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Dictionary of Mathematical Terms

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Dictionary of Mathematical Terms DICTIONARY OF MATHEMATICAL TERMS DR. ASHOT DJRBASHIAN in cooperation with PROF. PETER STATHIS 2 i INTRODUCTION changes in how students work with the books and treat them. More and more students opt for electronic books and "carry" them in their laptops, tablets, and even cellphones. This is This dictionary is specifically designed for a trend that seemingly cannot be reversed. two-year college students. It contains all the This is why we have decided to make this an important mathematical terms the students electronic and not a print book and post it would encounter in any mathematics classes on Mathematics Division site for anybody to they take. From the most basic mathemat- download. ical terms of Arithmetic to Algebra, Calcu- lus, Linear Algebra and Differential Equa- Here is how we envision the use of this dictio- tions. In addition we also included most of nary. As a student studies at home or in the the terms from Statistics, Business Calculus, class he or she may encounter a term which Finite Mathematics and some other less com- exact meaning is forgotten. Then instead of monly taught classes. In a few occasions we trying to find explanation in another book went beyond the standard material to satisfy (which may or may not be saved) or going curiosity of some students. Examples are the to internet sources they just open this dictio- article "Cantor set" and articles on solutions nary for necessary clarification and explana- of cubic and quartic equations. tion. The organization of the material is strictly Why is this a better option in most of the alphabetical, not by the topic. There are ap- cases? First of all internet search usually proximately a total of 1200 entries in this results in multiple, sometimes hundreds of dictionary. Some of them are just simple choices and that already creates problems. one-two sentence definitions while others are Second, many of these sources are rather gen- pretty detailed articles with definitions and eral and do not have in mind specifically even worked examples. All the entries are in the level of the students in community col- lowercase letters unless they contain a proper leges. Many of them are actually designed name. In text, however, some terms might with specialists in mind and contain material appear in capitalized form in expressions such that might confuse the student rather than as Fundamental Theorem of Calculus, Mean help. As an example, if a student needs to value theorem and so on. There are also recall the topic "Partial fractions" and ends about eighty illustrations of major functions up in Wikipedia then the explanation would and geometric figures. Most of the pictures be a very long, general, and confusing one are created by the author and others are im- containing lots of unnecessary material. The ported from open internet sources such as total number of pages corresponding to that Google and Wikipedia. particular entry in Wikipedia (after convert- Because this is an electronic book and not a ing into Word format) is about 12 pages. In traditional hard copy book it is worth writing contrast, the explanation in this dictionary a few words how we think it should be used is very concise, contains only the necessary the best. The last few years brought lots of cases and fits on four Word pages, including ii worked examples of all cases. cause of these in many cases we just used the traditional approach of emphasizing the Traditional dictionaries use italics to indicate terms with italics and that also requires some that the given word has its own entry. Elec- explanation. If the word is in italics then that tronic books present additional opportunity means one of the following : for the same purpose and especially for quick 1) There is an article for that particular term finding of the indicated terms, namely hyper- but in author's view it is not crucial to jump links. In this book we use a combination of there immediately in order to understand the both methods and it is worth to clarify fur- current article; ther how and when each of these approaches 2) There may or may not be an article for used in our dictionary. As the reader goes that term but the definition is contained in through some article in the dictionary he or this current article the reader is on; she might need explanation, clarification, or 3) There is an article for that term but it is definition of certain terms used in that ar- so close (might be the very next or just the ticle. When we thought that it is impor- previous one) there is no need of hyperlink to tant and beneficial to the reader to refresh get there. the memory and clarify that term in order We hope that this hybrid approach would to understand the article at hand we cre- make the use of dictionary as easy as possible. ated a hyperlink allowing immediately jump to that term without scrolling through mul- tiple pages. This is similar to most internet sources, such as above mentioned Wikipedia. For example, if the reader sees the expres- Dr. Ashot Djrbashian sion characteristic equation, then by simply Mathematics Division Glendale Community College clicking at that word he will jump to page Glendale, CA 20 of the text where that term is defined and explained. It is important to note however that not all the terms are hyperlinked and there are multiple reasons for that. First of all, that would make many articles very diffi- cult to read because of abundance of hyper- links and would destruct reader from the ac- tual article itself. In addition, there are many terms so common that it is obvious that there are specific articles for them. These terms in- clude, but not limited to, words such as Num- ber, Function, Polynomial, Equation, Inte- gral, Derivative, Matrix, etc. In rare cases, however, even some of these terms are hyper- linked by the reason explained above. Be- 1 A Abel's formula Also called Abel's theorem, for differential equations. Let y1 and y2 be solutions of second order equation L[y] = y00 + p(t)y0 + q(t)y = 0; where both p and q are continuous functions on some absolute value For a real number r, the non- interval I. Then by Abel's theorem the Wronskian of negative number jrj given by the formula the equation is given by the formula n r if r ≥ 0 jrj = Z −r if r < 0 W (y ; y ) = c exp − p(t) dt ; 1 2 Absolute value of a real number indicates the distance between the point on the number line where c is a constant that