relative to any (100 000 newtons per square meter). point in the system or outside it. prime A prime number is a positive inte- principle of superposition The result ger which is not 1 and has no factors ex- for a system described by a linear equation cept 1 and itself. The set of prime numbers that if f and g are both solutions to the is {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …}. equation then f + g is also a solution. This There is an infinite number of prime num- principle applies to more than two solu- bers but no general formula for them. The tions. An example of the principle of su- prime factors of a number are the prime perposition is given in the theory of small numbers that divide into it exactly. For ex- (harmonic) oscillations about an equilibrium point. In the case of a vibrating string ample, the prime factors of 45 are 3 and 5 the behavior of the string can be described since 45 = 3 × 3 × 5. Each whole number by adding together all the harmonic modes has a unique set of prime factors. See Er- of oscillation, with there being no inter- atosthenes, sieve of. ference between these modes. The principle works in this example because the prime number theorem A result con- equation for small harmonic oscillations is cerning the proportion of numbers that are linear. However, the principle of super- prime numbers that is valid for large num- position does not apply to NON-LINEAR OS- bers. It states that if π(x) is the number of CILLATIONS or, more generally, systems prime numbers less than or equal to x then governed by non-linear equations. The in- →∞ π → as x , [ (x)lnx]/x 1. This means that applicability of the principle of superposi- as x becomes a large number it becomes a tion in such cases is one of the main π good approximation that (x) is given by reasons why analyzing such systems is the ratio x/lnx, i.e. the ratio of the number much more difficult than for systems de- to its natural logarithm. This result, which scribed by linear equations. was first proposed in the late 19th century, The principle of superposition also ap- has not been given a simple proof in terms plies to states in quantum-mechanical sys- of elementary mathematics. tems. primitive A particular type of n-th root of printer A computer output device that unity in which all the n roots of unity are produces hard copy as a printout. There powers of that n-th root. For example, in are various types, including daisy-wheel, the case of the fourth roots of unity i and –i dot-matrix, ink-jet, and laser. are primitive fourth roots of unity but the other two fourth roots of unity, 1 and –1, printout The computer output, in the are not primitive. form of characters printed on a continuous sheet of paper, produced by a printer. principal A sum of money that is borrowed, on which interest is charged. See prism /priz-ăm/ A polyhedron with two compound interest; simple interest. parallel opposite faces, called bases, that

171 probability

right triangular prism oblique triangular prism right hexagonal prism Prism: examples of prisms

are congruent polygons. All the other Here, 0 ≤ P(A) ≤ 1 for all A in S. If A and B faces, called lateral faces, are parallelo- are separate independent events, i.e., if grams formed by the straight parallel lines A∩B = 0, then P(A∪B) = P(A) + P(B). If between corresponding vertices of the A∩B ≠ 0 then P(A∪B) = P(A) + P(B) – bases. If the bases have a center, the line P(A∩B). joining the centers is the axis of the prism. The conditional probability is the prob- If the axis is at right angles to the base, the ability that A occurs when it is known that prism is a right prism (in which case the lat- B has occurred. It is written as eral faces are rectangles); otherwise it is an P(A|B) = P(A∩B)/P(B) oblique prism. A triangular prism has tri- If A and B are independent events, P (A|B) angular bases and three lateral faces. This = P(A) and P(A∩B) = P(A) P(B). If A and B is the shape of many of the glass prisms cannot occur simultaneously, i.e. are mutu- used in optical instruments. A quadran- ally exclusive events, P(A∩B) = 0. gular prism has a quadrilateral base and four lateral faces. The cube is a special case probability density function See ran- of this with square bases and square lateral dom variable. faces. probability function See probability. probability The likelihood of a given event occurring. If an experiment has n procedure See subroutine. possible and equally likely outcomes, m of which are event A, then the probability of processor /pros-ess-er/ See central proces- A is P(A) = m/n. For example, if A is an sor. even number coming up when a die is thrown, then P(A) = 3/6. When the proba- product The result obtained by multipli- bilities of the different possible results are cation of numbers, vectors, matrices, etc. not already known, and event A has oc- See also Cartesian product. curred m times in n trials, P(A) is defined as the limit of m/n as n becomes infinitely product formulae See addition formu- large. lae. In set theory, if S is a set of events (called the sample space) and A and B are product notation Symbol Π. The symbol events in S (i.e. subsets of S), the probabil- (a capital Greek letter pi) used to denote ity function P can be represented in set no- the product of either a finite number of tation. P(A) = 1 and P(0) = 0 mean that A terms in the sequence a1, a2, ..., an or an in- is 100% certain and the probability of finite number of terms in the sequence a1, none of the events in S occurring is zero. a2, .... 172 projective geometry program A complete set of instructions to programming language See program. a computer, written in a programming language. (The word is also used as a verb, progression See sequence. meaning to write such instructions.) These instructions, together with the facts (usu- progressive wave See wave. ally called data) on which the instructions operate, enable the computer to perform a projectile An object falling freely in a wide variety of tasks. For example, there gravitational field, having been projected are instructions to do arithmetic, to move at a speed v and at an angle of elevation θ data between the main store and the cen- to the horizontal. In the special case that θ tral processor of the computer, to perform = 90°, the motion is linear in the vertical di- logical operations, and to alter the flow of rection. It may then be treated using the control in the program. equations of motion. In all other cases the The instructions and data must be ex- vertical and horizontal components of ve- pressed in such a way that the central locity must be treated separately. In the ab- processor can recognize and interpret the sence of friction, the horizontal component instructions and cause them to be carried is constant and the vertical motion may be out on the right data. They must in fact be treated using the equations of motion. The in binary form, i.e. in a code consisting of path of the projectile is then an arc of a the binary digits 0 and 1 (bits). This binary parabola. Some useful relations are given code is known as machine code (or ma- below. chine language). Each type of computer Time to reach maximum height: has its own machine code. t = vsinθ/g It is difficult and time-consuming for Maximum height: people to write programs in machine code. h = v2sin2θ/2g Instead programs are usually written in a Horizontal range: source language, and these source pro- R = v2sin2θ/g grams are then translated into machine The last result means that the angle for code. Most source programs are written in maximum horizontal range is θ = 45°. See a high-level language and are converted also orbit. into machine code by a complicated program called a compiler. High-level lan- projection A geometrical transformation guages are closer to natural language and in which one line, shape, etc., is converted mathematical notation than to machine into another according to certain geometri- code, with the instructions taking the form cal rules. A set of points (the object) is con- of statements. They are fairly easy to use. verted into another set (the image) by the They are designed to solve particular sorts projection. See central projection; Merca- of problems and are therefore described as tor’s projection; orthogonal projection; ‘problem-orientated’. stereographic projection. It is also possible to write source programs in a low-level language. These lan- projective geometry /prŏ-jek-tiv/ The guages resemble machine code more study of how the geometric properties of a closely than natural language. They are de- figure are altered by projection. There is a signed for particular computers and are one-to-one correspondence between points thus described as ‘machine-orientated’. As- in a figure and points in its projected sembly languages are low-level languages. image, but often the ratios of lengths will A program written in an assembly lan- be changed. In central projection for exam- guage is converted into machine code by ple, a triangle maps into a triangle and a means of a special program known as an quadrilateral into a quadrilateral, but the assembler. See also routine; software; sub- sides and angles may change. See also pro- routine. jection.

173 prolate prolate /proh-layt/ Denoting a spheroid proof by contradiction See reductio ad that has a polar diameter that is greater absurdum. than the equatorial diameter. Compare oblate. See ellipsoid. proper fraction See fraction. proof A logical argument showing that a proper subset A subset S of a set T is a statement, proposition, or mathematical proper subset if there are elements of T that are not in S, i.e. S has fewer elements than formula is true. A proof consists of a set of T (if T is a finite set). basic assumptions, called axioms or premisses, that are combined according to log- proportion The relation between two sets ical rules, to derive as a conclusion the of numbers when the ratio between their formula that is being proved. A proof of a corresponding members is constant. For proposition or formula P is just a valid ar- example, the sets {2, 6, 7, 11} and {6, 18, gument from true premises to give P as a 21, 33} are in proportion (because 2:6 = conclusion. See also direct proof; indirect 6:18 = 7:21 = 11:33). See also propor- proof. tional.

center of projection

A C

image B’ plane A’ C’ image figure central projection of a triangle ABC onto a triangle A’B’C’

A C

B’ A’ C’

orthogonal projection of a triangle ABC onto a triangle A’B’C’

Projection from one plane onto another

174 propositional logic proportional Symbol: ∝ Varying in a false but not both. Any logical argument constant ratio to another quantity. For ex- consists of a succession of propositions ample, if the length l of a metal bar in- linked by logical operations with a propo- creases by 1 millimeter for every 10°C rise sition as conclusion. in its temperature T, then the length is pro- Propositions may be simple or com- portional to temperature and the constant pound. A compound proposition is one of proportionality k is 1/10 millimeter per that is made up of more than one proposi- degree Celsius; l = l0 + kT, where l0 is the tion. For example, a proposition P might initial length. If two quantities a and b are consist of the constituent parts ‘if R, then S directly proportional then a/b = k, where k or Q’; i.e. in this case P = R → (S ∨ Q). A is a constant. If they are inversely propor- simple proposition is one that is not com- tional then their product is a constant; i.e. pound. See also logic; symbolic logic. ab = k, or a = k/b. propositional calculus See symbolic proposition A sentence or formula in a logic. logical argument. A proposition can have a truth value; that is, it can be either true or propositional logic See symbolic logic.

N P

P’

S Stereographic projection of a point P on the surface of a sphere onto a plane perpendicular to the line joining the poles N and S. The image of P is P’.

N Mercator’s ø projection of P( , ) a point P x onto a point P’ in an image plane or map. S

P’ x k log (tan ø/2) k

S The image P’ of a point P in Mercator’s projection.

Projection from a sphere onto a plane

175 protractor

apex apex

triangular base square base

Pyramids protractor A drawing instrument used depend on the relative arrangement of for marking out or measuring angles. It strings and wheels. The efficiency is not usually consists of a semicircular piece of usually very high as work must be done to transparent plastic sheet, marked with ra- overcome friction in the strings and the dial lines at one-degree intervals. wheel bearings and to lift any moving wheels. See machine. pseudoscalar /soo-doh-skay-ler/ A quantity that is similar to a SCALAR quantity but pulsatance /pul-să-tăns/ See angular fre- changes sign on going from a right-handed quency. coordinate system to a left-handed coordinate system or if the order of the vectors is pulse modulation See modulation. interchanged. An example of a pseudoscalar is the quantity formed in the punched card See card. TRIPLE SCALAR PRODUCT. The dot product of any AXIAL VECTOR and a POLAR VECTOR pure mathematics The study of mathe- (true vector) results in a pseudoscalar. By matical theory and structures, without nec- contrast, the dot product of two polar vec- essarily having an immediate application in tors or two axial vectors results in a true mind. For example, the study of the general scalar, i.e. a scalar that does not change its properties of vectors, considered purely as sign on going from a right-handed coordi- entities with certain properties, could be nate system to a left-handed coordinate considered as a branch of pure mathemat- system. ics. The use of vector algebra in mechanics to solve a problem on forces or relative ve- pseudovector / soo-doh-vek-ter/ See axial locity is a branch of applied mathematics. vector. Pure mathematics, then, deals with abstract entities, without any necessary refer- Ptolemy’s theorem /tol-ĕ-meez/ See ence to physical applications in the ‘real cyclic polygon. world’. pulley A class of machine. In any pulley pyramid A solid figure in which one of system power is transferred through the the faces, the base, is a polygon and the tension in a string wound over one or more others are triangles with the same vertex. If wheels. The force ratio and distance ratio the base has a center of symmetry, a line

176 Pythagorean triple

a c

Pythagoras’ theorem: c2 = a2 + b2.

from the vertex to the center is the axis of squares of the other two sides. There are the pyramid. If this axis is at right angles to many proofs of Pythagoras’ theorem. It is the base the pyramid is a right pyramid; named for the Greek mathematician and otherwise it is an oblique pyramid. A regu- philosopher Pythagoras (c. 580 BC–c. 500 lar pyramid is one in which the base is a BC). regular polygon and the axis is at right angles to the base. In a regular pyramid all the Pythagorean triple /pÿ-thag-ŏ-ree-ăn/ A lateral faces are congruent isosceles trian- set of three positive integers a, b, c that can gles making the same angle with the base. correspond to the lengths of sides in right- A square pyramid has a square base and angled triangles and hence, by PYTHAGO- four congruent triangular faces. The vol- RAS’ THEOREM satisfy the relation a2 + b2 = ume of a pyramid is one third of the area of c2. The most familiar example of a the base multiplied by the perpendicular Pythagorean triple is (3, 4, 5). Other exam- distance from the vertex to the base. ples of Pythagorean triples include (5, 12, 13), (7, 24, 25) and (8, 15, 17). It was es- Pythagoras’ theorem /pÿ-thag-ŏ-răs-iz/ tablished by the ancient Greeks that an in- A relationship between the lengths of the finite number of Pythagorean triples exist. sides in a right-angled triangle. The square If (a, b, c) is a Pythagorean triple then (ma, of the hypotenuse (the side opposite the mb, mc), where m is a positive integer is right angle) is equal to the sum of the also a Pythagorean triple.

177 Q

2nd 1st quadrant quadrant

x O•

3rd 4th quadrant quadrant

Quadrant: the four quadrants of a Cartesian coordinate system.

quadrangular prism /kwod-rang-gyŭ-ler/ perpendicular radii and a quarter of the cir- See prism. cumference. 3. A unit of plane angle equal to 90 degrees quadrant /kwod-rănt/ 1. One of four divi- (π/2 radians). A quadrant is a right angle. sions of a plane. In rectangular CARTESIAN COORDINATES, the first quadrant is the area quadrantal spherical triangle /kwod- to the right of the y-axis and above the ran-tăl/ See spherical triangle. x-axis, that is, where both x and y are positive. The second quadrant is the area quadratic equation /kwod-rat-ik/ A to the left of the y-axis and above the polynomial equation in which the highest x-axis, where x is negative and y is posi- power of the unknown variable is two. The tive. The third quadrant is below the x- general form of a quadratic equation in the axis and to the left of the y-axis, where variable x is both x and y are negative. The fourth ax2 + bx + c = 0 quadrant is below the x-axis and to the where a, b, and c are constants. It is also right of the y-axis, where x is positive and sometimes written in the reduced form y is negative. In polar coordinates, the first, x2 + bx/a + c/a = 0 second, third, and fourth quadrants occur In general, there are two values of x that when the direction angle, θ, is 0 to 90° satisfy the equation. These solutions (or (0 to π/2); 90° to 180° (π/2 to π); 180° to roots), are given by the formula 270° (π to 3π/2); and 270° to 360° (3π/2 to x = [–b ±√(b2 – 4ac)]/2a 2π), respectively. See also polar coordi- The quantity b2 – 4ac is called the DISCRIM- nates. INANT. If it is a positive number, there are 2. A quarter of a circle, bounded by two two real roots. If it is zero, there are two

178 quantum mechanics

rectangle square kite

quadrilateral rhombus parallelogram

Quadrilateral: six types of quadrilateral. equal roots. If it is negative there are no quadric surface See conicoid. real roots. The Cartesian coordinate graph of a quadratic function quadrilateral /kwod-ră-lat-ĕ-răl/ A plane y = ax2 + bx + c figure with four straight sides. For exam- is a parabola and the points where it ple, squares, kites, rhombuses, and trapez- crosses the x-axis are solutions to iums are all quadrilaterals. A square is a ax2 + bx + c = 0 regular quadrilateral. If it crosses the axis twice there are two real roots, if it touches the axis at a turning quantifier /kwon-tă-fÿ-er/ See existential point the roots are equal, and if it does not quantifier; universal quantifier. cross it at all there are no real roots. In this last case, where the discriminant is nega- quantum computer A computer that op- tive, the roots are two conjugate complex erates by using the principles of QUANTUM numbers. MECHANICS. In particular, a quantum computer could process alternative pathways quadratic graph A graph of the curve y = simultaneously. It is possible that for many ax2 + bx + c, where a, b, and c are con- problems, such as factorizing large inte- stants. If a is a positive number then the gers, a quantum computer could perform graph has a minimum when x = –b/2a. It calculations much more quickly than a has a maximum when x = –b/2a if a is a conventional computer. At the time of negative number. There is only one sta- writing, quantum computers have not been tionary point in a quadratic graph. This developed to the point of being technolog- can either be a minimum or a maximum, as ically useful. Several different quantum specified. See also quadratic equation. mechanical systems have been proposed and used as quantum computers but they quadric cone /kwod-rik/ A type of all have practical difficulties such as re- QUADRIC SURFACE that is described by the quiring very low temperatures for their op- equation: eration. x2/a2 + y2/b2 = z2/c2, using a particular coordinate system. If a quantum mechanics The theory that cross-section of this surface is taken paral- governs the behavior of particles at the lel to the xy-plane an ellipse results in gen- atomic and sub-atomic level. It bears the eral. In the specific case of a = b, a circle same relationship to Newtonian mechanics results. Cross-sections parallel to the as wave optics does to geometrical optics. planes of the other axes result in hyperbo- The development of quantum mechanics las. has had a substantial influence on several

179 quart branches of mathematics, notably the the- quartile One of the three points that di- ory of operators and group theory. Quan- vide a set of data arranged in numerical tum mechanics can be formulated order into four equal parts. The lower (or th mathematically in several ways including first) quartile, Q1, is the 25 PERCENTILE the use of differential equations, matrices, (P25). The middle (or second) quartile, Q2, variational principles, and information is the MEDIAN (P50). The upper (or third) th theory. Characteristic features of quantum quartile, Q3, is the 75 percentile (P75). mechanics are that many quantities have a set of discrete values rather than a range of quasicrystalline symmetry /kway-sÿ- continuous values and that particles such kriss-tă-lin, -lÿn, kway-zÿ-, kwah-see-, kwah-zee- as electrons cannot be ascribed precisely / The symmetry that is the three- dimensional analog of a PENROSE PATTERN, defined trajectories like the orbits of plan- i.e. there is order but not the periodicity of ets. A further feature is the existence of crystals. Quasicrystalline symmetry is real- non-commutative operators, i.e. pairs of ized in certain solids, called quasicrystals, operators in which the final result of the such as AlMn. This symmetry is made ap- operations depends on the order in which parent by the fact that quasicrystals give the operators operate. At the time of writ- clear x-ray diffraction images, characteris- ing, there is a great deal of activity in de- tic of there being order. Quasicrystals such veloping quantum, i.e. discrete and as AlMn have the point group symmetry of non-commutative, versions of mathematics an ICOSAHEDRON, a type of point group which have been previously formulated for symmetry that is incompatible with crys- continuous, commutative quantities. Some talline symmetry. of these branches of ‘quantum mathematics’ may well turn out to be important in quaternions /kwă-ter-nee-ŏnz/ General- physics. ized complex numbers invented by Hamil- ton. A quaternion is of the form a + bi + cj quart A unit of capacity equal to two + dk, where i2 = j2 = k2 = –1 and ij = –ji = k pints. In the USA a dry quart is equal to and a, b, c, d are real numbers. The most two US dry pints. striking feature of quaternions is that multiplication is not commutative. They have quartic equation /kwor-tik/ A polyno- applications in the study of the rotations of mial equation in which the highest power rigid bodies in space. of the unknown variable is four. The general form of a quartic equation in a vari- quintic equation /kwin-tik/ A polynomial able x is equation in which the highest power of the ax4 + bx3 + cx2 + dx + e = 0 unknown variable is five. The general form where a, b, c, d, and e are constants. It is of a quintic equation in a variable x is: ax5 + bx4 + cx3 + dx2 + ex + f = 0 also sometimes written in the reduced form where a, b, c, d, e, and f are constants. It is x4 + bx3/a + cx2/a + dx/a + e/a = 0 also sometimes written in the reduced form In general, there are four values of x that x5 + bx4/a + cx3/a + dx2/a satisfy a quartic equation. For example, 4 3 2 + ex/a + f/a = 0 2x – 9x + 4x + 21x – 18 = 0 In general, there are five values of x that can be factorized to satisfy a quintic equation. For example, (2x + 3)(x – 1)(x – 2)(x – 3) = 0 2x5 – 17x4 + 40x3 + 5x2 – 102x + 72 = and its solutions (or roots) are –3/2, 1, 2, 0 and 3. On a Cartesian coordinate graph, can be factorized to the curve (2x + 3)(x – 1)(x – 2)(x – 3)(x – 4) = 0 y = 2x4 – 9x3 + 4x2 + 21x – 18 = 0 and its solutions (or roots) are –3/2, 1, 2, 3, crosses the x-axis at x = –3/2; x = 1; x = 2; and 4. On a Cartesian coordinate graph, and x = 3. Compare cubic equation; qua- the curve dratic equation. y = 2x5 – 17x4 + 40x3 +

180 qwerty

5x2 – 102x + 72 may not be a remainder. For example, 16/3 crosses the x-axis at x = –3/2; x = 1; x = 2; gives a quotient of 5 and a remainder of 1. x = 3; and x = 4. Compare cubic equation; quadratic equation; quartic equation. qwerty /kwer-tee/ Describing the standard layout of ALPHANUMERIC characters on a quotient /kwoh-shĕnt/ The result of divid- typewriter or computer keyboard (named ing one number by another. There may or for the first six letters of the top letter row).

181 R

radial /ray-dee-ăl/ Along the direction of series converges for values of |x − a| < R. the radius. Here, R is the radius of convergence of the power series. radial symmetry Symmetry about a line or plane through the center of a shape. See radius of curvature See curvature. also bilateral symmetry; symmetrical. radius of gyration Symbol: k For a body radian Symbol: rad The SI unit for mea- of mass m and moment of inertia I about suring plane angle. It is the angle subtended an axis, the radius of gyration about that at the center of a circle by an arc equal in axis is given by π length to the radius of the circle. radians k2 = I/m ° = 180 . In other words, a point mass m rotating at a distance k from the axis would have the radical /rad-ă-kăl/ An expression for a same moment of inertia as the body. root. For example, √2, where √ is the radical sign. radius vector The vector that represents the distance and direction of a point from radical axis The radical axis of two cir- the origin in a polar coordinate system. cles x2 + y2 + 2a x + 2b y + c = 0 1 1 1 radix /ray-diks/ (pl. radices or radixes) and The base number of any counting system, x2 + y2 + 2a x + 2b y + c = 0 2 2 2 also known as the base. For example, the is the straight line obtained by eliminating decimal system has the radix 10, whereas the square terms between the equations of the radix of the binomial system is 2. See the circles, i.e. also base. 2(a1 – a2)x + 2(b1 – b2)y + (c1 – c2) = 0 When the circles intersect, the radical axis RAM Random-access memory. See ran- passes through their two points of intersec- dom access. tion. random access A method of organizing radius /ray-dee-ŭs/ (pl. radii or radiuses) information in a computer storage device The distance from the center of a circle to so that one piece of information may be any point on its circumference or from the reached directly in about the same time as center of a sphere to its surface. In polar any other. Main store and disk units oper- coordinates, a radius r (distance from a ate by random access and are thus known fixed origin) is used with angular position as random-access memory (RAM). In con- θ to specify the positions of points. trast a magnetic tape unit operates more slowly by serial access: a particular piece of radius of convergence For a power se- information can only be retrieved by work- 2 ries a0 + a1(x – a) + a2(x – a) + … + an(x – ing through the preceding blocks of data a)n + … there is a value R such that the on the tape.

182 rate of change random error See error. distance of the point from the starting point after some fixed time. Technically a randomness The lack of a pattern or random walk is a sequence order in either a physical system or a set of Sn = X1 + X2 + … + Xn numbers. Precise mathematical analysis of where {Xi} is a sequence of independent the concept of randomness is closely asso- random variables. ciated with CHAOS THEORY, ENTROPY, and INFORMATION THEORY. range 1. The difference between the largest and smallest values in a set of data. random number table A table consisting It is a measure of dispersion. In terms of of a sequence of randomly chosen digits percentiles, the range is (P100 – P0). Com- from 0 to 9, where each digit has a proba- pare interquartile range. bility of 0.1 of appearing in a particular po- 2. A set of numbers or quantities that form sition, and choices for different positions possible results of a mapping. In algebra, are independent. Random numbers are the range of a function f(x) is the set of val- used in statistical random sampling. ues that f(x) can take for all possible values of x. For example, if f(x) is taking the random sampling See sampling. square root of positive rational numbers, then the range would be the set of real random variable (chance variable; sto- numbers. See also domain. chastic variable) A quantity that can take any one of a number of unpredicted values. rank A method of ordering a set of objects A discrete random variable, X, has a defi- according to the magnitude or importance nite set of possible values x , x , x , … x , 1 2 3 n of a variable measured on them, e.g. with corresponding probabilities p , p , p , 1 2 3 arranging ten people in order of height. If … p . Since X must take one of the values n the objects are ranked using two different in this set, variables, the degree of association be- p + p + … + p = 1 1 2 n tween the two rankings is given by the co- If X is a continuous random variable, it efficient of rank correlation. See also can take any value over a continuous Kendall’s method. range. The probabilities of a particular value x occurring is given by a probability density function f(x). On a graph of f(x) raster graphics See computer graphics. against x, the area under the curve between two values a and b is the probability that X rate of change The rate at which one lies between a and b. The total area under quantity y changes with respect to another the curve is 1. quantity x. If y is written as y = f(x) and if f(x) is a differentiable function, the rate of random walk A succession of move- change of y with respect to x is described ments along line segments where the direc- by differential calculus. This rate of change ′ tion and the length of each move is is denoted by dy/dx or f (x). The value of randomly determined. The problem is to the rate of change varies with x since dy/dx determine the probable location of a point is a function of x. The value of dy/dx at a subject to such random motions given the specific value of x is found by expressing probability of moving some distance in dy/dx as a function of x and substituting some direction, where the probabilities are the value of x into the function for the rate the same at each step. Random walks can of change. be used to obtain probability distributions Frequently in physical systems the rate to practical problems. Consider, for exam- of change of a function with respect to time ple, a drunk man moving a distance of one is of interest. For example, the velocity of a unit in unit time, the direction of motion body is the rate of change of the displace- being random at each step. The problem is ment of the body with respect to time and to find the probability distribution of the the acceleration of a body is the rate of

183 ratio change of the velocity of the body with re- specified by position vectors a and b, rela- spect to time. tive to some origin O, and the point C divides the line AB in the ratio l:m then the ratio /ray-shee-oh/ One number or quan- position vector c for C is given by: c = (ma tity divided by another. The ratio of two + lb)/(l + m). By defining l and m so that l + variable quantities x and y, written as x/y m = 1, the expression c = ma + (1 – m)b is or x:y, is constant if y is proportional to x. obtained. See also fraction. ray A set consisting of all the points on a rational function A REAL FUNCTION, f(x), given line to the left or to the right of a of a real variable x in some domain, which given point, and including that point itself. is frequently taken to be the set of all real numbers, that can be expressed as the ratio reaction Newton’s third law of force of two polynomial functions g(x) and h(x). states that whenever object A applies a In this definition it is assumed that the two force on object B, B applies the same force polynomials do not have a common factor on A. An old word for force is ‘action’; ‘re- with a degree that is greater than or equal action’ is thus the other member of the to 1. A rational function defined in this pair. Thus in the interaction between two way is continuous except for values of x for electric charges, each exerts a force on the which the value of the denominator h(x) is other. Thus, in general, action and reaction zero. The concept of a rational function have little meaning. The word ‘reaction’ is can be extended to a function, f(z), of a still sometimes used in restricted cases, complex variable z. such as the reaction of a support on the object it supports. In this case the ‘action’ is rationalize /rash-ŏ-nă-lÿz/ To remove the effect of the weight of the object on the radicals (such as square root, or √) from an support. algebraic expression without changing its value, thus making the equation easier to reader A device used in a computer sys- deal with. For example, by squaring both tem to sense the information recorded on sides, the equation √(2x + 3) = x rational- some source and convert it into another izes to 2x + 3 = x2, equivalent to x2 – 2x – form. 3 = 0. read-only memory (ROM) See store. rationalized units A system of units in which the equations have a logical form re- read–write head See disk; drum; mag- lated to the shape of the system. SI units netic tape. form a rationalized system of units. For example, in it formulae concerned with circu- real analysis The branch of analysis con- lar symmetry contain a factor of 2π; those cerned with REAL FUNCTIONS. concerned with radial symmetry contain a factor of 4π. real axis An axis of the complex plane on which the points represent the real num- rational numbers /rash-ŏ-năl/ Symbol: Q bers. It is customary to draw the real axis The set of numbers that includes integers as the x-axis of the complex plane. and fractions. Rational numbers can be written down exactly as ratios or as finite real function A function f that maps or repeating decimals. For example, ⅓ (= members of the set R of real numbers (or a 0.333…) and ¼ (= 0.25) are rational. The subset of R) to members of R. This means square root of 2 (= 1.414 213 6…) is not. that if x is a real number in R or its subset Compare irrational numbers. then f(x) is also a real number. ratio theorem A result concerning vec- real line The line that represents the real tors stating that if two points A and B are numbers on the complex plane. The real

184 reductio ad absurdum line is drawn as a horizontal line. A specific rectangular hyperbola /rek-tang-gyŭ-ler/ point O is chosen to be the origin and an- See hyperbola. other specific point A, which is to the right of O on the horizontal line, is chosen so rectangular parallelepiped See paral- that the length of OA represents one unit. lelepiped. All positive real numbers are represented by points that lie to the right of O on the rectilinear /rek-tă-lin-ee-er/ Describing real line and all negative real numbers are motion in a straight line. represented by points that lie to the left of O, with the origin being the real number recurring decimal A repeating DECIMAL. with value zero. recursion formula A formula that relates real numbers Symbol: R The set of num- some quantity Qn, where n is a non-nega- bers that includes all rational and irra- tive integer, to quantities such as Qn–1, tional numbers. Qn–2, .... Formulas of this type are used extensively to calculate integrals involving real part Symbol Rez. The part x of a powers of trigonometric functions. For ex- ∫ n complex number z defined by z = x + iy, ample if In = sin xdx, there is a recursion where x and y are both real numbers. See formula for In which can be found by inte- also imaginary part. grating In by parts. This gives: In = ≥ [(n–1)/n]In–2, where n 2. The cases of I0 real time The actual time in which a phys- and I1 are readily found by direct integra- ical process takes place or in which a phys- tion. The recursion formula means that In can be determined for all even and odd val- ical process, machine, etc., is under the ues of n. Another example of a recursion direct control of a computer. A real-time formula is the expression Γ(n + 1) = nΓ(n) system is able to react sufficiently rapidly for the GAMMA FUNCTION. so that it may control a continuing process, A recursion formula is sometimes called making changes or modifications when a reduction formula. necessary. Air-traffic control and airline reservations require real-time systems. reduced form (of a polynomial) The Compare batch processing. See also time EQUATION of the form sharing. xn + (b/a)xn–1 + (c/a)xn–2 + … = 0 that is derived from a POLYNOMIAL of the reciprocal /ri-sip-rŏ-kăl/ The number 1 form divided by a quantity. For example, the axn + bxn–1 + cxn–2 + … = 0 ½ 2 reciprocal of 2 is . The reciprocal of (x + For example, 2 1) is 1/(x + 1). The product of any expres- 2x2 – 10x + 12 = 0 sion and its reciprocal is 1. For any func- is equivalent to the reduced form tion, the reciprocal is the multiplicative x2 – 5x + 6 = 0 inverse. See also quadratic equation. rectangle /rek-tang-găl/ A plane figure reductio ad absurdum /ri-duk-tee-oh ad with four straight sides, two parallel pairs ăb-ser-dŭm/ A method of proof which proof equal length forming four right angles. ceeds by assuming the falsity of what we The area of a rectangle is the product of the wish to prove and showing that it leads two different side lengths, the length times to a contradiction. Hence the statement the breadth. A rectangle has two axes of whose falsity we assumed must be true. symmetry, the two lines joining the mid- The following proof that √2 is irrational is points of opposite sides. It can also be su- a simple example of proof by this method. perimposed on itself after rotation through Assume √2 is rational. In that case it can 180° (π radians). The two diagonals of a be expressed in the form a/b where a and b rectangle have equal lengths. are integers. Assume that this fraction is in

185 reduction formulae its lowest terms and so a and b have no would be point P′ at the same distance common factor. Since a/b = √2 then a2/b2 = from the line but on the other side. The 2. Hence a2 = 2b2. This means that a2 is line, the axis of reflection, is the perpendic- even and hence a itself is even. In that case ular bisector of the line PP′. In a symmetri- we can write a as 2m where m is some in- cal plane figure there is an axis of teger. But then since a2 = 2b2 we have reflection, also called the axis of symmetry, (2m)2 = 2b2, or 4m2 = 2b2. Dividing by 2 in which the figure is reflected onto itself. we get 2m2 = b2. But this means that b2 and An equilateral triangle, for example, has hence b is also even. Hence, a and b do three axes of symmetry. In a circle, any di- have a common factor, namely 2. But we ameter is an axis of symmetry. Similarly, a assumed they had no common factor. Since solid may undergo reflection in a plane. In we have reached a contradiction, our start- a sphere, any plane passing through the ing-point – the assumption that √2 is ratio- sphere’s center would be a plane of sym- nal – must be false. metry. In a Cartesian coordinate system, re- reduction formulae In trigonometry, the flection in the x-axis changes the sign of the equations that express sine, cosine, and y-coordinate. A point (a,b) would become tangent functions of an angle in terms of an (a,–b). Reflection in the y-axis changes the angle between 0 and 90° (π/2). For exam- sign of the x-coordinate, making (a,b) be- ple: come (–a,b). In three dimensions, changing sin(90° + α) = cosα the sign of the z-coordinate is equivalent to sin(180° + α) = –sinα reflection in the plane of the x and y axes. sin(270° + α) = –cosα Reflection in a point is equivalent to rota- cos(90° + α) = –sinα tion through 180°. Each point P is moved tan(90° + α) = –cotα to a position P′ so that the point of reflection bisects the line PP′. Reflection in the re-entrant polygon A many-sided figure origin of plane Cartesian coordinates (polygon) with one of its internal angles a changes the signs of all the coordinates. It reflex angle (i.e. greater than 180°). is equivalent to reflection in the x-axis followed by reflection in the y-axis, or vice reflection /ri-flek-shŏn/ The geometrical versa. See also rotation. transformation of a point or a set of points from one side of a point, line, or plane to a reflex /ree-fleks/ Describing an angle that symmetrical position on the other side. On is greater than 180° (but less than 360°). reflection in a line, the image of a point P Compare acute; obtuse.

A´ A

C´ C

B´ B

O x Reflection in the y-axis

186 relativity, theory of reflexive /ri-fleks-iv/ A relation R defined relative velocity If two objects are mov- on a given set is said to be reflexive if every ing at velocities vA and vB in a given direc- member of the set has this relation to itself. tion the velocity of A relative to B is vA – vB Equality is an example of a reflexive rela- in that direction. In general, if two objects tion. are moving in the same frame at nonrela- tivistic speeds their relative velocity is the register See central processor. vector difference of the two velocities. regression line A line y = ax + b, called relativistic mass /rel-ă-ti-vis-tik/ The the regression line of y on x, which gives mass of an object as measured by an ob- the expected value of a random variable y server at rest in a frame of reference in conditional on the given value of a random which the object is moving with a velocity variable x. The regression line of x on y v. It is given by √ 2 2 is not in general the same as that of y on x. m = m0/ (1 – v /c ) If a SCATTER DIAGRAM of data points where m0 is the REST MASS, c is the velocity of light in free space, and m is the relativis- (x1,y1), …, (xn,yn) is drawn and a linear relationship is shown up, the line can be tic mass. The equation is a consequence of drawn by hand. The best line is drawn the special theory of relativity, and is in ex- cellent agreement with experiment. No ob- using the LEAST-SQUARES METHOD. See also correlation. ject can travel at the speed of light in free space because its mass would then be infi- regular Describing a figure that has all nite. See also relativity, theory of. faces or sides of equal size and shape. See relativistic mechanics A system of me- polygon; polyhedron. chanics based on relativity theory. See also classical mechanics. relation A property that holds for ordered pairs of elements of some set, for example relativistic speed (relativistic velocity) being greater than. We can think of a rela- Any speed (velocity) that is sufficiently tion abstractly as the set of all ordered high to make the mass of an object signifi- pairs in which the two members have the cantly greater than its REST MASS. It is usu- given relation to one another. ally expressed as a fraction of c, the speed of light in free space. At a speed of c/2 the relative Expressed as a difference from or RELATIVISTIC MASS of an object is about as a ratio to, some reference level. Relative 15% greater than the rest mass. density, for example, is the mass of a substance per unit volume expressed as a frac- relativity, theory of /rel-ă-tiv-ă-tee/ A tion of a standard density, such as that of theory put forward in two parts by Albert water at the same temperature. Compare Einstein. The spcecial theory (1905) re- absolute. ferred only to nonaccelerated (inertial) frames of reference. The general theory relative error The ERROR or uncertainty (1915) is also applicable to accelerated sys- in a measurement expressed as a fraction of tems. the measurement. For example, if, in mea- The special theory was based on two suring a length of 10 meters, the tape mea- postulates: sures only to the nearest centimeter, then 1. That physical laws are the same in all in- the measurement might be written as 10 ± ertial frames of reference. 0.01 meters. The relative error is 0.01/10 = 2. That the speed of light in a vacuum is 0.001. Compare absolute error. constant for all observers, regardless of the motion of the source or observer. relative maximum See local maximum. The second postulate seems contrary to ‘common sense’ ideas of motion. Einstein relative minimum See local minimum. was led to the theory by considering the

187 remainder problem of the ‘ether’ and the relation be- f(x) = (x – a)g(x) + f(a) tween electric and magnetic fields in rela- This means that if a polynomial in x, f(x), tive motion. The theory accounts for the is divided by (x – a), where a is a constant, negative result observed in the MICHELSON– the remainder term is equal to the value of MORLEY EXPERIMENT and shows that the the polynomial when x = a. For example, if Lorentz–Fitzgerald contraction is only an 2x3 + 3x2 – x – 4 apparent effect of motion on an object rel- is divided by (x – 4), then the remainder ative to an observer, not a ‘real’ contrac- term is tion. It leads to the result that the mass of f(4) = 128 + 48 – 4 – 4 = 168 an object moving at a speed v relative to an The remainder theorem is useful for find- observer is given by: ing the factors of a polynomial. In this ex- √ 2 2 m = m0/ (1 – v /c ) ample, where c is the speed of light and m0 the f(1) = 2 + 3 – 1 – 4 = 0 mass of the object when at rest relative to Thus, there is no remainder so (x – 1) is a the observer. The increase in mass is signif- factor. icant at high speeds. Another consequence of the theory is that an object has an energy repeated root A root of an equation that content by virtue of its mass, and similarly occurs more than once. that energy has inertia. Mass and energy are related by the famous equation E = repeating decimal See decimal. mc2. The general theory of relativity seeks to representation of a group A set of oper- explain the difference between accelerated ators that correspond to the elements of the and nonaccelerated systems and the nature GROUP. The operators are defined in a VEC- of the forces acting in both of them. For ex- TOR SPACE V. The dimension of the rep- ample, a person in a spacecraft far out in resentation is the dimensionality of V. space would not be subject to gravitational Frequently, the set of operators is ex- forces. If the craft were rotating, he would pressed in terms of matrices, with this type be pressed against the walls of the craft and of representation being called a matrix rep- would consider that he had weight. There resentation of the group. Group represen- would not be any difference between this tations are of great importance in QUAN- force and the force of gravity. To an out- TUM MECHANICS. In particular, irreducible side observer the force is simply a result of representations, i.e. representations that the tendency to continue in a straight line; cannot be reduced to representations of i.e. his inertia. This type of analysis of lower dimensions, characterize quantum forces led Einstein to a principle of equiva- mechanical systems since the quantum lence that inertial forces and gravitational numbers that characterize the energy levels forces are equivalent, and that gravitation of a system correspond to the irreducible can be a consequence of the geometrical representations of the group that describes properties of space. He visualized a four- the symmetry of the system. For instance, dimensional space–time continuum in the irreducible representations of the ROTA- which the presence of a mass affects the TION GROUP characterize the energy levels geometry – the space-time is ‘curved’ by the of atoms and the irreducible representa- mass with the geometry being non-Euclid- tions of POINT GROUPS characterize the en- ean. ergy levels of molecules. remainder The number left when one representative fraction A fraction used number is divided into another. Dividing to express the SCALE of a map in which the 12 into 57 gives 4 remainder 9 (4 × 12 = 48; numerator represents a distance on the 57 – 48 = 9). map and the denominator represents the corresponding distance on the ground. As remainder theorem The theorem ex- a fraction is a ratio, the units of the numer- pressed by the equation ator and denominator must be the same.

188 Riemann sum

For example, a scale of 1 cm = 1 km would resultant of a set of vectors can be found by be given as a representative fraction of different methods. See parallel forces; par- 1/100 000, because there are 100 000 cm allelogram of vectors; principle of mo- in 1 km. ments. 2. See eliminant. residue /rez-ă-dew/ If there exists an x such that xn ≡ a(mod p), i.e. xn is congruent revolution, solid of A solid generated by to a modulo p, then a is called a residue of revolving a plane area about a line called p of order n. the axis of revolution. For example, rotating a rectangle about an axis joining the resolution of vectors The determination midpoints of two opposite sides produces a of the components of a vector in two given cylinder as the solid of revolution. directions at 90°. The term is sometimes used in relation to finding any pair of com- rhombic dodecahedron /rom-bik/ A ponents (not necessarily at 90° to each type of polyhedron in which there are 12 other). faces, each of which is a RHOMBUS. resonance The large-amplitude vibration rhombohedron /rom-bŏ-hee-drŏn/ A of an object or system when given impulses solid figure bounded by six faces, each one at its natural frequency. For instance, a a parallelogram, with opposite faces con- pendulum swings with a natural frequency gruent. that depends on its length. If it is given a periodic ‘push’ at this frequency – for ex- rhomboid /rom-boid/ A parallelogram ample, at each maximum of a complete os- that is neither a rhombus nor a rectangle. cillation – the amplitude is increased with See parallelogram. little effort. Much more effort would be required to produce a swing of the same am- rhombus /rom-bŭs/ (pl. rhombuses or plitude at a different frequency. rhombi) A plane figure with four straight sides of equal length; i.e. a parallelogram restitution, coefficient of /res-tă-tew- with equal sides. Its area is equal to half the shŏn/ Symbol: e For the impact of two product of the lengths of its two diagonals, bodies, the elasticity of the collision is mea- which bisect each other perpendicularly. sured by the coefficient of restitution. It is The rhombus is symmetrical about both of the relative velocity after collision divided its diagonals and also has rotational sym- by the relative velocity before collision metry, in that it can be superimposed on (with the velocities measured along the line itself after rotation through 180° (π radi- of centers). For spheres A and B: ans). ′ ′ vA – vB = e(vA – vB) v indicates velocity before collision; v′ ve- Riemannian geometry /ree-mah-nee-ăn/ locity after collision. Kinetic energy is con- A type of geometry that describes the served only in a perfectly elastic collision. higher-dimensional analogs of curved surfaces. When extended from space to space– rest mass Symbol: m0 The mass of an ob- time, Riemannian geometry can be used to ject at rest as measured by an observer at describe the effect of gravitation in the rest in the same frame of reference. See also general theory of relativity. This type of relativistic mass. geometry is named for the German mathematician (Georg Friedrich) Bernhard Rie- resultant /ri-zul-tănt/ 1. A vector with the mann (1826–66). same effect as a number of vectors. Thus, the resultant of a set of forces is a force that Riemann integral /ree-mahn/ See definite has the same effect; it is equal in magnitude integral; Riemann sum. and opposite in direction to the equilibrium. Depending on the circumstances, the Riemann sum The series that approxi-

189 Riemann zeta function mates the area between the curve of a func- ing a coordinate system such that rotating tion f(x) and the x-axis: the x-axis by 90° into the y-axis corre- n sponds to rotating a right-handed screw in ∑ ξ ∆ f( i) xi the positive z direction. i =1 When discussing VECTOR PRODUCTS it is ∆ ξ where x is an increment of x, i is any necessary to specify whether a right- value of f(x) within that interval, and n is handed or left-handed coordinate system is the number of intervals. The definite (or being used. Riemann) integral is the limit of the sum as n becomes infinitely large and ∆x infinites- right solid A solid geometrical figure that imally small. is upright; for example, a cone, cylinder, pyrimid, or prism that has an axis at right Riemann zeta function /zay-tă/ A func- angles to the base. Compare oblique solid. tion of a complex variable z = x + iy, where x and y are real numbers, defined by the in- rigid body In mechanics, a body for finite series: which any change of shape produced by ∞ forces on the body can be neglected in the ζ(z) = ∑ n–z calculations. n =1 The Riemann zeta function can also be de- ring A set of entities with two binary op- fined for real numbers. There is a conjec- erations called addition and multiplication ture concerning the Riemann zeta function and denoted by + and • respectively, such for a complex variable z that has remained that: unproved since Riemann postulated it in 1. the set is a commutative group under ad- the nineteenth century. If Riemann’s condition; jecture is correct it would establish a firm 2. for every pair of elements a,b, in the ring result about the distribution of prime num- the product a•b is unique, multiplication is bers. associative, i.e. (a•b)•c = a•(b•c), and multiplication is distributive with respect to right angle An angle that is 90° or π/2 ra- addition, i.e. a•(b + c) = a•b + a•c and (b + dians. It is the angle between two lines or c)•a = b•a + c•a for each a, b, and c in the planes that are perpendicular to each set. other. The corner of a square, for example, If multiplication is also commutative, is a right angle. the ring is called a commutative ring. For example, the set of real numbers, the set of right-handed system A way of specify- integers, and the set of rational numbers

z z

k k

i j x y

j i

y x Right-handed system Left-handed system

190 Roman numerals are rings with respect to ordinary addition sections made of different woods. A light and multiplication. rod is a rod in which the mass of the rod is neglected. The concept of a light rod is use- robot A feedback-controlled mechanical ful when considering two bodies connected device. Robotics is the study of the design, by a rod, with the masses of both the bod- applications, and control and sensory sys- ies being much greater than the mass of the tems of robots; for example, the design of rod. robot arms that can approach an object from any orientation and grip it. A robot’s Rolle’s theorem /rol-ĕz/ A curve that in- control system may be simple and consist tersects the x-axis at two points a and b, is of only a sequencing device so that the de- continuous, and has a tangent at every vice moves in a repetitive pattern, or more point between a and b, must have at least sophisticated so that the robot’s move- one point in this interval at which the tan- ments are generated by computer from gent to the curve is horizontal. For a curve data about the environment. The robot’s y = f(x), it follows from Rolle’s theorem sensory system gathers information needed that the function f(x) has a turning point (a by the control system, usually visually by maximum or minimum value) between f(a) using a television camera. and f(b), where the derivative f′(x) = 0. The theorem is named for the French mathe- rod An object that is considered to have a length but no breadth or depth, with all the matician Michel Rolle (1652–1719). See mass of the object concentrated along the also turning point. length. Although this definition is an ideal- ization it is a reasonably good approxima- rolling friction See friction. tion for objects in which the length is much bigger than both the breadth and depth. Roman numerals The system of writing The pole of a broom can be modelled fairly integers that was used by the ancient Ro- accurately by regarding it as a rod. In a uni- mans, in which I denotes 1, V denotes 5, X form rod equal lengths of the rod have denotes 10, L denotes 50, C denotes 100, D equal masses. A broom pole is an example denotes 500, and M denotes 1000. The in- of a uniform rod if it is made of the same tegers are written using the following rules: wood and has the same cross-sectional (1) the values of the letters are added if a area all along the length of the rod. A non- letter is repeated or immediately followed uniform rod is a rod in which equal lengths by a letter of lesser value; of the rod do not have equal masses. An ex- (2) the value of the letter of smallest value ample would be an object in two connected is subtracted from the value of the letter of

horizontal C• tangent at C

x O ab

Rolle’s theorem for a function f(x) that is continuous between x = a and x = b and for which f(a) = f(b) = 0

191 root greater value when a letter is immediately roots are both real numbers, even if the followed by a letter of greater value. roots are complex numbers. This is the case There is no symbol for zero. The inte- because complex roots occur in pairs that gers from 1 to 10 are written I, II, III, IV, are complex conjugates of each other. The V, VI, VII, VIII, IX, X; and, for example, solutions of cubic and higher power equa- 1987 is written MCMLXXXVII. tions cannot, in general, be found by using the relationships between roots and coeffi- root In an equation, a value of the inde- cients. pendent variable that satisfies the equation. In general, the degree of a rose A curve obtained by plotting the POLYNOMIAL is equal to the number of equation roots. A QUADRATIC EQUATION (one of de- r = asinnθ gree two) has two roots, although in some in polar coordinates (a is a real-number circumstances they may be equal. For a constant and n is an integer constant). It number a, an nth root of a is a number that has a number of petal-shaped loops, or satisfies the equation xn = a leafs. When n is even there are 2n loops See also discriminant. and when n is odd there are n loops. For example, the graph of r = asin2θ is a four- root-mean-square (rms) For a number of leafed rose. values, an average equal to the square root of (the sum of the squares of the values di- rotation A geometrical transformation in vided by the number of values). For exam- which a figure is moved rigidly around a ple, the rms of 2, 3, 4, 5 is √(54/4) = 3.674. fixed point. If the point, the center of rotation, is labelled O, then for any point P in roots and coefficients Relations be- the figure, moving to point P′ after rota- tween the roots of a polynomial equation tion, the angle POP′ is the same for all and the coefficients of that equation. If the points in the figure. This angle is the angle roots of the quadratic equation ax2 + bx + of rotation. Some figures are unchanged by c = 0 are denoted by α and β then one has certain rotations. A circle is not affected by the relations α + β = –b/a and αβ = c/a. If any rotation about its center. A square the roots of the cubic equation ax3 + bx2 + does not change if it is rotated through 90° α β γ cx + d = 0 are written , , and , then b/a about the point at which its diagonals α β γ αβ βγ γα = –( + + ), c/a = + + , d/a = cross. Similarly an equilateral triangle is αβγ α β γ – . If one uses the notation + + = unchanged by rotation through 120° about Σα αβ βγ γα Σαβ and + + = these results can its centroid. These properties are known as be summarized in the form: the rotational symmetry of the figure. See Σα = –b/a, Σαβ = c/a. also rotation of axes; transformation. If the roots of the quartic equation ax4 + bx3 + cx2 + dx + e = 0 are denoted by α, β, rotational motion Motion of a body γ and δ, then using the Σ notation, one has turning about an axis. The physical quan- the relations: tities and laws used to describe linear mo- Σα = –b/a, Σαβ = c/a, Σαβγ = –d/a, αβγδ = e/a. tion all have rotational analogs; the If the roots of the quintic equation ax5 + equations of rotational motion are the bx4 + cx3 + dx2 + ex + f = 0 are written α, analogs of the equations of motion (linear). β, γ, δ, ε, then Σα = –b/a, Σαβ = c/a, Σαβγ = As well as the kinematic equations, the –d/a, Σαβγδ = e/a, αβγδε = –f/a. This pat- equations of rotational motion include T = tern of relations between roots and coeffi- Iα, the analog of F = ma. Here T is the cients extends to higher degree. turning-force, or torque (the analog of If the coefficients are all taken to be real force), I is the moment of inertia (analo- numbers then these results mean that the gous to mass), and α is the angular acceler- sum of the roots and the product of the ation (analogous to linear acceleration).

192 Russell’s paradox

The kinematic equations relate the an- rounding (rounding off) The process of ω gular velocity, 1, of the object at the start adjusting the least significant digit or digits ω of timing to its angular velocity, 2, at in a number after a required number of dig- some later time, t, and thus to the angular its has been truncated (dropped). This re- displacement θ. They are: duces the error arising from truncation but ω ω α = 1 + t still leaves a rounding error so that the ac- θ ω ω = ( 1 + 2)/2t curacy of, say, the result of a calculation θ ω α 2 = 1t + t /2 will be decreased. For example, the num- θ ω α 2 = 2t – t /2 ber 2.871 329 71 could be truncated to ω 2 ω 2 αθ 2 = 1 + 2 2.871 32 but would be rounded to 2.871 33. rotation group The group consisting of the set of all possible rotations about an routine A sequence of instructions used in axis. This group is a continuous group, i.e. computer programming. It may be a short it has an infinite number of members. The program or sometimes part of a program. rotation group in two dimensions is an See also subroutine. ABELIAN GROUP but the rotation group in three dimensions is a non-Abelian group. row matrix See row vector. In physical systems the rotation group is closely associated with the angular mo- row vector (row matrix) A number, (n), mentum of the system. There are many ap- of quantities arranged in a row; i.e. a 1 × n plications of the rotation group in the matrix. For example, the coordinates of a quantum theory of atoms, molecules, and atomic nuclei. point in a Cartesian coordinate system with three axes is a 1 × 3 row vector, rotation of axes In coordinate geometry, (x,y,z). the shifting of the reference axes so that they are rotated with respect to the original Runge–Kutta method /rûng-ĕ kût-ă/ An axes of the system by an angle (θ). If the iterative technique for solving ordinary DIF- new axes are x′ and y′ and the original axes FERENTIAL EQUATIONS, used in computer x and y, then the coordinates (x,y) of a analysis. See also iteration. The method is point with the original axes are related to named for the German mathematicians the new coordinates (x′,y′) by: Carl Runge (1856–1927) and Martin x = x′cosθ – y′sinθ Kutta (1867–1944). y = x′sinθ – y′cosθ Russell’s paradox /russ-ĕlz/ A PARADOX rough In mechanics, describing a system at the foundations of set theory which was in which frictional effects have to be taken pointed out by the British philosopher into consideration in the calculations. Bertrand Russell (1872–1970) in 1901.

193 S

saddle point A stationary point on a testing a new production process against curved surface, representing a function of an old. The population can be finite or in- two variables, f(x,y), that is not a turning finite. In sampling with replacement, each point, i.e. it is neither a maximum nor a individual chosen is returned to the popu- minimum value of the function. At a saddle lation before the next choice is made. In point, the partial derivatives ∂f/∂x and random sampling every member of the ∂f/∂y are both zero, but do not change sign. population has an equal chance of being The tangent plane to the surface at the sad- chosen. In stratified random sampling the dle point is horizontal. Around the saddle population is divided into strata and the point the surface is partly above and partly random samples drawn from each are below this tangent plane. pooled. In systematic sampling the population is ordered, the first individual chosen sample space See probability. at random and further individuals chosen at specified intervals, for example, every sampling The selection of a representa- 100th person on the electoral roll. If a ran- tive subset from a whole population. dom sample of size n is the set of numerical Analysis of the sample gives information values {x1,x2,…xn}, the sample mean is: about the whole population. This is called n ∑ - statistical inference. For example, popula- x = xi/n tion parameters (such as the population 1 mean and variance) can be estimated using The sample variance is: ∑ - 2 sample statistics (such as the sample mean (xi – x) /(n – 1) ∑ - 2 and variance). Significance (or hypothesis) (xi – x) /n tests are used to test whether observed dif- for a normal distribution. If µ is the popu- ferences between two samples are due to lation mean, the sample variance is: ∑ µ chance variation or are significant, as in (xi – )/n

saddle point

Saddle point on a surface

194 scatter diagram sampling distribution The distribution a1b1 + a2b2 of a sample statistic. For example, when See also vector product. different samples of size n are taken from the same population the means of each scalar projection The length of an or- sample form a sampling distribution. If the thogonal projection of one vector on an- population is infinite (or very large) and other. For example, the scalar projection of sampling is with replacement, the mean of A on B is Acosθ, or (A.B)/b where θ is the µ µ the sample means is x = and the stan- smaller angle between A and B and b is the σ dard deviation of the sample means is x = unit vector in the direction of B. Compare σ/√n, where µ and σ are the population vector projection. mean and standard deviation. When n ≥ 30, sampling distributions are approxi- scale 1. The markings on the axes of a mately normal and large-sampling theory graph, or on a measuring instrument, that is used. When n < 30, exact sample theory correspond to values of a quantity. Each is used. See also sampling. unit of length on a linear scale represents the same interval. For example, a ther- satisfy To be a solution of. For example, mometer that has markings 1 millimeter 3 and –3 as values of x satisfy the equation apart to represent 1°C temperature inter- x2 = 9. vals has a linear scale. See also logarithmic scale. scalar /skay-ler/ A number or a measure in 2. The ratio of the length of a line between which direction is unimportant or mean- two points on a map to the distance repre- ingless. For instance, distance is a scalar sented. For example, a map in which two quantity, whereas displacement is a vector. points 5 kilometers apart are shown 5 cen- Mass, temperature, and time are scalars – timeters apart has a scale of 1/100 000. they are each quoted as a pure number with a unit. See also vector. scale factor The multiplying factor for each linear measurement of an object when scalar product A multiplication of two it is to be enlarged about a given center of vectors to give a scalar. The scalar product enlargement. A scale factor can be positive of A and B is defined by A.B = ABcosθ, or negative. If the scale factor is positive, where A and B are the magnitudes of A and the image is on the same side of the center B and θ is the angle between the vectors. of enlargement as the object. If the scale An example is a force F displaced s. Here factor is negative, the image will be on the the scalar product is energy transferred (or opposite side of the center of enlargement work done): and will be inverted. Fractional scale fac- W = F.s tors can be used – these give images that W = Fscosθ are smaller than the objects. where θ is the angle between the line of action of the force and the displacement. A scalene /skay-leen/ Denoting a triangle scalar product is indicated by a dot be- with three unequal sides. tween the vectors and is sometimes called a dot product. The scalar product is commu- scanner A computer input device that tative produces a digital electronic version of an A.B = B.A image (or text), so that it can be manipu- and is distributive with respect to vector lated using the computer. addition A.(B + C) = A.B + A.C scatter diagram (Galton graph) A graph- If A is perpendicular to B, A.B = 0. In two- ical representation of data from a bivariate dimensional Cartesian coordinates with distribution (x,y). The variables are mea- unit vectors i and j in the x- and y-direc- sured on n individuals giving data (x2,y1), tions respectively, …, (xn,yn); e.g. xi and yi are the height and th A.B = (a1i + a2j).(b1i + b2j) = weight of the i individual. If yi is plotted 195 Schwarz inequality

y y

x x

Scatter diagrams: the left diagram shows weak correlation; the right shows strong correlation.

against xi the resultant scatter diagram will Even so, the force ratio (F2/F1) can be very indicate any relationship between x and y high. The distance ratio is given by 2πr/p, by showing if a smooth curve can be drawn where r is the radius and p the pitch of the through the points. If the points seem to lie screw (the angle between the thread and a near a straight line they are said to be lin- plane at right angles to the barrel of the early correlated. If they lie near another screw). type of curve they are non-linearly corre- 2. A symmetry that can occur in a crystal, lated. Otherwise they are uncorrelated. See formed by a combination of a translation also regression line. and a rotation. See also space group.

Schwarz inequality /shworts, shvarts/ An scruple A unit of mass equal to 20 grains. inequality that applies to both series and It is equivalent to 1.295 978 grams. integrals. In the case of series the Schwarz inequality is frequently known as the sec /sek/ See secant. CAUCHY INEQUALITY. For functions f(x) and g(x) the Schwarz inequality states that for secant /see-kănt/ 1. A line that intersects a definite integrals given curve. The intercept is a chord of the ∫f(x)g(x)dx ≤∫f2(x)dx ∫g2(x)dx curve. The case of equality holds if and only if 2. (sec) A trigonometric function of an g(x) = cf(x), where c is a constant, i.e. g(x) angle equal to the reciprocal of its cosine; is directly proportional to f(x). i.e. secα = 1/cosα. See also trigonometry. The Schwarz inequality can be deduced from the Cauchy inequality by taking the sech /sech, shek, sek-aych/ A hyperbolic integrals as limits of series. secant. See hyperbolic functions. scientific notation (standard form) A second 1. Symbol: s The SI unit of time. It number written as the product of a number is defined as the duration of 9 192 631 770 between 1 and 10 with a power of 10. For cycles of a particular wavelength of radia- example, 2342.6 in scientific notation is tion corresponding to a transition between 2.342 6 × 103, and 0.0042 is written as two hyperfine levels in the ground state of 4.2 × 10–3. the cesium-133 atom. 2. A unit of plane angle equal to one sixti- screw 1. A type of machine, related to eth of a minute. the inclined plane, and, in practice, to the second-order lever. The efficiency of screw second-order determinant See determi- systems is very low because of friction. nant.

196 series second-order differential equation A but, unlike regular polyhedra (Platonic DIFFERENTIAL EQUATION in which the high- solids), there is more than one kind of poly- est derivative of the dependent variable is a gon. Generally, there are two types of poly- second derivative. gon. There are 13 semi-regular polyhedra. All of them can be inscribed inside a section See cross-section. sphere. An example of a semi-regular polyhedron is the polyhedron with faces of 12 sector Part of a circle formed between two pentagons and 20 hexagons arranged like radii and the circumference. The area of a the panels on a soccer ball. Semi-regular sector is ½r2θ, where r is the radius and θ polyhedra are also referred to as is the angle, in radians, formed at the cen- Archimedean polyhedra or Archimedean ter of the circle between the two radii. A solids. See also tetrakaidekahedron. minor sector has θ less than π; a major sector has θ greater than π. separation of variables A method of solving ordinary differential equations. In segment Part of a line or curve between a first-order DIFFERENTIAL EQUATION, two points, part of a plane figure cut off by dy/dx = F(x,y) a straight line, or part of a solid cut off by if F(x,y) can be written as f(x),g(y), the a plane. For example, on a graph, a line variables in the function are separable and segment may show the values of a function the equation can therefore be solved by within a certain range. The area between a writing it as chord of a circle and the corresponding arc dy/g(y) = f(x)dx is a segment of the circle. A cut through a and integrating both sides. For example cube parallel to one of its faces forms two dy/dx = x2y cuboid segments. can be written (1/y)dy = x2dx. semantics /si-man-tiks/ In computing or logic, the semantics of a language or system sequence (progression) An ordered set of is the meaning of particular forms or ex- numbers. Each term in a sequence can be pressions in the language or system. Com- written as an algebraic function of its posi- pare syntax. tion. For example, in the sequence (2, 4, 6, 8, …) the general expression for the nth semicircle // Half a circle, bounded by a n term is an = 2 . A finite sequence has a def- diameter and half of the circumference. inite number of terms. An infinite sequence has an infinite number of terms. Compare semiconductor memory See store. series. See also arithmetic sequence; convergent sequence; divergent sequence; geo- semigroup /sem-ee-groop/ A set G with a metric sequence. binary operation, •, that maps G×G, the set of ordered pairs of members of G, into serial access See random access. G, and that is associative, i.e. a•(b•c) = (a•b)•c for all a, b, and c in G. A semi- series The sum of an ordered set of num- group is Abelian or commutative if a•b = bers. Each term in the series can be written b•a for all a,b in G. Sometimes a cancella- as an algebraic function of its position. For tion law is assumed, i.e. x = y if there is an example, in the series 2 + 4 + 6 + 8 … the element z such that xz = yz or zx = zy. general expression for the nth term, an, is Compare group. 2n. A finite series has a finite fixed number of terms. An infinite series has an infinite semilogarithmic graph /sem-ee-lôg-ă- number of terms. A series with m terms, or rith-mik/ See log-linear graph. the sum of the first m terms of an infinite series, can be written as Sm or semi-regular polyhedron A POLYHE- m ∑ a DRON that is bounded by regular polygons n=1 n 197 set

Compare sequence. See also arithmetic se- shear A TRANSFORMATION of a shape or ries; convergent series; divergent series; body in which a line or plane of the shape geometric series. or body is left unchanged. The unchanged line and plane are called the invariant line set Any collection of things or numbers and invariant plane respectively. Physi- that belong to a well-defined category. For cally, shear of a body can occur if a force is example, ‘dog’ is a member, or element, of applied to the body in which the force ei- the set of ‘types of four-legged animal’. The ther lies in the plane of an area of a surface set of ‘days in the week’ has seven ele- of the body or a plane which is parallel to ments. In a set notation, this would be such a surface. written as {Monday, Tuesday, …}. This kind of set is a finite set. Some sets, such as s.h.m. See simple harmonic motion. the set of natural numbers, N = {1, 2, 3, … }, have an infinite number of elements. A short-wavelength approximation An line segment also is an infinite set of points. approximate technique for solving differ- Another way of writing a set of num- ential equations or evaluating integrals in- bers is by defining it algebraically. The set volving waves in which the starting point is of all numbers between 0 and 10 could be the solution for the problem in the limit of written as {x:0siemens /see-mĕnz/ (mho) Symbol: S The sextic /sex-tik/ An equation of degree six, SI unit of electrical conductance, equal to a i.e. conductance of one ohm–1. The unit is ax6 + bx5 + cx4 + dx3 + ex2 + fx + g = 0 named for the German electrical engineer There is no general method of solution for Ernst Werner von Siemens (1816–92). such an equation. sieve of Eratosthenes See Eratosthenes; shares See stocks and shares. sieve of. sheaf A sheaf of planes is a set of planes sigma /sig-mă/ The Greek letter Σ (upper that all pass through a given point, called case) or σ (lower case). In mathematics, the center of the sheaf. Compare pencil. upper-case sigma (Σ) is used to denote sum-

198 simply connected mation; lower-case sigma (σ) often denotes T = 1/f the standard deviation in statistics. T = 2π/ω where f is frequency and ω pulsatance. sign A unit of plane angle equal to 30 de- Other relationships are: π ω grees ( /6 radian). x = x0sin t ±ω√ 2 2 dx/dt = (x0 – x ) signed number A number that is denoted d2x/dt2 = –ω2x as positive or negative. Here x0 is the maximum displacement; i.e. the amplitude of the vibration. In the case significance test See hypothesis test. of angular motion, as for a pendulum, θ is used rather than x. significant figures The number of digits A simple harmonic motion can be rep- used to denote an exact value to a specified resented by the motion of a point at con- degree of accuracy. For example, 6084.324 stant speed in a circular path. The foot of is a value accurate to seven significant fig- the perpendicular from the point to an axis ures. If it is written as approximately 6080, through a diameter describes a simple har- it is accurate to three significant figures monic motion. This is used in a method of The final 0 is not significant because it is representing simple harmonic motions by used only to show the order of magnitude rotating vectors (called phasors). of the number. simple interest The interest earned on similar Denoting two or more figures that capital when the interest is withdrawn as it differ in scale but not in shape. The condi- is paid, so that the capital remains fixed. If tions for two triangles to be similar are: the amount of money invested (the princi- 1. Three sides of one are proportional to pal) is denoted by P, the time in years by T, three sides of the other. and the percentage rate per annum by R, 2. The angle of one is equal to the angle of then the simple interest is PRT/100. Com- the other and the sides forming the angle in pare compound interest. one are proportional to the same sides in the other. simple proposition See proposition. 3. Three angles of one are equal to three angles of the other. simplify To use mathematical and alge- Compare congruent. braic techniques to contract an expression. For example, simplifying the following ex- simple harmonic motion (s.h.m.) Any pressions: motion that can be drawn as a sine wave. 115/184 = 5/8 Examples are the simple oscillation (vibra- 2x – 7x + 9x = 4x tion) of a pendulum (with small amplitude) 15x3y2z/5x2yz = 3xy or a sound source and the variation involved in a simple wave motion. Simple simply connected Describing a region harmonic motion is observed when the sys- for which any closed curve lying in the re- tem, moved away from the central posi- gion can be continuously deformed into a tion, experiences a restoring force that is single point without leaving the region, i.e. proportional to the displacement from this the region does not have any ‘holes’ in it. If position. a region is not simply connected it is said to The equation of motion for such a sys- be multiply connected, i.e. the region has tem can be written in the form: one or more ‘hole’. A doubly-connected md2x/dt2 = λx region has one hole, etc. In general, an λ being a constant. During the motion n-tuply-connected region has (n–1) holes. there is an exchange of kinetic and poten- The concepts of simply connected and tial energy, the sum of the two being con- multiply connected regions are important stant (in the absence of damping). The in topology and the theory of functions of period (T) is given by a complex variable. The concepts of simply

199 Simpson’s rule

y parabola approximating curve f((a + b)/2)

f(b) f(a)

y = f(x)

O x ab

Simpson’s rule and multiply connected regions also have unique value for each variable that satisfies important applications in many branches all the equations. For example, the equa- of physics. tions x + 2y = 6 Simpson’s rule A rule for finding the ap- and proximate area under a curve by dividing it 3x + 4y = 9 into pairs of vertical columns of equal have the solution x = –3; y = –4.5. The width with bases lying along the horizontal method of solving simultaneous equations axis. Each pair of columns is bounded by is to eliminate one of the variables by the vertical lines from the x-axis to three adding or subtracting the equations. For corresponding points on the curve and at example, multiplying the first equation the top by a parabola that goes through above by 2 and subtracting it from the sec- these three points, which approximates the ond gives: curve. For example, if the value of f(x) is 3x + 4y – 2x – 4y = 9 – 12 known at x = a, x = b, and at a value mid- i.e. x = –3. Substituting this into either way between a and b the integral of f(x)dx equation gives the value of y. Simultaneous between limits a and b is approximately equations can also be solved graphically. equal to: On a Cartesian coordinate graph, each h{f(a) + 4f[(a+b)/2] + f(b)}/3 equation would be shown as a straight line h is half the distance between a and b. As and the point at which the two cross is, in with the trapezium rule, which is less accu- this case, (–3,–4.5). See also substitution; rate, better approximations can be ob- inverse of a matrix. tained by subdividing the area into 4, 6, 8, … columns until further subdivision makes sin /sÿn/ See sine. no significant difference to the result. The rule is named for the British mathematician sine /sÿn/ (sin) A trigonometric function of Thomas Simpson (1710–61). Compare an angle. The sine of an angle α (sinα) in a trapezium rule. See also numerical integra- right-angled triangle is the ratio of the side tion. opposite the angle to the hypotenuse. This definition applies only to angles between simultaneous equations A set of two or 0° and 90° (0 and π/2 radian). more equations that together specify con- More generally, in rectangular Carte- ditions for two or more variables. If the sian coordinates, the y-coordinate of any number of unknown variables is the same point on the circumference of a circle of ra- as the number of equations, then there is a dius r centred at the origin is rsinα, where

200 SI units

π 2π 3π 4π 0 x

–1

cosine Sine curve: a graph of y = sin x, with x in radians.

α is the angle between the x-axis and the singular point A point on a curve y = f(x) radius to that point. In other words, the at which the derivative dy/dx has the inde- sine function depends on the vertical com- terminate form 0/0. The singular points on ponent of a point on a circle. Sinα is zero a curve are found by writing the derivative when α is 0°, rises to 1 when α = 90° (π/2) in the form falls again to zero when α = 180° (π), be- dy/dx = g(x)/h(x) comes negative and reaches –1 at α = 270° (3π/2), and then returns to zero at α = 360° π 2 1 (2 ). This cycle is repeated every complete 4 2 revolution. The sine function has the following properties: 2 1 = (2 x 2) – (4 x 1) = 0 sinα = sin(α + 360°) 4 2 sinα = –sin(180° + α) sin(90° – α) = sin(90° + α) Singular 2 x 2 matrix The sine function can also be defined as an infinite series. In the range between 1 and –1: and then finding the values of x for which sinx = x/1! – x3/3! + x5/5! – … g(x) and h(x) are both zero. See also trigonometry. sinh /shÿn, sinsh, sÿn-aych/ A hyperbolic sine rule In any triangle, the ratio of the sine. See hyperbolic functions. side length to the sine of the angle opposite that side is the same for all three sides. sinusoidal /sÿ-ŭ-soid-ăl/ Describing a Thus, in a triangle with sides of lengths a, quantity that has a waveform that is a sine b, and c and angles α, β, and γ (α opposite wave. a, β opposite b, and γ opposite c): a/sinα = b/sinβ = c/sinγ SI units (Système International d’Unités) The internationally adopted system of sine wave The waveform resulting from units used for scientific purposes. It is com- plotting the sine of an angle against the prised of seven base units (the meter, kilo- angle. Any motion for which distance plot- gram, second, kelvin, ampere, mole, and ted against time gives a sine wave is a sim- candela) and two supplementary units (the ple harmonic motion. radian and steradian). Derived units are formed by multiplication and/or division singular matrix A square matrix that has of base units; a number have special names. a determinant equal to zero and that has no Standard prefixes are used for multiples inverse matrix. See also determinant. and submultiples of SI units. The SI sys-

201 skew lines tem is a coherent rationalized system of also have fixed scales showing squares, units. cubes, and trigonometric functions. The accuracy of the slide rule is usually to three skew lines Lines in space that are not par- significant figures. Slide rules have gener- allel and do not intersect. Whether a pair of ally been replaced by electronic calcula- lines in space are skew lines rather than the tors. other two possibilities for lines in space, i.e. parallel lines and lines that are not parallel sliding friction See friction. and intersect, can be determined from the equations for the lines. If the direction ra- slope See gradient. tios of the two lines are equal then the two lines are parallel. If the equations of the small circle A circle, drawn on the surface two lines are: (x–x1)/a1 = (y–y1)/b1 = of a sphere, whose centre is not at the cen- (z–z1)/c1 = m, and (x–x1)/a2 = (y–y2)/b2 = tre of the sphere. Compare great circle. (z–z2/c2 = n, where a1, b1, c1, m, a2, b2, c2 and n are constants, then these two lines smooth In mechanics, describing a system can only intersect if these are unique values in which friction can be neglected in the of m and n such that x = x3, y = y3, z = z3 calculations. satisfy the equations for both lines. If such unique values of m and n cannot be found snowflake curve See fractal. the lines are skew lines. software The PROGRAMS that can be run skewness A measure of the degree of on a computer, together with any associ- asymmetry of a distribution. If the fre- ated written documentation. A software quency curve has a long tail to the right package is a professionally written pro- and a short one to the left, it is called gram or group of programs that is designed skewed to the right or positively skewed. If to perform some commonly required task, the opposite is true, the curve is skewed to such as statistical analysis or graph plot- the left, or negatively skewed. Skewness is ting. The availability of software packages measured by either Pearson’s first measure means that common tasks need not be proof skewness (mean minus mode) divided by grammed over and over again. Compare standard deviation, or the equivalent Pear- hardware. See also program. son’s second measure of skewness divided by standard deviation. solenoidal vector /soh-lĕ-noi-dăl/ A vector V for which divV = 0. It is possible to skew-symmetric matrix A matrix A write a solenoidal vector in the form V = which has its transpose equal to –A, i.e. if curlA, where A is a vector called the vector the entries of A are denoted by aij then aij = potential. If it is known that V is a sole- –aji for all i and j in the matrix. It follows noidal vector it is possible to construct an from the definition of a skew-symmetric infinite number of A’s that satisfy V = matrix that the entries in the main diagonal curlA. The concept of a solenoidal vector is of the matrix, i.e. all entries of the form aii, used extensively in the theory of electro- have to be zero. magnetism. slant height 1. The length of an element solid A three-dimensional shape or object, of a right cone; i.e. the distance from the such as a sphere or a cube. vertex to the directrix. 2. The altitude of the faces of a right pyra- solid angle Symbol: Ω The three-dimen- mid. sional analog of angle; the region subtended at a point by a surface (rather than slide rule A calculating device on which by a line). The unit is the steradian (sr), sliding logarithmic scales are used to mul- which is defined analogously to the radian tiply and divide numbers. Most slide rules – the solid angle subtending unit area at

202 solid angle

BASE AND DIMENSIONLESS SI UNITS

Physical quantity Name of SI unit Symbol for SI unit length meter m mass kilogram(me) kg time second s electric current ampere A thermodynamic temperature kelvin K luminous intensity candela cd amount of substance mole mol *plane angle radian rad *solid angle steradian sr *supplementary units

DERIVED SI UNITS WITH SPECIAL NAMES

Physical quantity Name of SI unit Symbol for SI unit frequency hertz Hz energy joule J force newton N power watt W pressure pascal Pa electric charge coulomb C electric potential difference volt V electric resistance ohm Ω electric conductance siemens S electric capacitance farad F magnetic flux weber Wb inductance henry H magnetic flux density tesla T luminous flux lumen lm illuminance (illumination) lux lx absorbed dose gray Gy activity becquerel Bq dose equivalent sievert Sv

DECIMAL MULTIPLES AND SUBMULTIPLES USED WITH SI UNITS

Submultiple Prefix Symbol Multiple Prefix Symbol 10–1 deci- d 101 deca- da 10–2 centi- c 102 hecto- h 10–3 milli- m 103 kilo- k 10–6 micro- µ 106 mega- M 10–9 nano- n 109 giga- G 10–12 pico- p 1012 tera- T 10–15 femto- f 1015 peta- P 10–18 atto- a 1018 exa- E 10–21 zepto- z 1021 zetta- Z 10–24 yocto- y 1024 yotta- Y

203 solid geometry

Ω x r 2 r S P Ω

Solid angle: the surface S subtends a solid angle Ω in steradians at point P. An area that forms part of the surface of a sphere of radius r, center P, and subtends the same solid angle Ω at P is equal to Ωr2.

unit distance. As the area of a sphere is tion; for example, x2 = 16 has two; x = –4 4πr2, the solid angle at its center is 4π stera- and x = +4. dians. See illustration overleaf. solution of triangles Calculating the un- solid geometry The study of geometric known sides and angles in triangles. Be- figures in three dimensions. cause the sum of angles in a triangle is always 180°, the third angle can be calcu- solid of revolution A solid figure that lated if two are known. All the sides and can be produced by revolution of a line or angles can be calculated when two sides curve (the generator) about a fixed axis. and the angle between them are known, For instance, rotating a circle about a di- but if two sides and another angle are ameter generates a sphere. Rotating a circle known there are two possible solutions. about an axis that does not cut the circle Any two angles and one side are sufficient generates a torus. to solve a triangle. See also trigonometry. solid-state memory See store. source language (source program) See program. soliton /sol-ă-ton/ A type of kink solution that can occur as a solution to certain non- space curve A curved line in a volume, linear equations for propagation. A char- defined in three-dimensional Cartesian co- acteristic feature of solitons is that they are ordinates by three functions: very stable and can pass through each x = f(t) other unchanged. Since solitons are very y = g(t) atypical of solutions to nonlinear equa- z = h(t) tions there has to be some mathematical or by two equations of the form: structure in the nonlinear equation to per- F(x,y,z) = 0 mit their existence, and they are frequently G(x,y,z) = 0 associated with the TOPOLOGY of a system. There are many physical applications of space group The set of all symmetry op- solitons. erations of a crystal, i.e. the possible rotations, reflections, and translations. There solution A value of a variable that satis- are 230 space groups in three dimensions. fies an algebraic equation. For example, Space groups can also be analyzed in two the solution of 2x + 4 = 12 is x = 4. An dimensions (ornamental or ‘wallpaper’ equation may have more than one solu- symmetry) and in higher dimensions.

204 spectrum

P(r, , φ) •

O φ

Spherical polar coordinates of point P are (r, θ, φ).

Of the 230 space groups 73 of these garded as separate continua, they should groups consist only of point group opera- be replaced by a single continuum of four tions and the translations of the crystal dimensions, called space–time. In lattice. Such space groups are called sym- space–time the history of an object’s mo- morphic. The remaining 157 space groups tion in the course of time is represented by consist of point-group operations and a line called the world curve. See also rela- translations which are not individually in tivity, theory of. the space group. Those space groups contain GLIDE planes and SCREW axes, i.e. Spearman’s method /speer-mănz/ A way symmetry operations involving a trans- of measuring the degree of association be- lation which is not a translation of the crys- tween two rankings of n objects using two tal. Such space groups are said to be different variables x and y which give data non-symmorphic. (x1,y1), …, (xn,yn). The objects are ranked The concept of a space group can be ex- using first the x’s and then the y’s, and the tended from crystals to PENROSE PATTERNS difference, D, between the RANKS calcu- and QUASICRYSTALLINE SYMMETRY. One lated for each object. way in which this can be done is by pro- Spearman’s coefficient of rank correla- jecting down to the Penrose pattern or qua- tion sicrystal from the space group of a crystal ρ = 1–(6ΣD2/[n(n2 – 1)]) in a higher dimension. The method is named for the British behavioral scientist Charles Spearman space–time In Newtonian (pre-relativity) (1863–1945). physics, space and time are separate and absolute quantities; that is they are the special theory See relativity. same for all observers in any FRAME OF REF- ERENCE. An event seen in one frame is also spectrum /spek-trŭm/ (pl. spectra) The seen in the same place and at the same time spectrum of an operator A is the set of all by another observer in a different frame. complex numbers λ such that the operator After Einstein had proposed his theory A – λI, where I is the identity operator, i.e. of relativity, Minkowski suggested that I(x) ≡ x, does not have an inverse. For ex- since space and time could no longer be re- ample, if A is a matrix, its spectrum is the

205 speed set of its eigenvalues since if λ is an eigen- points on the vertical axis below O, θ = value of A, det(A – λI) = 0 and A – λI is 180°. φ is the angle that the radius vector therefore not invertible. makes with an axis in the equatorial plane. It is called the azimuth. For all points lying speed Symbol: c Distance moved per unit in the axial plane, that is, on, vertically time: c = d/t. Speed is a scalar quantity; the above, or vertically below this axis, φ = 0 equivalent vector is velocity – a vector on the positive side of O and φ = 180° on quantity equal to displacement per unit the negative side. This plane corresponds time. Usage can be confusing and it is com- to the plane y = 0 in rectangular CARTESIAN mon to meet the word ‘velocity’ where COORDINATES. For points in the vertical ° ‘speed’ is more correct. For instance c0 is plane at 90 to this, (x = 0 in rectangular the speed of light in free space, not its ve- Cartesian coordinates) φ = 90° in the posi- locity. tive half-plane and 270° in the negative half-plane. For a point P(r,θ,φ), the corre- sphere A closed surface consisting of the sponding rectangular Cartesian coordi- locus of points in space that are at a fixed nates (x,y,z) are: distance, the radius r, from a fixed point, x =rcosφsinθ the center. A sphere is generated when a y = rsinφsinθ circle is turned through a complete revolu- z =rcosθ tion about an axis that is one of its diame- Compare cylindrical polar coordinates. See ters. The plane cross-sections of a sphere also polar coordinates. are all circles. The sphere is symmetrical about any plane that passes through its spherical sector The solid generated by center and the two mirror-image shapes on rotating a sector of a circle about a diame- each side are called hemispheres. In Carte- ter of the circle. The volume of a spherical sian coordinates, the equation of a sphere sector generated by a sector of altitude h of radius r with its center at the origin is (parallel to the axis of rotation) in a circle x2 + y2 + z2 = r2 of radius r is The volume of a sphere is 4πr3/3 and its (⅔)πr2h surface area is 4πr2. spherical segment A solid formed by cut- spherical harmonic A type of function ting in one or two parallel planes through that frequently occurs as a solution of dif- a sphere. The volume of a spherical seg- ferential equations for systems that have ment bounded by circular plane cross-sec- spherical symmetry. Spherical harmonics tions of radii r1 and r2 a distance h apart, can be regarded as the analog for the is: π 2 2 2 sphere of CIRCULAR FUNCTIONS for the cir- h(3r1 + 3r2 + h )/6 cle. They occur in the solutions of the equa- If the segment is bounded by one one plane tions of quantum mechanics for electrons of radius r and the curved surface of the in atoms. sphere, then the volume is: πh(3r2 + h2)/6 spherical polar coordinates A method of defining the position of a point in space spherical triangle A three-sided figure on by its radial distance, r, from a fixed point, the surface of a sphere, bounded by three the origin O, and its angular position on great circles. A right spherical triangle has the surface of a sphere centred at O. The at least one right angle. A birectangular angular position is given by two angles θ spherical triangle has two right angles and and φ. θ is the angle that the radius vector a trirectangular spherical triangle has three makes with a vertical axis through O (from right angles. If one of the sides of a spheri- the south pole to the north pole). It is called cal triangle subtends an angle of 90° at the the co-latitude. For points on the vertical center of the sphere, then it is called a axis above O, θ = 0. For points lying in the quadrantal spherical triangle. An oblique ‘equatorial’ horizontal plane, θ = 90°. For spherical triangle has no right angles.

206 spherical triangle

small circle

great circle

great and small circles on a sphere spherical wedge

spherical wedge

trirectangular spherical triangle

right

birectangular

spherical triangles Spherical trigonometry

207 spherical trigonometry spherical trigonometry The study and ments in this diagonal is called the trace (or solution of spherical triangles. spur) of the matrix. spheroid /sfeer-oid/ A body or curved sur- square pyramid See pyramid. face that is similar to a sphere but is length- ened or shortened in one direction. See square root For any given number, an- ellipsoid. other number that when multiplied by itself equals the given number. It is denoted spiral A plane curve formed by a point by the symbol √ or the index (power) ½. winding about a fixed point at an increas- For example, the square root of 25, written ing distance from it. There are many kinds √25 = 5. of spirals, e.g. the Archimedean spiral is θ given by r = a , the logarithmic spiral is squaring the circle The attempt to con- given by log(r) = aθ, and the hyperbolic spi- struct a square that has the same area as a ral is given by rθ = a, where in each case a particular circle, using a ruler and com- is a constant, and r and θ are polar coordi- passes. An exact solution is impossible be- nates. cause there is no exact length for the edge, spline A smooth curve passing through a which is a multiple of the transcendental √π fixed number of points. Splines are used in number . computer graphics. stability A measure of how hard it is to spur See square matrix. displace an object or system from equilibrium. square 1. The second power of a number Three cases are met in statics differing or variable. The square of x is x × x = x2 (x in the effect on the center of mass of a small squared). The square of the square root of displacement. They are: a number is equal to that number. 1. Stable equilibrium – the system returns 2. In geometry, a plane figure with four to its original state when the displacing equal straight sides and right angles be- force is removed. tween the sides. Its area is the length of one 2. Unstable equilibrium – the system of the sides squared. A square has four axes moves away from the original state when of symmetry – the two diagonals, which displaced a small distance. are of equal length and bisect each other 3. Metastable or neutral equilibrium – perpendicularly, and the two lines joining when displaced a small distance, the sys- the mid-points of opposite sides. It can be tem is at equilibrium in its new position. superimposed on itself after rotation An object’s stability is improved by: (a) through 90°. lowering the center of mass; or (b) increasing the area of support; or by both. square matrix A matrix that has the same number of rows and columns, that is, stable equilibrium See stability. a square array of numbers. The diagonal from the top left to the bottom right of a square matrix is called the leading diagonal standard Established as a reference. (or principal diagonal). The sum of the ele- 1. Writing an equation in standard form enables comparison with other equations of the same type. For example, x2/42 + y2/22 = 1 1 2 4 and 3 5 11 x2/32 + y2/52 = 1 4 8 9 are equations of hyperbolas in rectangular Cartesian coordinates, both written in Square 3 x 3 matrix: the trace is 1 + 5 + 9 = 15. standard form.

208 stationary point

2. A standard measuring instrument is one mainly when discussing the weather, is against which other instruments are cali- 100 kPa exactly. See also STP. brated. 3. Standard form of a number. See scien- standard temperature An internation- tific notation. ally agreed value for which many measurements are quoted. It is the melting standard deviation A measure of the dis- temperature of water, 0°C (273.15 K). See persion of a statistical sample, equal to the also STP. square root of the variance. In a sample of standing wave See stationary wave. n observations, x1,x2,x3, … xn, the sample standard deviation is: static friction See friction. n ͙ ∑ - 2 s = [ (xi – x) / (n – 1)] 1 static pressure The pressure on a surface due to a second solid surface or to a fluid where x- is the sample mean. If the mean µ that is not flowing. of the whole population from which the sample is taken is known, then statics A branch of MECHANICS dealing n with the forces on an object or in a system ͙ ∑ µ 2 s = [ (xi – ) /n] 1 in equilibrium. In such cases there is no resultant force or torque and therefore no re- standard form See scientific notation. sultant acceleration. standard form of a number See scien- stationary point A point on a curved line tific notation. at which the slope of the tangent to the curve is zero. All turning points (maximum standard pressure An internationally points and minimum points) are stationary agreed value; a barometric height of points. In this case the slope of the tangent 760 mmHg at 0°C; 101 325 Pa (approxi- passes through zero and changes its sign. mately 100 kPa). Some stationary points are not turning This is sometimes called the atmosphere points. In such cases, the curve levels out (used as a unit of pressure). The bar, used and then continues to increase or decrease

absolute maximum • y = f(x) local maximum • •

0 x horizontal inflection point • local minimum • absolute minimum

Five kinds of stationary points for a function

209 stationary wave as before. At a stationary point the deriva- tion can be identified with the DIRAC DELTA tive dy/dx of y = f(x) vanishes (is zero). At FUNCTION. a maximum, the second derivative, d2y/dx2, is negative; at a minimum it is pos- steradian /sti-ray-dee-ăn/ Symbol: sr The itive. At a horizontal inflection point the SI unit of solid angle. The surface of a second derivative is zero. Not all inflection sphere, for example, subtends a solid angle points have dy/dx = 0; i.e. not all are sta- of 4π at its center. The solid angle of a cone tionary points. is the area intercepted by the cone on the At a stationary point on a curved sur- surface of a sphere of unit radius. face, representing a function of two variables, f(x,y), the partial derivatives ∂f/∂x stereographic projection /ste-ree-oh- and ∂f/∂y are both zero. This may be a graf-ik, steer-ee-/ A geometrical transfor- maximum point, a minimum point, or a mation of a sphere onto a plane. A point is SADDLE POINT. taken on the surface of the sphere – the pole of the PROJECTION. The projection of stationary wave (standing wave) The in- points on the sphere onto a plane is ob- terference effect resulting from two waves tained by taking straight lines from the of the same type moving with the same fre- pole through the points, and continuing quency through the same region. The effect them to the plane. The plane taken does is most often caused when a wave is re- not pass through the pole and is perpendic- flected back along its own path. The re- ular to the diameter of the sphere through sulting interference pattern is a stationary the pole. wave pattern. Here, some points always show maximum amplitude; others show Stirling approximation /ster-ling/ An minimum amplitude. They are called an- approximate expression for the value of tinodes and nodes respectively. The dis- the factorial of a number or the GAMMA tance between neighboring node and FUNCTION. The expression for n! is n!~nn antinode is a quarter of a wavelength. exp(–n)√(2πn), with the expression for Γ(n + 1) being Γ(n + 1) ∼ nnexp(–n)√(2πn), statistical inference See sampling. where ∼ denotes ‘is asymptotic to’. This means that the ratio n!/[nn exp(–n)√(2πn)] statistical mechanics The evaluation of tends to 1 as n→∞, i.e. Stirling’s approxi- data that applies to a large number of enti- mation becomes a better approximation as ties by the use of statistics. It finds its main n increases, with the ratio of the error to applications in chemistry and physics. the value of n! tending to zero as n→∞. The approximation is named for the British statistics The methods of planning exper- mathematician James Stirling (1692– iments, obtaining data, analyzing it, draw- 1770). ing conclusions from it, and making decisions on the basis of the analysis. In stochastic process /stoh-kas-tik/ A statistical inference, conclusions about a process that generates a series of random population are inferred from analysis of a values of a variable and builds up a partic- sample. In descriptive statistics, data is ular statistical distribution from these. For summarized but no inferences are made. example, the POISSON DISTRIBUTION can be built up by a stochastic process that starts step function A function θ(x) defined by: with values taken from tables of random θ(x) = 1, for x > 0 and θ(x) = 0, for x < 0. numbers. In physical problems the variable x is sometimes the time t, with the step func- Stokes’ theorem A result in vector calcu- tion representing a sudden ‘turning on’ of lus that states that the surface integral of some effect at t = 0. The step function is the curl of a vector function is equal to the also sometimes known as Heaviside unit line integral of that vector function around function. The derivative of the step func- a closed curve. If the vector function is de-

210 strain noted V, Stokes’ theorem can be written in can in some types be changed by special the form: techniques. Stores may be magnetic or electronic in Ύcurlv•dS - Ώv•dl s character. The electronic memory now where S represents surface and l represents widely used in main store consists of highly line. Stokes’ theorem has many physical complex integrated circuits. This semicon- applications, particularly to the theory of ductor memory (or solid-state memory) electricity and magnetism. There are exten- stores an immense amount of information sions of Stokes’ theorem to higher dimen- in a very small space; items of information sions. The theorem is named for the British can be extracted at very high speed. mathematician Sir George Gabriel Stokes (1819–1903). STP (NTP) Standard temperature and pressure. Conditions used internationally when measuring quantities that vary with store (memory) A system or device used in both pressure and temperature (such as the computing to hold information (programs density of a gas). The values are and data) in such a way that any piece of 101 325 Pa (approximately 100 kPa) and ° information can automatically be retrieved 0 C (273.15 K). See also standard pres- by the computer as required. The main sure; standard temperature. store (or internal store) of a computer is under the direct control of the central straight line The curve that gives the processor. It is the area in which programs, shortest distance between two points in or parts of programs, are stored while they Euclidean space. are being run on the computer. Data and In two dimensions, i.e. a plane, a straight line can be represented using program instructions can be extracted ex- Cartesian coordinates by a linear equation tremely rapidly by random access. The of the form ax +by + c = 0, where a, b, and main store is supplemented by backing c are constants and a and b cannot both be store, in which information can be perma- zero. The equation for a straight line in two nently stored. The two basic forms of back- dimensions can be written in several differing store are those in which magnetic tape ent forms. If the line has a gradient m and is used (i.e. magnetic tape units) and those cuts the y-axis at the intercept point (0,c) in which disks, or some other random-ac- the equation for the line can be written as cess device are used. y = mx + c. If the line has a gradient m and The main store is divided into a huge passes through the point (x ,y ) then its number of locations, each able to hold one 1 1 equation can be written in the form y–y1 = unit of information, i.e. a word or a byte. m(x–x1). If the line passes through the The number of locations, i.e. the number of ≠ points (x1, y1) and (x2, y2) then, if x1 x2, words or bytes that can be stored, gives the the equation of the line can be written y–y1 capacity of the store. Each location is iden- = [(y2–y1)/(x2–x1)](x–x1). tified by a serial number, known as its ad- In three-dimensional space a straight dress. line can be found by the intersection of two There are many different ways in which planes. This means that the straight line memory can be classified. Random-access can be given in terms of two equations: a1x memory (RAM) and serial-access memory + b1y + c1z + d1 = 0 and a2x + b2y + c2z + differ in the manner in which information d2 = 0, where these two equations are equa- is extracted from a store. With volatile tions for two planes that intersect. See also memory, stored information is lost when vector equation of a plane. the power supply is switched off, unlike nonvolatile memory. With read-only mem- strain A measure of how a solid body is ory (ROM), information is stored perma- deformed when a force is applied to it. The nently or semipermanently; it cannot be linear strain, sometimes called the tensile altered by programmed instructions but strain, is the ratio of the change of the body

211 strange attractor

α ≤ α α to its original length. The linear strain ap- The values tn( ) for which P(t tn( )) = plies to wires. The bulk strain, sometimes for various values of n are available in ta- called the volume strain, is the ratio of the bles. See also mean, standard deviation; change in the volume of the body to its Student’s t-test. original volume. When a body is subject to a shear force there is another type of strain Student’s t-test A hypothesis test for called a shear strain, which is the angular accepting or rejecting the hypothesis that distortion of the body measured in radians. the mean of a normal distribution wih un- µ Since linear strain and bulk strain are ratios known variance is 0, using a small sample. - µ √ of length and volume respectively they are The statistic t = (x – 0) n/s is computed - dimensionless. See also Poisson’s ratio, from the data (x1,x2, … xn), where x is the stress, Young modulus. sample mean, s is the sample standard deviation, and n < 30. If the hypothesis is true strange attractor A path in phase space t has a tn–1 distribution. If t lies in the crit- α that is not closed. Strange attractors are ical region |t| > tn–1(1 – /2) the hypothesis characteristic of chaotic behavior. See at- is rejected at significance level α. See also tractor. hypothesis test; Student’s t-distribution. stratified random sampling See sam- subgroup A subgroup S of a GROUP G is a pling. subset of G that is also a group under the same law of combination of elements as G, stress The force per unit area acting on a i.e. a group whose members are members body and tending to deform that body. It is of another group. a term used when a solid body is subject to forces of tension, compression, or shear. subject The main independent variable in The unit of stress is the pascal (Pa), or an algebraic formula. For example, in the Nm–2. function If the solid body is stretched the stress is y = f(x) = 2x2 + 3x, tensile stress. If the body is compressed the y is the subject of the formula. stress is compressive stress. If the force tends to shear a body the stress is shear submatrix /sub-may-triks, -mat-riks/ A stress. The biggest force per unit area that matrix that is obtained from another ma- a solid body can withstand without frac- trix M by deleting some rows and columns ture is called the breaking stress of that from M. body. The stress–strain graph of a solid is im- subnormal The projection on the x-axis portant both in the theory of solids and in of a line normal to a curve at point P0 engineering. See also strain; Young modu- (x0,y0) and extending from P0 to the x-axis. lus. The length of the subnormal is my0, where m is the gradient of the tangent to the curve Student’s t-distribution The distribu- at P0. tion, written tn, of a random variable t = (x- – µ)√n/σ subroutine /sub-roo-teen/ (procedure) A where a random sample of size n is taken section of a computer program that per- from a normal population x with mean µ forms a task that may be required several and standard deviation σ. n is called the times in different parts of the program. In- number of degrees of freedom. The mean stead of inserting the same sequence of in- of the distribution is 0 for n > 1, and the structions at a number of different points, variance is n/(n – 2) for n > 2. When n is control is transferred to the subroutine and large t has an approximately standard nor- when the task is complete it is returned to mal distribution. The probability density the main part of the program. See also rou- function, f(t), has a symmetrical graph. tine.

212 substitution

A B

B ⊂ A

Subset: the set B (shaded) in the Venn diagram is a subset of A. subscript /sub-skript/ A small letter or tionship of inclusion, and N ⊂ I can be number written below, and usually to the read: N is included in I. Another symbol right of, a letter for various purposes, such used is ⊃, which means ‘contains as a sub- as to identify a particular element of a set, set’ as in I ⊃ N. See also Venn diagram. See ∈ e.g. x1,x2 X, to denote a constant, e.g. illustration overleaf. a1,a2, or to distinguish between variables, e.g. f(x1,x2,…,xn). Double subscripts are substitution /sub-stă-stew-shŏn/ A also used, for example to write a determi- method of solving algebraic equations that nant with general terms; aij denotes the el- involves replacing one variable by an ement in the ith row and jth column. See equivalent in terms of another variable. also superscript. For example, to solve the SIMULTANEOUS EQUATIONS subset Symbol: ⊂ A set that forms part of x + y = 4 another set. For example, the set of natural and numbers, N = {1, 2, 3, 4, …} is a subset of 2x + y = 9 the set of integers I = {… –2, –1, 0, 1, 2, … we can first write x in terms of y, that is: }, written as N ⊂ I. ⊂ stands for the rela- x = 4 – y

P(x0,y0)

normal tangent

x O T S N subtangent subnormal

Subtangent and subnormal of a curve at a point P (x0,y0).

213 subtangent

The substitution of 4 – y for x in the second two numbers depends on the order of sub- equation gives: traction, the sense of the vector difference 2(4 – y) + y = 9 depends on the sense of the angle between or y = –1, and therefore, from the first the two vectors. Matrix subtraction, like equation, x = 5. Another use of substitu- matrix addition, can only be carried out tion of variables is in integration. See also between matrices with the same number of integration by substitution. rows and columns. Compare addition. See also matrix. subtangent /sub-tan-jĕnt/ The projection on the x-axis of the line tangent to a curve subtraction of fractions See fractions. at a point P0 (x0,y0) and extending from P0 to the x-axis. The length of the subtangent subtrahend /sub-tră-hend/ A quantity is y0/m, where m is the gradient of the tan- that is to be subtracted from another given gent. quantity. subtend /sŭb-tend/ To lie opposite and successor The successor of a member of a mark out the limits of a line or angle. For series is the next member of the series. In example, each arc of a circle subtends a particular, the successor of an integer is the particular angle at the center of the circle. next integer, i.e. the successor of n is n + 1.

sufficient condition See condition.

The result obtained by adding two or A sum more quantities together.

summation notation Symbol Σ. A sym-

O bol used to indicate the sum of a series. The B number of terms to be summed is indicated by putting i, the number of the first term to be summed, beneath the ∑ sign and the number of the last term to be summed is put above the Σ sign. The sum to infinity of a series is indicated by putting the infinity sign ∞ above the ∑ sign. Subtend: the arc AB subtends an angle θ at the center O of the circle. sum to infinity In a CONVERGENT SERIES, the value that the sum of the first n terms, Sn approaches as n becomes infinitely subtraction /sŭb-trak-shŏn/ Symbol: – large. The binary operation of finding the DIFFER- ENCE between two quantities. In arith- sup See supremum. metic, unlike addition, the subtraction of two numbers is neither commutative (4 – 5 superelastic collision /soo-per-i-las-tik/ ≠ 5 – 4) nor associative [2 – (3 – 4) ≠ (2 – A collision for which the restitution coeffi- 3) – 4]. The identity element for arithmetic cient is greater than one. In effect the rela- subtraction is zero only when it comes on tive velocity of the colliding objects after the right-hand side (5 – 0 = 5, but 0 – 5 ≠ the interaction is greater than that before. 5). In vector subtraction, two vectors are The apparent energy gain is the result of placed tail-to-tail forming two sides of a transfer from energy within the colliding triangle. The length and direction of the objects. For example, if a collision between third side gives the VECTOR DIFFERENCE. two trolleys causes a compressed spring in Just as the sign of the difference between one to be released against the other, the

214 symmetrical collision may be superelastic. See also resti- including the calculation of surface areas, tution; coefficient of. the center of mass and moment of inertia, the flow of fluids through a surface and superscript /soo-per-skript/ A number or applications of VECTOR CALCULUS to the letter written above and to the right or left theory of electricity and magnetism. of a letter. A superscript usually denotes a power, e.g. x3, or a derivative, e.g. f4(x) = surface of revolution A surface gener- d4f/dx4, or is sometimes used in the same ated by rotating a curve about an axis. For sense as a SUBSCRIPT. example, rotating a parabola about its axis of symmetry produces a PARABOLOID of rev- supplementary angles A pair of angles olution. that add together to make a straight line (180° or π radians). Compare complemen- syllogism /sil-ŏ-jiz-ăm/ In logic, a deduc- tary angles; conjugate angles. tive argument in which a conclusion is derived from two propositions, the major premiss and the minor premiss, the conclusion necessarily being true if the premisses are true. For example, ‘Tim wants a car or a bicycle’, ‘Tim does not want a car’, therefore ‘Tim wants a bicycle’. A hypothetical αβ syllogism is a particular type of syllogism of the form ‘A implies B’, ‘B implies C’, Supplementary angles: α + β = 180°. therefore ‘A implies C’.

symbol A letter or character used to represent an object, operation, quantity, rela- supplementary units The dimensionless tion, or function. See the appendix for a list units – the radian and the steradian – used of mathematical symbols. along with base units to form derived units. See SI units. symbolic logic (formal logic) The branch of logic in which arguments, the terms used supremum /soo-pree-mŭm/ (sup) The in them, the relationships between them, least upper BOUND of a set. and the various operations that can be performed on them are all represented by surd /serd/ See irrational number. symbols. The logical properties and IMPLI- CATIONS of arguments can then be more surface Any locus of points extending in easily studied strictly and formally, using two dimensions. It is defined as an area. A algebraic techniques, proofs, and theorems surface may be flat (a plane surface) or in a mathematically rigorous way. It is curved and may be finite or infinite. For ex- sometimes called mathematical logic. ample, the plane z = 0 in three-dimensional The simplest system of symbolic logic is Cartesian coordinates is flat and infinite; propositional logic (sometimes called the outside of a sphere is curved and finite. propositional calculus) in which letters, e.g. P, Q, R, etc., stand for propositions or surface integral An integral in which the statements, and various special symbols integration takes place along a surface. It is stand for relationships that can hold be- possible to define a surface integral in tween them. See also biconditional; con- terms of a parametric representation for junction; disjunction; negation; truth table. the surface and then show that the value of the integral is independent of the paramet- symmetrical Denoting any figure that ric representation under general condi- can be divided into two parts that are mir- tions. There are many mathematical and ror images of each other. The letter A, for physical applications of surface integrals example, is symmetrical, and does not

215 symmetric matrix change when viewed in a mirror, but the sometimes enables conclusions to be letter R is not. A symmetrical plane figure drawn without performing a calculation. has at least one line that is an axis of sym- Symmetry is systematically analyzed in metry, which divides it into two mirror im- mathematics (and its applications) using ages. GROUP theory. In the specific context of graphs of symmetric matrix /să-met-rik/ A matrix functions y = f(x) there are symmetries as- A that has its transpose equal to A, i.e. if sociated with EVEN FUNCTIONS and ODD FUNCTIONS, with even functions having a the entries of A are denoted by aij then aij = symmetry about the y-axis while odd func- aji for all and j in the matrix. tions have a symmetry about the origin. symmetry /sim-ĕ-tree/ The property that figures and bodies can have that these syntax /sin-taks/ In logic, syntax concerns entities appear unchanged after certain the properties of formal systems which do not depend on what the symbols actually transformations, called symmetry transfor- mean. It deals with the way the symbols mations, are performed on them. Examples can be connected together and what com- of symmetry transformations include rota- binations of symbols are meaningful. The tions about a fixed point, reflection about syntax of a formal logical language will a mirror plane, and translation along a lat- specify precisely and rigorously which for- tice. For example, in a square rotation mulae are well-formed within the lan- ° about the center of the square by 90 is a guage, but not their intuitive meaning. symmetry transformation since the square Compare semantics. appears unchanged after the rotation, whereas rotation about the center by 45° is systematic error See error. not a symmetry transformation since the position of the square is different after the systematic sampling See sampling. transformation. Taking symmetry into account fre- Système International d’Unités /see- quently simplifies the mathematical analy- stem an-tair-nas-yo-nal doo-nee-tay/ See SI sis of a problem considerably and units.

216 T

tan See tangent. 0°, and becomes an infinitely large positive number as α approaches 90°. At + 90°, tangent 1. A straight line or a plane that tanα jumps from +∞ to –∞ and then rises to touches a curve or a surface without cut- zero at α = 180°. See also trigonometry. ting through it. On a graph, the slope of the tangent to a curve is the slope of the curve tangent plane The plane through a point at the point of contact. In Cartesian coor- P on a smooth surface that is perpendicular dinates, the slope is the derivative dy/dx. If to the normal to the surface at P. All the θ is the angle between the x-axis and a tangent lines at P lie in the tangent plane, straight line to the point (x,y) from the ori- with a tangent line at P being a tangent at gin, then the trigonometric function tanθ = P to any curve on the surface that goes y/x. through P. 2. (tan) A trigonometric function of an It is frequently the case that all points angle. The tangent of an angle α in a right- on the surface near P are on the same side angled triangle is the ratio of the lengths of of the tangent plane at P. However, at a the side opposite to the side adjacent. This SADDLE POINT some of the points close to P definition applies only to angles between are on one side of the tangent plane while 0° and 90° (0 and π/2 radians). More gen- other points close to P are on the other side erally, in rectangular Cartesian coordi- of the tangent plane. nates, with origin O, the ratio of the y-coordinate to the x-coordinate of a point tanh /th’an, tansh, tan-aych/ A hyperbolic P (x,y) is the tangent of the angle between tangent. See hyperbolic functions. the line OP and the x-axis. The tangent function, like the sine and cosine functions, tautology /taw-tol-ŏ-jee/ In LOGIC, a is periodic, but it repeats itself every 180° proposition, statement, or sentence of a and is not continuous. It is zero when α = form that cannot possibly be false. For ex-

3 – 2 – 1 – 0 x – – – – –π/2 π/2 π 3π/2 –1 – –2 – –3 –

Tangent: graph of y = tan x, with x in radians.

217 Taylor series ample, ‘if all pigs eat mice then some pigs often at some distance from it, and is eat mice’ and ‘if I am coming then I am linked to it by electric cable, telephone, or coming’ are both true regardless of some other transmission channel. A mouse whether the component propositions ‘I am or keyboard, similar to that on a type- coming’ and ‘all pigs eat mice’ are true or writer, is used to feed information (data) to false. More strictly, a tautology is a com- the computer. The output can either be pound proposition that is true no matter printed out or can be displayed on a screen, what truth values are assigned to the sim- as with a VISUAL DISPLAY UNIT. An interac- ple propositions that it contains. A tautol- tive terminal is one connected to a com- ogy is true purely because of the laws of puter, which gives an almost immediate logic and not because of any fact about the response to an enquiry from the user. An World (the laws of thought are tautolo- intelligent terminal can store information gies). A tautology therefore contains no in- and perform simple operations on it with- formation. Compare contradiction. out the assistance of the computer’s central processor. See also input/output. Taylor series (Taylor expansion) A formula for expanding a function, f(x), by terminating decimal A DECIMAL that has writing it as an infinite series of derivatives a finite number of digits after the decimal for a fixed value of the variable, x = a: point, also called a finite decimal. f(x) = f(a) + f′(a)(x – a) + ″ 2 ′″ 3 f (a)(x – a) /2! + f (a)(x – a) /3! + … ternery /ter-nĕ-ree/ Describing a number If a = 0, the formula becomes: system to the base (radix) 3. It uses the f(x) = f′(0) + f′(0)x = f″(0)x2/2! + … numbers 0, 1, and 2 (with place values … The Taylor series is named for the 243, 81, 27, 9, 3, 1). See also binary. British mathematician Brook Taylor (1685–1731) and the Maclaurin series, or tesla /tess-lă/ Symbol: T The SI unit of Maclaurin expansion (a special case of the magnetic flux density, equal to a flux den- Taylor series) is named for the Scottish sity of one weber of magnetic flux per mathematician Colin Maclaurin (1698– square meter. 1 T = 1 Wb m–2. The unit is 1746). See also expansion. named for the Croatian–American physi- t-distribution See Student’s t-distribu- cist Nikola Tesla (1856–1943). tion. tessellation /tess-ă-lay-shŏn/ A regular tension A force that tends to stretch a pattern of tiles that can cover a surface body (e.g. a string, rod, wire, etc.). without gaps. Regular polygons that will, on their own, completely tessellate a sur- tensor /ten-ser/ A mathematical entity that face are equilateral triangles, squares and is a generalization of a vector. Tensors are regular hexagons. used to describe how all the components of a quantity in an n-dimensional system be- tetrahedral numbers /tet-ră-hee-drăl/ have under certain transformations, just as For a positive integer n, a number defined a VECTOR can describe a translation from as the sum of the first n TRIANGULAR NUM- one point to another in a plane or in space. BERS. This definition means that the nth tetrahedral number is given by (n + 1)(n + tera- Symbol: T A prefix denoting 1012. 2)(n/6). The first four triangular numbers For example, 1 terawatt (TW) = 1012 watts are 1, 4, 10, and 20. (W). tetrahedron /tet-ră-hee-drŏn/ (triangular terminal A point at which a user can com- pyramid) A solid figure bounded by four municate directly with a computer both for triangular faces. A regular tetrahedron has the input and output of information. It is four congruent equilateral triangles as its situated outside the computer system, faces. See also polyhedron; pyramid.

218 topological space

B A D

D C Top view C Tetrahedron tetrakaidekahedron /tet-ră-kÿ-dek-ă- nate system using three variables, for ex- hee-drŏn/ A polyhedron bounded by six ample, three-dimensional Cartesian coor- squares and eight hexagons. It can be dinates with an x-axis, y-axis, and z-axis. formed by truncating the six corners of an Compare two-dimensional. octahedron in a symmetrical way. It is possible to fill the whole of space without thrust A force tending to compress a body either gaps or overlaps with tetrakai- (e.g. a rod or bar) in one direction. Thrust dekahedra. The tetrakaidekahedron is an acts in the opposite direction to tension. example of a SEMI-REGULAR POLYHEDRON. time sharing A method of operation in theorem /th’ee-ŏ-rĕm, th’eer-ĕm/ The con- computer systems in which a number of clusion which has been proved in the jobs are apparently executed simultane- course of an argument upon the basis of ously instead of one after another (as in certain given assumptions. A theorem must BATCH PROCESSING). This is achieved by be a result of some general importance. transferring each program in turn from Compare lemma. backing store to main store and allowing it to run for a short time. theorem of parallel axes If I0 is the moment of the inertia of an object about an ton /tun/ 1. A unit of mass equal to 2240 axis, the moment of inertia I about a par- pounds (long ton, equivalent to 1016.05 allel axis is given by: kg) or 2000 pounds (short ton, equivalent 2 I = I0md to 907.18 kg). where m is the mass of the object and d is 2. A unit used to express the explosive the separation of the axes. power of a nuclear weapon. It is equal to an explosion with an energy equivalent of theory of games See game theory. one ton of TNT or approximately 5 × 109 joules. therm A unit of heat energy equal to 105 British thermal units (1.055 056 joules). tonne /tun/ (metric ton) Symbol: t A unit of mass equal to 103 kilograms. third-order determinant See determinant. topologically equivalent See topology. thou /th’ow/ See mil. topological space /top-ŏ-loj-ă-kăl/ A non-empty set A together with a fixed col- three-dimensional Having length, lection (T) of subsets of A satisfying: breadth, and depth. A three-dimensional 1. Ø ∈T, A ∈T; figure (solid) can be described in a coordi- 2. if U ∈T and V ∈T then U∩V ∈ T;

219 topology

∈ 3. if Ui T, where {Ui} is a finite or infinite torque /tork/ Symbol: T A turning force ∩ ∈ collection of sets, then Ui T. (or MOMENT). The torque of a force F The set of subsets T is called a TOPOLOGY about an axis (or point) is Fs, where s is the for A, and the members of T are called the distance from the axis to the line of action open sets of the topological space. Com- of the force. The unit is the newton meter. pare metric space. Note that the unit of work, also the newton meter, is called the joule. Torque is not, topology /tŏ-pol-ŏ-jee/ A branch of geom- however, measured in joules. The two etry concerned with the general properties physical quantities are not in fact the same. of shapes and space. It can be thought of as Work (a scalar) is the scalar product of the study of properties that are not force and displacement. Torque is the vec- changed by continuous deformations, such tor product F × s and is a vector at 90° to as stretching or twisting. A sphere and an the plane of the force and displacement. ellipsoid are different figures in solid (Eu- See also couple. clidean) geometry, but in topology they are considered equivalent since one can be torr A unit of pressure equal to a pressure transformed into the other by a continuous of 101 325/760 pascals (133.322 Pa). It is deformation. A torus, on the other hand, is equal to the mmHg. The unit is named for not topologically equivalent to a sphere – it the Italian physicist Evangelista Torricelli would not be possible to distort a sphere (1608–47). into a torus without breaking or joining surfaces. A torus is thus a different type of torsional wave /tor-shŏ-năl/ A wave mo- shape to a sphere. Topology studies types tion in which the vibrations in the medium of shapes and their properties. A special are rotatory simple harmonic motions case of this is the investigation of networks around the direction of energy transfer. of lines and the properties of knots. In fact Euler’s study of the KÖNIGSBERG torus /tor-ŭs, toh-rŭs/ (anchor ring) A BRIDGE PROBLEM was one of the earliest re- closed curved surface with a hole in it, like sults in topology. A modern example is in a donut. It can be generated by rotating a the analysis of electrical circuits. A circuit circle about an axis that lies in the same diagram is not an exact reproduction of the plane as the circle but does not cut it. A paths of the wires, but it does show the connections between different points of the cross-section of the torus in a plane per- circuit (i.e. it is topologically equivalent to pendicular to the axis is two concentric cir- the circuit). In printed or integrated circuits cles. A cross-section in any plane that it is important to arrange connections so contains the axis is a pair of congruent cir- that they do not cross. cles at equal distances on both sides of the π 2 Topology uses methods of higher alge- axis. The volume of the torus is 4 dr and π2 bra including group theory and set theory. its surface area is 3 dr, where r is the ra- An important notion is that of sets of dius of the generating circle and d is the points in the neighborhood of a given point distance of its center from the axis. (i.e. within a certain distance of the point). An open set is a set of points such that each total derivative A derivative that can be point in the set has a neighborhood con- expressed in terms of a series of partial de- taining points in the set. A topological rivatives. For example, if the function z = transformation occurs when there is a one- f(x,y) is a continuous function of x and y, to-one correspondence between points in and both x and y are continuous functions one figure and points in another so that of another variable t, then the total deriva- open sets in one correspond to open sets in tive of z with respect to t is: the other. If one figure can be transformed dz/dt = (∂z/∂x)(dx/dt) + into another by such a transformation, the (∂z/∂y)(dy/dt) sets are topologically equivalent. See also chain rule; total differential.

220 translatory motion total differential An infinitesimal change transformation of coordinates 1. in a function of one or more variables. It is Changing the position of the reference axes the sum of all the partial differentials. See in a cordinate system by translation, rota- differential. tion, or both, usually to simplify the equation of a curve. See rotation of axes; trace See square matrix. translation of axes. 2. Changing the type of coordinate system track See disk; drum; magnetic tape; in which a geometrical figure is described; paper tape. for example, from Cartesian coordinates to POLAR COORDINATES. trajectory /tră-jek-tŏ-ree/ 1. The path of a moving object, e.g. a projectile. transitive relation A relation *on a set 2. A curve or surface that satisfies some A such that if a*b and b*c then a*c for all given set of conditions, such as passing a, b, and c in A. ‘Greater than’, ‘less than’, through a given set of points, or having a and ‘equals’ are examples of transitive re- given function as its gradient. lations. A relation that is not transitive is an intransitive relation. transcendental number /tran-sen-den- tăl/ See irrational number; pi. translation /trans-lay-shŏn, tranz-/ The moving of a geometrical figure so that only transfinite /trans-fÿ-nÿt/ A transfinite its position relative to fixed axes is number is a cardinal or ordinal number changed, but not its orientation, size, or that is not an integer. See cardinal number; shape. See also translation of axes. See il- ordinal number; aleph. lustration overleaf. Transfinite induction is the process by which we may reason that if some proposi- translation of axes In coordinate geom- tion is true for the first element of a well- etry, the shifting of the reference axes so ordered set S, and that if the proposition is that each axis is parallel to its original po- true for a given element it is also true for sition and each point is given a new set of the next element, then the proposition is coordinates. For example, the origin O of a true for every element of S. system of x- and y-axes, may be shifted to the point O′, (3, 2,) in the original system. transformation /trans-fer-may-shŏn/ 1. The new axes x′ and y′ are at x = 3 and y = In general, any FUNCTION or mapping that 2, respectively. This is sometimes done to changes one quantity into another. simplify the equation of a curve. The circle 2. The changing of an algebraic expression (x – 3)2 + (y – 2)2 = 4 can be described by or equation into an equivalent with a dif- new coordinates x′ = (x – 3) and y′ = (y – ferent form. For example, the equation 3): (x′)2 + (y′)2 = 4. The origin O′ is then at (x – 3)2 = 4x + 2 the center of the circle. See also rotation of can be transformed into axes. x2 – 10x + 7 = 0 See also changing the subject of a formula. translatory motion /trans-lă-tor-ee, -toh- 3. In geometry, the changing of one shape ree, tranz-/ (translation) Motion involving into another by moving each point in it to change of position; it compares with rota- a different position by a specified proce- tory motion (rotation) and vibratory mo- dure. For example, a plane figure may be tion (vibration). Each is associated with moved in relation to two rectangular axes. kinetic energy. In an object undergoing Another example is when a figure is en- translatory motion, all the points move in larged. See translation. See also deforma- parallel paths. Translatory motion is usu- tion; dilatation; enlargement; projection; ally described in terms of (linear) speed or rotation. velocity, and acceleration.

221 transpose of a matrix

A transformation can be represented by a 2 x 2 matrix. A point (x, y) is tranformed to a point (x’, y’) by multiplying the column vector of (x, y) by a matrix M x i.e. M y

The transformation matrices are:

1 0 reflection in x-axis 0 –1

reflection in y-axis –1 0 0 1

k 0 enlargement scale factor k 0 k

k 0 stretch in x-direction 0 1

1 0 stretch in y-direction 0 k

1 k shear in x-direction by k 0 1

rotation by α cos α –sin α (anticlockwise positive) sin α cos α

y y

PP’ P’

x x stretch in x-direction shear in x-direction

Transformations

222 transpose of a matrix

6 – k j

5 – 3i

4 – B’

B 3 – A’

2 – A C’

1 – C

0 x – – – – – – – 1 234 56

The translation of a triangle ABC The translation vector k = 3i + j

A translation by a in the x direction and b in the y direction transforms a point (x, y) to (x’, y’).

In matrix terms x’ =+x a y’ y b

Translation

transpose of a matrix /trans-pohz/ The trix. The transpose of a row vector is a col- matrix that results from interchanging the umn vector and vise versa. If two matrices rows and columns of a given matrix. The A and B are comformable (can be multi- determinant of the transpose of a square plied together), then the transpose of the ~ ~ ~ ~ ~ matrix is equal to that of the original ma- matrix product AB = C is C = (AB) = BA. In

223 transversal

trapezoid /trap-ĕ-zoid/ A quadrilateral A = a b → Ã = a a a 1 1 1 2 3 with two parallel sides. It is sometimes part a b b b b 2 2 1 2 3 of the definition that the other sides are not a3 b3 parallel. The parallel sides are called the ~ Transpose A of a matrix A. bases of the trapezium and the perpendicular distance between the bases is called the altitude. The area of a trapezium is the product of the sum of the parallel side other words, in taking the transpose of a lengths and half the perpendicular distance matrix product the order must be reversed. between them. transversal /trans-ver-săl, tranz-/ A line trapezoid rule A rule for finding the ap- that intersects two or more other lines. proximate area under a curve by dividing it into pairs of trapezium-shaped sections, transverse axis /trans-vers, tranz-, trans- forming vertical columns of equal width vers, tranz-/ See hyperbola. with bases lying on the horizontal axis. The trapezium rule is used as a method of NU- transverse wave A wave motion in which MERICAL INTEGRATION. For example, if the the motion or change is perpendicular to value of a function f(x) is known at x = a, the direction of energy transfer. Electro- x = b, and at a value mid-way between a magnetic waves and water waves are ex- and b, the integral is approximately: amples of transverse waves. Compare (h/2){f(a) + 2f[(a + b)/2] + f(b)} longitudinal waves. h is half the distance between a and b. If this does not give a sufficiently accurate re- trapezium /tră-pee-zee-ŭm/ (pl. trapezi- sult, the area may be subdivided into 4, 6, ums or trapezia) A quadrilateral, none of 8, etc., columns until further subdivision whose sides are parallel. Note that in the makes no significant difference to the re- UK the terms ‘trapezium’ and ‘trapezoid’ sult. See also Simpson’s rule. have the opposite meanings to those in the US. travel graph A graph of displacement

y f(b) f((a + b)/2) •y = f(x) •

f(a•)

h h x O ab

Trapezium rule

224 tree diagram

• equilateral isosceles scalene

ϭ Ϫ Ϫ ϭ

• Ϫ •

Types of triangle C

ϭ CD Ϫ

D E ϭ Ϫ

A AB B

The line joining the mid-points of two sides Triangles on the same base and between the of a triangle is parallel to the third side and same parallels are equal in area (area ABC = equal to half of it (AB = 2DE). area ABD).

Triangles

(usually plotted on a y-axis) against time lowest value. However, for n towns the (usually plotted on the x-axis) which can number of routes is (n–1)!/2, which means be used to illustrate a journey of a person. that, except for small values of n, this num- The velocity of a person at any time is the ber becomes too large for any computer to gradient of the curve at that time. The con- handle. It is possible to find algorithms that cept of travel graph can be extended to find roots that are near to the minimum, graphs of velocity against time, with accel- but not exactly at the minimum, which per- eration being the gradient of the curve and form the calculation fairly quickly. displacement being the area under the curve. traversable network A network that can be drawn by beginning at a point on the traveling wave See wave. network and not lifting the pen from the paper, without going over any line traveling salesman problem The prob- twice. lem of minimizing the cost or time to visit a number of towns, with the route from tree diagram A type of diagram used as town a to town b being given a value of cab. an aid to solve problems in probability the- This problem can, in principle, be solved ory. The different ways in which an event, by calculating the cost or time for all possi- such as drawing balls with various colours ble routes and seeing which route has the out of a bag, can occur, are drawn as

225 trial solution

a11 a12 a13 a11 0 0

0 a22 a23 a21 a22 0

0 0 a33 a31 a32 a33

Triangular matrix: these are 3 x 3 matrices.

branches of a tree, with each branch show- forces. Similarly a triangle of velocities can ing the probability of that particular event be constructed. occurring. triangle of velocities See triangle of vec- trial solution A guess at a solution to a tors. problem such as a differential equation. The trial solution is substituted into the triangular matrix A square matrix in problem to be solved. If it does not solve which either all the elements above the the problem exactly then the trial solution leading diagonal or all the elements below is improved upon until it is a satisfactory the leading diagonal are zero. The determi- solution to the problem. nant of a triangular matrix is equal to the product of its diagonal elements. triangle A plane figure with three straight sides. The area of a triangle is half the triangular numbers The set of numbers length of one side (the base) times the {1, 3, 6, 10, …} generated by triangular ar- length of the perpendicular line (the alti- rays of dots. Each array has one more row tude) from the base to the opposite vertex. than the preceding one, the additional row The sum of the internal angles in a triangle having one more dot than the longest in the is 180° (or π radians). In an equilateral tri- preceding array. The nth triangular num- angle, all three sides have the same length ber is n(n + 1)/2. and all three angles are 60°. An isosceles triangle has two sides of equal length and two equal angles. A scalene triangle has no • 1 equal sides or angles. In a right-angled tri- 3 angle, one angle is 90° (or π/2 radians), and •• the others therefore add up to 90°. In an ••• 6 acute angled triangle, all the angles are less 10 than 90°. In an obtuse-angled triangle, one • ••• of the angles is greater than 90°. See illus- ••••• 15 tration overleaf. •••••• 21 triangle inequality For any triangle ••••••• 28 ABC, the length of one side is always less than the sum of the other two: Triangular numbers AB < BC + CA triangle of forces See triangle of vectors. triangular prism See prism. triangle of vectors A triangle describing three coplanar VECTORS acting at a point triangular pyramid See tetrahedron. with zero resultant. When drawn to scale – shown correctly in size, direction, and trichotomy /trÿ-kot-ŏ-mee/ The property sense, but not in position – they form a of an ordering that exactly one of the state- closed triangle. Thus three forces acting on ments x < y, x = y, x > y is true for any x an object at equilibrium form a triangle of and y. See linearly ordered.

226 triple vector product trigonometric functions /trig-ŏ-nŏ-met- 1 + cot2α = cosec2α rik/ See trigonometry. See also addition formulae; cosine rule; sine rule. trigonometry /trig-ŏ-nom-ĕ-tree/ The study of the relationships between the sides trillion In the US and Canada, a number and angles in a triangle, in terms of the represented by 1 followed by 12 zeros trigonometric functions of angles (sine, co- (1012). In the UK, 1 followed by 18 zeros sine, and tangent). Trigonometric func- (1018). tions, can be defined by the ratio of sides in a right-angled triangle. In a right-angled trinomial /trÿ-noh-mee-ăl/ An algebraic α triangle with an angle , o is the length of expression with three variables in it. For α the side opposite , a the length of side ad- example, 2x + 2y + z and 3a + b = c are tri- jacent to α, and h the length of the hy- nomials. Compare binomial. potenuse. For such a triangle the three trigonometric functions of α are defined triple integral The result of integrating by: the same function three times. For exam- sinα = o/h cosα = a/h ple, if a function f(x,y,z) is integrated first tanα = o/a with respect to x, holding y and z constant, The following relationships hold for all and the result is then integrated with re- values of the variable angle α: spect to y, holding x and z constant, and cosα = sin(α + 90°) the resultant double integral is then inte- tanα = sinα/cosα grated with respect to z holding x and y The trigonometric functions of an angle constant, the triple integral is can also be defined in terms of a circle ∫∫∫f(x,y,z)dzdydx (hence they are sometimes called circular See also double integral. functions). A circle is taken with its center at the origin of Cartesian coordinates. If a triple product A product of three vectors. point P is taken on this circle, the line OP See triple scalar product; triple vector makes an angle α with the positive direc- product. tion of the x-axis. Then, the trigonometric functions are: triple scalar product A product of three tanα = y/x vectors, the result of which is a scalar, de- sinα = y/OP fined as: α cos = x/OP A.(B × C) = ABCsinθcosφ Here, (x,y) are the coordinates of the where φ is the angle between A and the vec- √ 2 2 point P and OP = (x + y ). The signs of x tor product (B × C), θ being the angle be- and y are taken into account. For example, tween B and C. The scalar triple product is for an angle β° between 90° and 180°, y equal to the volume of the parallelepiped of will be positive and x negative. Then: which A, B, and C form nonparallel edges. tanβ = –tan(180 – β) If A, B, and C are coplanar, their triple sinβ = + sin(180 – β) cosβ = –cos(180 – β) scalar product is zero. Similar relationships can be written for the trigonometric functions of angles be- triple vector product A product of three tween 180° and 270° and between 270° vectors, the result of which is a vector. It is and 360°. a vector product of two vectors, one of The functions secant (sec), cosecant which is itself a vector product. That is: (cosec), and cotangent (cot), which are the A × (B × C) = (A.C)B – (A.B)C reciprocals of the cosine, sine, and tangent Similarly functions respectively, obey the following (A × B) × C = (A.C)B – (B.C)A rules for all values of α: These are equal only when A, B, and C tan2α + 1 = sec2α are mutually perpendicular.

227

trirectangular ∧ ∧ trirectangular /trÿ-rek-tang-gyŭ-ler/ P Q (P Q) ~P Having three right angles. See spherical tri- T T T T F T F F F F angle. F T F T T F F F T T trisection /trÿ-sek-shŏn/ Division into three equal parts. Truth table trivial solution A solution to an equation or set of equations that is obvious and gives no useful information about the relation- assigns a value to the whole. The truth- ships between the variables involved. For table definitions for conjunction, disjunc- example, x2 + y2 = 2x + 4y has a trivial so- tion, negation, and implication are shown lution x = 0; y = 0. at those headwords. An example of a truth table for a com- trochoid /troh-koid/ The curve described plex, or compound proposition is shown in by a fixed point on the radius or extended the illustration. The assigning of values radius of a circle as the circle rolls (in a proceeds in this way: on the basis of the plane) along a straight line. Let r be the ra- truth values of P and Q, the simple propo- dius of the circle and a the distance of the sitions are given truth values, written fixed point from the center of the circle. If under the sign ∧ (in P∧Q) and under ∼P. a > r the curve is called a prolate cycloid, if Using these, truth values can then be as- a < r the curve is called a curtate cycloid, signed to the whole; in the example this is and if a = r the curve is a CYCLOID. in effect a complex disjunction and the values are written under the sign ∨. truncated Describing a solid generated Thus in the case where P is true and Q from a given solid by two non-parallel is false, P∧Q will be false, ∼P will be false, planes cutting the given solid. and therefore the whole will be false. See also symbolic logic. truncation The dropping of digits from a number, used as an approximation when truth value The truth or falsity of a that number has more digits than it is con- proposition in logic. A true statement or venient or possible to deal with. This is dif- proposition is indicated by T and a false ferent to ROUNDING since in rounding the one by F. In computer logic, the digits 1 number with the required digits is the near- and 0 are often used to denote truth values. est number with the required digits, which See also truth table. is not necessarily the number obtained by truncation. For example, if truncated to Turing machine /tewr-ing/ An abstract one decimal place 2.791 and 2.736 both model of a computer that consists of a con- give 2.7 whereas rounding of the first num- trol or processing unit and an infinitely ber gives 2.8 and rounding of the second long tape divided into single squares along number gives 2.7. its length. At any given time each square of the tape is either blank or contains a single truth table In logic, a mechanical proce- symbol from a fixed finite list of symbols dure (sometimes called a matrix) that can s1, s2, …, sn. The Turing machine moves be used to define certain logical operations, along the tape square by square and reads, and to find the truth value of complex erases, and prints symbols. At any given propositions or statements containing time the Turing machine is in one of a fixed combinations of simpler ones. A truth finite number of states represented by q1, table lists, in rows, all the possible combi- q2, …, qn. The ‘program’ for the machine is nations of truth values (T = ‘true’, F = made up of a finite set of instructions of the ‘false’) of a PROPOSITION or statement, and form qisjskXqj, where X is either R (move given an initial assignment of truth or fal- to the right), L (move to the left), or N sity to the constituent parts, mechanically (stay in the same position). Here, qi is the 228 two-dimensional state of a machine reading sj, which it function at which the slope (gradient) of changes to sk, then moves left, right, or the tangent to a continuous curve changes stays and completes the operation by going its sign. If the slope changes from positive into state qj. A Turing machine can be used to negative, that is, the y-coordinate stops to define computability. The machine is increasing and starts decreasing, it is a named for the British mathematician Alan maximum point. If the slope changes from Mathison Turing (1912–54). See com- negative to positive it is a minimum point. putability. Turning points may be local maxima and minima or absolute maxima and minima. turning effect The ability of a force to All turning points are STATIONARY POINTS. cause a body to rotate about an axis of ro- At a turning point the derivative, dy/dx, of tation. It is described quantitatively by the the curve y = f(x) is zero. torque, sometimes called the moment of the force, which is the product of the mag- two-dimensional Having length and nitude of the force, denoted F, and the per- breadth but not depth. Flat shapes, such as pendicular distance d of the force from the circles, squares, and ellipses, can be de- axis of rotation. The unit of torque is the scribed in a coordinate system using newton meter (Nm). only two variables, for example, two- dimensional Cartesian coordinates with an turning point A point on the graph of a x-axis and a y-axis. See also plane.

229 U

unary operation /yoo-nă-ree/ A mathe- underdamping /un-der-damp-ing/ See matical procedure that changes one num- damping. ber into another. For example, taking the square root of a number is a unary opera- unicursal /yoo-nă-ker-săl/ Describing a tion. Compare binary operation. closed curve that, starting and finishing at the same point, can be traced in one sweep, unbounded An unbounded function is a without having any part retraced. function that is not bounded – it has no bounds or limits. Intuitively this means uniform acceleration Constant accelera- that for any number N there is a value of tion. the function numerically greater than N, i.e. there is a point x such that |f(x)| > N. uniform distribution See distribution For example, the function f(x) = x is un- function. bounded on the real axis, and f(x) = 1/x is unbounded on the interval 0 < x ≤ 1. uniform motion A vague phrase, usually taken to mean motion at constant velocity undecidability /un-di-sÿ-dă-bil-ă-tee/ In (constant speed in a straight line). logic, if a formula or sentence can be proved within a given system the formula uniform speed Constant speed. or sentence is said to be decidable. If every formula in a system can be proved then the uniform velocity Constant velocity, de- whole system is decidable. Otherwise the scribing motion in a straight line with zero system is undecidable. The process of de- acceleration. termining which systems are decidable and which are not is an important branch of union Symbol: ∪ The combined set of all logic. the elements of two or more sets. If A = {2,

A B

A ∪ B

Union: the shaded area in the Venn diagram is the union of sets A and B. upthrust

4, 6} and B = {3, 6, 9}, then A ∪ B = {2, 3, tion for a unit circle in Cartesian coordi- 4, 6, 9}. See also Venn diagram. nates is x2 + y2 = 1. A unit circle in the complex plane gives the set of complex unique factorization theorem A result numbers z for which the modulus |z| has in number theory which states that a posi- the value 1. tive integer can be expressed as a product of prime numbers in a way which is unit fraction See fraction. unique, except for the order in which the prime factors are written. This theorem fol- unit matrix (identity matrix) Symbol: I A lows from fundamental axioms for prime square MATRIX in which the elements in the numbers. The factorization of a number leading diagonal are all equal to one, and into prime numbers (and powers of prime the other elements are zero. If a maxtrix A numbers) is called the prime decomposi- with m rows and n columns is multiplied tion of that number. by an n × n unit matrix, I, it remains unchanged, that is IA = A. The unit matrix is uniqueness theorem /yoo-neek-nis/ A the identity matrix for matrix multiplica- theorem which shows that there can only tion. be a single entity satisfying a given condition, for example only one solution to a unit vector A vector with a magnitude of given equation. The proof of such a theo- one unit. Any vector r can be expressed in rem usually proceeds by assuming that terms of its magnitude, the scalar quantity there exist two distinct entities satisfying r, and the unit vector r′ which has the same the condition and showing that this leads direction as r. The vector r = rr′, where r is to a contradiction. the magnitude of r′. In three-dimensional Cartesian coordinates with origin O unit unique solution The only possible value vectors i, j, and k are used in the x-, y-, and of a variable that can satisfy an equation. z-directions respectively. For example, x + 2 = 4 has the unique solution x = 2, but x2 = 4 has no unique solu- universal quantifier In logic, a symbol tion because x = +2 and x = –2 both satisfy meaning ‘for all’ and usually written ∀ . the equation. For example (∀ x)Fx means ‘for all x, the property F is true’. unit A reference value of a quantity used to express other values of the same quan- universal set Symbol: E The set that con- tity. See also SI units tains all possible elements. In a particular problem, E will be defined according to the unitary matrix A matrix in which the in- scope of the problem. For example, in a verse of the matrix is the complex conju- calculation involving only positive num- gate of the transpose of the matrix. If the bers, the universal set, E, is the set of all determinant of a unitary matrix has a value positive numbers. See also Venn diagram. 1 then the matrix is said to be a special unitary matrix. The set of all unitary N × N unstable equilibrium Equilibrium such matrices form a group called the unitary that if the system is disturbed a little, there group, denoted U(N), with the group of all is a tendency for it to move further from its special unitary N × N matrices forming a original position rather than to return. See group called the special unitary group, de- stability. noted SU(N). There are many important physical applications of unitary groups and upper bound See bound. matrices. upthrust An upward force on an object in unit circle A circle in which the radius is a fluid. In a fluid in a gravitational field the one unit and the origin of the coordinate pressure increases with depth. The pres- system is the center of the circle. The equa- sures at different points on the object will

231 utility programs therefore differ and the resultant is verti- Utility programs have a number of uses. cally upward. See also Archimedes’ princi- They can be used, for example, to make ple. copies of files (organized collections of data) and to transfer data from one storage utility programs PROGRAMS that help in device to another, as from a magnetic tape the general running of a computer system. unit to a disk store. See also program.

232 V

validity In logic, a property of arguments, variational principle A mathematical inferences, or deductions. An argument is principle stating that the value of some valid if it is impossible that the conclusion quantity either has to be a minimum or be false while the premisses are true. That (more rarely) a maximum out of all possi- is, to assert the premisses and deny the con- ble values. Many physical laws can be clusion would be a contradiction. stated as variational principles, with FER- MAT’S PRINCIPLE of geometrical optics being variable A changing quantity, usually de- an example of this. Variational principles noted by a letter in algebraic equations, are also used to calculate quantities of in- that might have any one of a range of pos- terest in physical science and engineering sible values. Calculations can be carried approximately. out on variables because certain rules apply to all the possible values. For exam- VDU See visual display unit. ple, to carry out the operation of squaring all the integers between 0 and 10, and vector /vek-ter/ A measure in which direc- equation can be written in terms of an in- tion is important and must usually be spec- teger variable n, : y = n2, with the condition ified. For instance, displacement is a vector that n is between 0 and 10 (0

233 vector difference

b r r d

c a b a The parallelogram law: r is the resultant of The polygon of vectors: r is the resultant. a and b.

rrr

Resolution of the vector r into different pairs of components

Vectors: addition and resolution of vectors types of operator associated with differen- The vector difference may also be cal- tiation in vector calculus: CURL, DIV, GRAD. culated by taking the difference in the mag- Some of the key theorems for integrals in nitudes of the corresponding components vector calculus are GAUSS’S THEOREM, of each vector. For example, for two plane GREEN’S THEOREM, and STOKES’ THEOREM. vectors in a Cartesian coordinate system. Vector calculus was originally formulated A = 4i + 2j for vectors in two and three dimensions in B = 2i + j Euclidean space, but can be generalized to where i and j are the unit vectors parallel to higher dimensions and curved manifolds. the x- and y-axes respectively, There are many physical applications of A – B = 2i + j vector calculus, particularly in the theory See also vector sum. of electricity and magnetism. vector equation of a line An equation vector difference The result of subtract- for a straight line in space when it is ex- ing two VECTORS. On a vector diagram, pressed in terms of vectors. If a is the posi- two vectors, A and B, are subtracted by tion vector of a point A on the line and v is drawing them tail-to-tail. The difference, A any vector which has its direction along the – B is the vector represented by the line line, then the line is the set of points P from the head of B to the head of A. If A which have their position vector p given by and B are parallel, the magnitude of the dif- p = a + cv, for some value of c. This type of ference is the difference of the individual equation is a vector equation of a line. The magnitudes. If they are antiparallel, it is the form of the equation means that the point sum of the individual magnitudes. P can only be on the line if p–a is propor-

234 vector product

P(x, y, z)

k j O y

Basic vectors: the vector OP can be represented as ix + jy + kz .

b c

Vector product c = a x b.

Vectors: base vectors and vector product. tional to v, i.e. the two vectors are in the vector graphics See computer graphics. same direction (or in opposite directions). If a line is specified by two points A and vector multiplication Multiplication of B with position vectors a and b respectively two or more vectors. This can be defined in then defining v = b–a produces a vector two ways according to whether the result is equation of the line in the form p = (1– c) a vector or a scalar. See scalar product; vec- a + cb. tor product; triple scalar product; triple vector product. vector equation of a plane An equation for a plane when it is expressed in terms of vector product A multiplication of two vectors. If a point A in the plane has a po- vectors to give a vector. The vector product sition vector a and n is a normal vector, i.e. of A and B is written A × B. It is a vector of a vector which is perpendicular to the magnitude ABsinθ, where A and B are the plane, then the plane is the set of all points magnitudes of A and B and θ is the angle P with position vector p which satisfies the between A and B. The direction of the vec- equation (p–a).n = 0. This type of equation tor product is at right angles to A and B. It is a vector equation of a plane. This equa- points in the direction in which a right- tion can be written in the form p.n = c, hand screw would move turning from A to- where c is a constant. The vector forms of ward B. An example of a vector product is equations of a plane can be converted into the force F on a moving charge Q in a field an equation for a plane in terms of the B with velocity v (as in the motor effect). Cartesian coordinates x, y, and z by taking Here components of the vectors. F = QB × v

235 vector projection

Another example is the product of a The scalars may be real numbers, com- force and a distance to give a moment plex numbers, or elements of some other (turning effect), which can be represented field. by a vector at right angles to the plane in which the turning effect acts. The vector vector sum The result of adding two VEC- product is sometimes called the cross prod- TORS. On a vector diagram, vectors are uct. The vector product is not commutative added by drawing them head to tail. The because sum is the vector represented by the A × B = –(B × A) straight line from the tail of the first to the It is distributive with respect to vector head of the last. If they are parallel, the addition: magnitude of the sum is the sum of the C × (A + B) = (C × A) + (C × B) magnitudes of the individual vectors. If The magnitude of A × B is equal to the two vectors are anti-parallel, then the mag- area of the parallelogram of which A and B nitude of the sum is the difference of the in- form two non-parallel sides. In a three-di- dividual magnitudes. mensional Cartesian coordinate system The vector sum may also be calculated with unit vectors i, j, and k in the x, y, and by summing the magnitudes of the corre- z directions respectively, sponding components of each individual A × B = (a i + a j + a k) × vector. For example, for two plane vectors, 1 2 3 A = 2i + 3j and B = 6i + 4j, in a Cartesian (b1i + b2j + b3k) This can also be written as a determi- coordinate system with unit vectors i and j parallel to the x- and y-axes respectively, nant. See also scalar product. the vector sum, A + B, equals 8i + 7j. See also vector difference. vector projection The vector resulting from an orthogonal projection of one vec- velocities, parallelogram of See paral- tor on another. For example, the vector lelogram of vectors. projection of A on B is bAcosθ where θ is the smaller angle between A and B, and b is velocities, triangle of See triangle of vec- the unit vector in the direction of B. Com- tors. pare scalar projection. velocity Symbol: v Displacement per unit vectors, parallelogram of See parallelo- time. The unit is the meter per second gram of vectors. (m s–1). Velocity is a vector quantity, speed being the scalar form. If velocity is con- vectors, triangle of See triangle of vec- stant, it is given by the slope of a posi- tors. tion/time graph, and by the displacement divided by the time taken. If it is not con- vector space A vector space V over a stant, the mean value is obtained. If x is the scalar field F is a set of elements x, y, …, displacement, the instantaneous velocity is called vectors, together with two algebraic given by operations called vector addition and mul- v = dx/dt tiplication of vectors by scalars, i.e. by ele- See also equations of motion. ments of F, such that: 1. the sum of two elements is written x + y, velocity ratio See distance ratio. and V is an Abelian group with respect to addition; Venn diagram /ven/ A diagram used to 2. the product of a vector, x, and a scalar, show the relationships between SETS. The a, is written ax and a(bx) = (ab)x, 1x = x universal set, E, is shown as a rectangle. In- for all a and b in F and all x in V; side this, other sets are shown as circles. In- 3. the distributive laws hold, i.e. a(x + y) = tersecting or overlapping circles are ax + ay, (a + b)x = ax + bx for all a and b intersecting sets. Separate circles are sets in F and all x and y in V. that have no intersection. A circle inside

236 vulgar fraction another is a subset. A group of elements, or visual display unit (VDU) A computer a subset, defined by any of these relation- terminal with which the user can commu- ships may be indicated by a shaded area in nicate with the computer by means of a the diagram. The diagram was devised by keyboard; this input and also the output the British mathematician John Venn from the computer appears on a cathode- (1834–1923). ray screen. A VDU can operate both as in input and output device. vertex (pl. vertexes or vertices) 1. A point at which lines or planes meet in a figure; volt Symbol: V The SI unit of electrical for example, the top point of a cone or potential, potential difference, and e.m.f., pyramid, or a corner of a polygon or poly- defined as the potential difference between hedron. two points in a circuit between which a 2. One of the two points at which an axis constant current of one ampere flows when of a conic cuts the conic. See ellipse; hyper- the power dissipated is one watt. One volt bola; parabola. is one joule per coulomb (1 V = 1 J C–1). The unit is named for the Italian physicist vertical A direction that is at right angles Count Alessandro Volta (1745–1827). to the horizontal (see horizontal); in everyday terms, upright. In geometry, measure- volume Symbol: V The extent of the space ments PERPENDICULAR to the base line are occupied by a solid or bounded by a closed often described as being vertical. surface, measured in units of length cubed. The volume of a box is the product of its One of the vertically opposite angles length, its breadth, and its height. The SI two pairs of equal angles formed when two unit of volume is the cubic meter (m3). straight lines cross each other. volume integral The integral of a function over a volume. To emphasize the three-dimensional nature of the integral, a volume integral is sometimes written with three integral signs and dxdydz. There are many physical applications of physical in- X tegrals, particularly to the theory of elec- •• tricity and magnetism. X volume of revolution The volume V obtained by rotating the region bounded by y Vertically opposite angles formed at the inter- = f(x), the x-axis and the lines x = a and x section of two straight lines. = b through one complete revolution, i.e. 360%, about the x-axis (with it being assumed that f(x) is continuous in this interval). This volume is given by vibration (oscillation) Any regularly re- ∫ b π 2 aab y dx. peated to-and-fro motion or change. Ex- amples are the swing of a pendulum, the The expression can be derived by divid- vibration of a sound source, and the ing the volume of revolution into very thin change with time of the electric and mag- slabs. The volume of each slab is πy2δx, netic fields in an electromagnetic wave. where δx is the difference between x values in a slab. Summing over all the slabs be- virtual work The work done if a system tween x = a and x = b gives the desired ex- is displaced infinitesimally from its posi- pression for V. tion. The virtual work is zero if the system is in equilibrium. vulgar fraction See fraction.

237 W

Wallis’s formula an expression for π/2 in For the simple case of a plane progres- terms of an infinite product: sive wave the displacement at a point can π ∞ 2 = 2.2.4.4.6.6.8.8… = Π 4n be represented by an equation: 2 = 1 3 3 5 5 7 7 9 n = 1 (4n2–1) y = asin2π(ft – x/λ) This product is sometimes known as Wal- where a is the amplitude, f the frequency, x lis’s product. The formula is named for the the distance from the origin, and λ the English mathematician and theologian wavelength. Other relationships are: John Wallis (1616–1703). y = asin2π(vt – x)/λ where v is the speed, and wallpaper symmetry A name given to y = asin2π(t/T – x/λ) the types of regular repeating patterns that where T is the period. Note that if the are possible in two dimensions, i.e. the minus sign is replaced by a plus sign in the two-dimensional analog of SPACE GROUPS. above equations it implies a similar wave There are 17 different types of possible moving in the opposite direction. For a sta- wallpaper symmetry. tionary wave resulting from two waves in opposite directions, the displacement is wot watt / / Symbol: W The SI unit of given by: power, defined as a power of one joule per Y = 2acos2πx/λ second. 1 W = 1 J s–1. The unit is named for See also longitudinal wave; transverse the British instrument maker and inventor wave; phase. James Watt (1736–1819). wave equation A second-order partial wave A method of energy transfer involv- differential equation that describes wave ing some form of vibration. For instance, waves on the surface of a liquid or along a motion. The equation ∂2 ∂ 2 2 ∂2 ∂ 2 stretched string involve regular to-and-fro u/ x = (1/c ) u/ t motion of particles about a mean position. might represent, for example, the vertical Sound waves carry energy by alternate displacement u of the water surface as a compressions and rarefactions of air (or plane wave of velocity c passes along the other media). In electromagnetic waves, surface, the horizontal position and the electric and magnetic fields vary at right time being given by x and t respectively. angles to the direction of propagation of The general solution of this one-dimen- the wave. At any particular instance, a sional wave equation is a periodic function graph of displacement against distance is a of x and t. There exist analogous wave regular repeating curve – the waveform or equations for two and three dimensions. wave profile of the wave. In a traveling (or progressive) wave the whole periodic dis- waveform See wave. placement moves through the medium. At any point in the medium the disturbance is wavefront A continuous surface associ- changing with time. Under certain condi- ated with a wave radiation, in which all the tions a STATIONARY (or standing) WAVE can vibrations concerned are in phase. A paral- be produced in which the disturbance does lel beam has plane wavefronts; the output not change with time. of a point source has spherical wavefronts.

238 world curve wavelength Symbol: λ The distance be- erence of the spacecraft the person feels tween the ends of one complete cycle of a that he or she has no weight. wave. Wavelength relates to the wave speed (c) and its frequency (v) thus: wheel and axle A simple MACHINE con- c = vλ sisting of a wheel on an axle that has a rope around it. An effort applied to the wheel is wave motion Any form of energy trans- transmitted to a load exerted at the axle fer that may be described as a wave rather rope. The force ratio (mechanical advan- than as a stream of particles. The term is tage) is equal to rW/rA where rW is the ra- also sometimes used to mean any harmonic dius of the wheel and rA that of the axle. motion. whole numbers Symbol: W The set of in- wave number Symbol: σ The reciprocal tegers {1,2,3,…}, excluding zero. of the wavelength of a wave. It is the number of wave cycles in unit distance, and is word The basic unit in which information often used in spectroscopy. The unit is the is stored and manipulated in a computer. meter–1 (m–1). The circular wave number Each word consists usually of a fixed num- (Symbol: k) is given by ber of BITS. This number, known as the k = 2πσ word length, varies according to the type of computer and may be as few as eight weber /vay-ber/ Symbol: Wb The SI unit or as many as 60. Each word is given a of magnetic flux, equal to the magnetic unique address in store. A word may rep- flux that, linking a circuit of one turn, pro- resent an instruction to the computer or a duces an e.m.f. of one volt when reduced to piece of data. An instruction word is coded zero at a uniform rate in one second. 1 Wb to give the operation to be performed and = 1 V s. The unit is named for the German the address or addresses of the data on physicist Wilhelm Eduard Weber (1804– which the operation is to be performed. See 91). also byte. weight Symbol: W The force by which a word processor A microcomputer that is mass is attracted to another, such as the programmed to help in preparing text for Earth. It is proportional to the body’s mass printing or data transmission. A general- (m), the constant of proportionality being purpose computer can be used as a word the gravitational field strength (i.e. the ac- processor by means of a suitable applica- celeration of free fall). Thus W = mg, where tions program. g is the acceleration of free fall. The mass of a body is normally constant, but its work Symbol: W The work done by a weight varies with position (because it de- force is the product of the force and the dis- pends on g). tance moved in the same direction: Although mass and weight are often work = force × displacement used interchangeably in everyday lan- Work is in fact a process of energy guage, they are different in scientific lan- transfer and, like energy, is measured in guage and must not be confused. joules. If the directions of force (F) and motion are not the same the component of the weighted mean See mean. force in the direction of the motion is used. W = Fscosθ weightlessness An apparent loss of where s is displacement and θ the angle be- weight experienced by an object in free fall. tween the directions of force and motion. Thus for a person in an orbiting spacecraft, Work is the scalar product of force and dis- the weight in the Earth’s frame of reference placement. is the centripetal force necessary to maintain the circular orbit. In the frame of ref- world curve See space–time.

239 write head

write head A device, part of a computer f1(x)f2(x) . . . fn(x) system, that records data onto a magnetic ‘ ‘ ‘ f 1(x) f 2(x) . . . f 1(x) storage medium such as tape or disk. See W = . . also input device. . (n−1) (n−1) (n−1) f1 (x) f2 (x)fn (x) Wronskian /vrons-kee-ăn/ A determinant which can be used to examine whether the dependent but if W ≠ 0 the functions are functions f1(x), f2(x), fn(x) of x, each of linearly dependent. which has the non-vanishing derivatives The concept of the Wronskian is used in are linearly independent. The Wronskian the theory of differential equations. The W is defined to be the determinant shown. Wronskian is named for the Polish mathe- If W ≠ 0 then the n functions are linearly in- matician J. M. H. Wronski (1778–1853).

240 Y

yard A unit of length now defined as There are several other types of elastic 0.9144 meter. y-coordinate See ordinate. modulus i.e. the ratio of stress to strain, depending on the nature of the deformation yield The income produced by a stock or of the body. The bulk modulus is the ratio share expressed as a percentage of its mar- of the pressure applied to a body to the ket price. fraction by which the volume of the body has decreased. The shear modulus, also yocto- Symbol: y A prefix denoting 10–24. known as the rigidy is the ratio of the shear For example, 1 yoctometer (ym) = 10–24 force per unit area divided by the deforma- meter (m). tion of the body, measured in radians. Since these elastic moduli all have ratios of yotta- Symbol: Y A prefix denoting 1024. stress to strain they have the dimensions of For example, 1 yottameter (Ym) = 1024 force per unit area since stresses have the meter (m). dimensions of force per unit area and strains are dimensionless numbers. The Young modulus The ratio of the stress to modulus is named for the British physicist the strain in a deformed body in the case of and physician Thomas Young (1773– elongation or compression of the body. 1829).

241 Z

zepto- Symbol: z A prefix denoting 10–21. zetta- Symbol: Z A prefix denoting 1021. For example, 1 zeptometer (zm) = 10–21 For example, 1 zettameter (Zm) = 1021 meter (m). meter (m). zero (0) The number that when added to zone A part of a sphere produced by two another number gives a sum equal to that parallel planes cutting the sphere. other number. It is included in the set of integers but not in the set of whole numbers. Zorn’s lemma /zornz/ (Kuratowski–Zorn The product of any number and zero is lemma) If a set S is partially ordered and zero. Zero is the identity element for addi- each linearly ordered subset has an upper tion. bound in S, then S contains at least one zero function A function f(x), for which maximal element, i.e. an element x such < f(x) = 0 for all values of x, where x belongs that there is no y in S for x y. The lemma to the set of all real numbers. was first discovered in 1922 by the Polish mathematician Kazimierz Kuratowski zero matrix See null matrix. (1896–1980) and independently in 1935 by the German-born American mathemati- zeta function See Riemann zeta function. cian Max Zorn (1906–93).

242 APPENDIXES

Appendix I Symbols and Notation Arithmetic and algebra equal to = not equal to ≠ identity ≡ approximately equal to ϴ approaches → proportional to ∝ less than < greater than > less than or equal to ≤ greater than or equal to ≥ much less than << much greater than >> plus, positive + minus, negative – plus or minus ± multiplication a × b a.b division a ÷ b a/b magnitude of a |a| factorial aa! logarithm (to base b) logba common logarithm log10a natural logarithm logea or lna summation ∑ continued product Π

Geometry and trigonometry angle Є triangle ᭝ ٗ square circle ᭺ parallel to || perpendicular to ֊ congruent to ≡ similar to ~ sine sin

245 Appendix I Symbols and Notation cosine cos tangent tan cotangent cot, ctn secant sec cosecant cosec, csc inverse sine sin–1, arc sin inverse cosine cos–1, arc cos inverse tangent tan–1, arc tan Cartesian coordinates (x, y, z) spherical coordinates (r, θ, φ) cylindrical coordinates (r, θ, z) direction numbers or cosines (l, m, n)

Sets and logic implies that ⇒ is implied by ⇐ implies and is implied by (if and only if) ⇔ set a, b, c,... {a, b, c, ...} is an element of ∈ is not an element of ∉ such that : number of elements in set Sn(S) universal set E or Ᏹ empty set ∅ complement of SS′ union ∪ intersection ∩ is a subset of ⊂ corresponds one-to-one with ↔ x is mapped onto yx→y conjunction ∧ disjunction ∨ negation (of p)~p or ¬p implication → or ⊃ biconditional (equivalence) ≡ or ↔ the set of natural numbers ގ the set of integers ޚ the set of rational numbers ޑ the set of real numbers ޒ the set of complex numbers ރ

246 Appendix I Symbols and Notation Calculus increment of x ᭝x or δx limit of function of x as x approaches a Lim f(x) x→a derivative of f(x) df(x)/dx or f′(x) second derivative of f(x)d2f(x)/dx2 or f″(x) etc. etc. indefinite integral of f(x) with respect to x Ύf(x)dx definite integral with a limits a and b Ύ f(x)dx b partial derivative of function f(x,y) with respect to x ∂f(x,y)/∂x

247 Appendix II Symbols for Physical Quantities acceleration a moment of inertia I angle θ, φ, α, momentum p β, etc. period τ α angular acceleration potential energy Ep, V angular frequency, 2πf ω power P angular momentum L pressure p angular velocity ω radius r area A reduced mass µ breadth b relative density d circular wavenumber k solid angle Ω, ω density ρ thickness d diameter d time t distance s, L torque T energy W, E velocity v force F viscosity η frequency f, ν volume V height h wavelength λ σ kinetic energy Ek, T wavenumber length l weight W mass m work W, E moment of force M

248 Appendix III Areas and Volumes

Plane figures Figure Dimensions Perimeter Area triangle sides a, b, and c, a + b + c ½bc.sinA angle A square side a 4aa2 rectangle sides a and b 2(a + b) a × b kite diagonals c and d ½c × d parallelogram sides a and b distances 2(a + b) a.c or b.d c and d apart circle radius r 2πr πr2 ellipse axes a and b 2π√[(a2 + b2)/2] πab

Solid figures Figure Dimensions Area Volume cylinder radius r, height h 2πr(h + r) πr2h cone base radius r, πrl πr2h/3 slant height l, height h sphere radius r 4πr2 4πr3/3

249 Appendix IV Expansions sin xx/1! – x3/3! + x5/5! – x7/7! + x9/9! – ... cos x 1 – x2/2! + x4/4! – x6/6! + x8/8! – ... ex 1 + x/1! + x2/2! + x3/3! + x4/4! + x5/5! + ... sinh xx+ x3/3! + x5/5! + x7/7! + x9/9! + ... cosh x 1 + x2/2! + x4/4! + x6/6! + x8/8! + ...

2 3 4 5 loge(1+x) x – x /2 + x /3 – x /4 + x /5 – ...

2 3 4 5 loge(1–x) –x – x /2 – x /3 – x /4 – x /5 – ...

(1 + x)n 1 + nx + n(n – 1)x2/2! + n(n – 1)(n – 2)x3/3! + ... for |x| < 1 f(a + x)f(a) + xf′(a) + (x2/2!)f′′(a) + (x3/3!)f′′′(a) + (x4/4!)f′′′′(a) + ... where f′(a) denotes the first derivative, f′′(a) the second derivative, etc. f(x) f(0) + xf′(0) + (x2/2!)f′′(0) + (x3/3!)f′′′(0) + (x4/4!)f′′′′(0) + ...

250 Appendix V Derivatives x is a variable, u is a function of x, and a and n are constants

Function f(x) Derivative df(x)/dx x 1 ax a axn naxn–1 eax aeax

loge x 1/x

loga x (1/x)loge a cos x –sin x sin x cos x tan x sec2 x cot x –cosec2 x sec x tan x.sec x cosec x–cot x.cosec x cos u –sin u.(du/dx) sin u cos u.(du/dx) tan u sec2 u.(du/dx)

loge u (1/u)(du/dx) sin–1 (x/a)1/√(a2 – x2) cos–1 (x/a) –1/√(a2 – x2) tan–1 (x/a) a/(a2 + x2)

251 Appendix VI Integrals x is a variable and a and n are constants. Note that a constant of integration C should be added to each integral.

Function f(x) Integral Ύf(x)dx aax xx2/2 xn xn+1/(n + 1)

1/x loge x eax eax/a

loge ax xloge ax – x cos x sin x sin x –cos x

tan x loge (cos x)

cot x loge (sin x)

sec x loge (sec x + tan x)

cosec x loge (cosec x – cot x) 1/√(a2 – x2) sin–1 (x/a) –1/√(a2 – x2) cos–1 (x/a) a/(a2 + x2) tan–1 (x/a)

252 Appendix VII Trigonometric Formulae Addition formulae: sin(x + y) = sinx cosy + cosx siny sin(x – y) = sinx cosy – cosx siny cos(x + y) = cosx cosy – sinx siny cos(x – y) = cosx cosy + sinx siny tan(x + y) = (tanx + tany)/(1 – tanx tany) tan(x – y) = (tanx – tany)/(1 + tanx tany)

Double-angle formulae: sin(2x) = 2 sinx cosx cos(2x) = cos2x – sin2x tan(2x) = 2tanx/(1 – tan2x)

Half-angle formulae: sin(x/2) = ±√[(1 – cosx)/2] cos(x/2) = ±√[(1 + cosx)/2] tan(x/2) = sinx/(1 + cosx) = (1 – cosx)/sinx

Poduct formulae: sinx cosy = ½[sin(x + y) + sin(x – y)] cosx siny = ½[sin(x + y) – sin(x – y)] cosx cosy = ½[cos(x + y) + cos(x – y)] sinx siny = ½[cos(x – y) – cos(x + y)]

253 Appendix VIII Conversion Factors Length To convert into multiply by inches meters 0.0254 feet meters 0.3048 yards meters 0.9144 miles kilometers 1.60934 nautical miles kilometers 1.85200 nautical miles miles 1.15078 kilometers miles 0.621371 kilometers nautical miles 0.539957 meters inches 39.3701 meters feet 3.28084 meters yards 1.09361

Area To convert into multiply by square inches square centimeters 6.4516 square inches square meters 6.4516 × 10–4 square feet square meters 9.2903 × 10–2 square yards square meters 0.836127 square miles square kilometers 2.58999 square miles acres 640 acres square meters 4046.86 acres square miles 1.5625 × 10–3 square centimeters square inches 0.155 square meters square feet 10.7639 square meters square yards 1.19599 square meters acres 2.47105 × 10–4 square meters square miles 3.86019 × 10–7 square kilometers square miles 0.386019

Volume To convert into multiply by cubic inches liters 1.63871 × 10–2 cubic inches cubic meters 1.63871 × 10–5 cubic feet liters 28.3168 cubic feet cubic meters 0.0283168 cubic yard cubic meters 0.764555 gallon (US) liters 3.785438 gallon (US) cubic meters 3.785438 × 10–3 gallon (US) gallon (UK) 0.83268

254 Appendix VIII Conversion Factors Mass To convert into multiply by pounds kilograms 0.453592 pounds tonnes 4.53592 × 10–4 hundredweight (short) kilograms 45.3592 hundredweight (short) tonnes 0.0453592 tons (short) kilograms 907.18 tons (short) tonnes 0.90718 kilograms pounds 2.204623 kilograms hundredweights (short) 0.022046 kilograms tons (short) 1.1023 × 10–3 tonnes pounds 2204.623 tonnes hundredweights (short) 22.0462 tonnes tons (short) 0.90718 The short ton is used in the USA and is equal to 2000 pounds. The short hundredweight (also known as the cental) is 100 pounds. The long ton, which is used in the UK, is equal to 2240 pounds (1016.047 kg). The long hundredweight is 112 pounds (50.802 kg). 1 long ton equals 20 long hundredweights.

Force To convert into multiply by pounds force newtons 4.44822 pounds force kilograms force 0.453592 pounds force dynes 444822 pounds force poundals 32.174 poundals newtons 0.138255 poundals kilograms force 0.031081 poundals dynes 13825.5 poundals pounds force 0.031081 dynes newtons 10–5 dynes kilograms force 1.01972 × 10–6 dynes pounds force 2.24809 × 10–6 dynes poundals 7.2330 × 10–5 kilograms force newtons 9.80665 kilograms force dynes 980665 kilograms force pounds force 2.20462 kilograms force poundals 70.9316 newtons kilograms 0.101972 newtons dynes 100000 newtons pounds force 0.224809 newtons poundals 7.2330

255 Appendix VIII Conversion Factors Work and energy To convert into multiply by British Thermal Units joules 1055.06 British Thermal Units calories 251.997 British Thermal Units kilowatt-hours 2.93071 × 10–4 kilowatt-hours joules 3600000 kilowatt-hours calories 859845 kilowatt-hours British Thermal Units 3412.14 calories joules 4.1868 calories kilowatt-hours 1.16300 × 10–6 calories British Thermal Units 3.96831 × 10–3 joules calories 0.238846 joules kilowatt hours 2.7777 × 10–7 joules British Thermal Units 9.47813 × 10–4 joules electron volts 6.2418 × 1018 joules ergs 107 electronvolts joules 1.6021 × 10–19 ergs joules 10–7

Pressure To convert into multiply by atmospheres pascals* 101325 bars pascals 100000 pounds per square pascals 68894.76 inch pounds per square kilograms per square 703.068 inch meter pounds per square atmospheres 0.068046 inch kilograms per square pascals 9.80661 meter kilograms per square pounds per square 1.42234 × 10–3 meter inch kilograms per square atmospheres 9.67841 × 10–5 meter pascals kilograms per square 0.101972 meter pascals pounds per square 1.45038 × 10–4 inch pascals atmospheres 9.86923 × 10–6

*1 pascal = 1 newton per square meter

256 Appendix IX Powers and Roots

3 nn2 n3 √n √n

1 1 1 1.000 1.000 2 4 8 1.414 1.260 3 9 27 1.732 1.442 4 16 64 2.000 1.587 5 25 125 2.236 1.710

6 36 216 2.449 1.817 7 49 343 2.646 1.913 8 64 512 2.828 2.000 9 81 729 3.000 2.080 10 100 1 000 3.162 2.154

11 121 1 33` 3.317 2.224 12 144 1 728 3.464 2.289 13 169 2 197 3.606 2.351 14 196 2 744 3.742 2.410 15 225 3 375 3.873 2.466

16 256 4 096 4.000 2.520 17 289 4 913 4.123 2.571 18 324 5 832 4.243 2.621 19 361 6 859 4.359 2.668 20 400 8 000 4.472 2.714

21 441 9 261 4.583 2.759 22 484 10 648 4.690 2.802 23 529 12 167 4.796 2.844 24 576 13 824 4.899 2.844 25 625 15 625 5.000 2.924

26 676 17 576 5.099 2.962 27 729 19 683 5.196 3.000 28 784 21 952 5.292 3.037 29 841 24 389 5.385 3.072 30 900 27 000 5.477 3.107

31 961 29 791 5.568 3.141 32 1 024 32 768 5.657 3.175

257 Appendix IX Powers and Roots

3 nn2 n3 √n √n

33 1 089 35 937 5.745 3.208 34 1 156 39 304 5.831 3.240 35 1 225 42 875 5.916 3.271

36 1 296 46 656 6.000 3.302 37 1 369 50 653 6.083 3.332 38 1 444 54 872 6.164 3.362 39 1 521 59 319 6.245 3.391 40 1 600 64 000 6.325 3.420

41 1 681 68 921 6.403 3.448 42 1 764 74 088 6.481 3.476 43 1 849 79 507 6.557 3.503 44 1 936 85 184 6.633 3.530 45 2 025 91 125 6.708 3.557

46 2 116 97 336 6.782 3.583 47 2 209 103 823 6.856 3.609 48 2 304 110 592 6.928 3.634 49 2 401 117 649 7.000 3.659 50 2 500 125 000 7.071 3.684

51 2 601 132 651 7.141 3.708 52 2 704 140 608 7.211 3.733 53 2 809 148 877 7.280 3.756 54 2 916 157 464 7.348 3.780 55 3 025 166 375 7.416 3.803

56 3 136 175 616 7.483 3.826 57 3 249 185 193 7.550 3.849 58 3 364 195 112 7.616 3.871 59 3 481 205 379 7.681 3.893 60 3 600 216 000 7.746 3.915

61 3 721 226 981 7.810 3.936 62 3 844 238 328 7.874 3.958 63 3 969 250 047 7.937 3.979 64 4 096 262 144 8.000 4.000 65 4 225 274 625 8.062 4.021

258 Appendix IX Powers and Roots

3 nn2 n3 √n √n

66 4 356 287 496 8.124 4.041 67 4 489 300 763 8.185 4.062 68 4 624 314 432 8.246 4.082 69 4 761 328 509 8.307 4.102 70 4 900 343 000 8.367 4.121

71 5 041 357 911 8.426 4.141 72 5 184 373 248 8.485 4.160 73 5 329 389 017 8.544 4.179 74 5 476 405 224 8.602 4.198 75 5.625 421 875 8.660 4.217

76 5 776 438 976 8.718 4.236 77 5 929 456 533 8.775 4.254 78 6 084 474 552 8.832 4.273 79 6 241 493 039 8.888 4.291 80 6 400 512 000 8.944 4.309

81 6 561 531 441 9.000 4.327 82 6 724 551 368 9.055 4.344 83 6 889 571 787 9.110 4.362 84 7 056 592 704 9.165 4.380 85 7 225 614 125 9.220 4.397

86 7 396 636 056 9.274 4.414 87 7 569 658 503 9.327 4.431 88 7 744 681 472 9.381 4.448 89 7 921 704 969 9.434 4.465 90 8 100 729 000 9.487 4.481

91 8 281 753 571 9.539 4.498 92 8 464 778 688 9.592 4.514 93 8 649 804 357 9 644 4.531 94 8 836 830 584 9.695 4.547 95 9 025 857 375 9.747 4.563

96 9 216 884 736 9.798 4.579 97 9 409 912 673 9.849 4.595 98 9 604 941 192 9.899 4.610 99 9 801 970 299 9.950 4.626 100 10 000 1 000 000 10.000 4.642

259 Appendix X

The Greek Alphabet

A α alpha N ν nu B β beta Ξξxi Γγgamma O ο omikron ∆δdelta Ππ pi E ε epsilon P ρ rho Z ζ zeta Σσsigma H η eta T τ tau Θθ theta Υυupsilon I ι iota Φφ phi K κ kappa X χ chi Λλlambda Ψψpsi M µ mu Ωωomega

260 Appendix XI

Web Sites

Organizations:

American Mathematical Society www.ams.org

Mathematical Association of America www.maa.org

International Mathematical Union www.mathunion.org

London Mathematical Society www.lms.ac.uk

Mathematics Foundation of America www.mfoa.org

General resources:

Mathematics on the Web www.ams.org/mathweb

Mathematics WWW Virtual Library euclid.math.fsu.edu/Science/math.html

MathGate Homepage www.mathgate.ac.uk

Mathematical Resources www.ama.caltech.edu/resources.html

Math-Net Links www.math-net.de/links/ show?collection=math

Mathematics Resources on the WWW mthwww.uwc.edu/wwwmahes/ homepage.htm

History and biography:

History of Mathematics Archive www-history.mcs.st-and.ac.uk/history

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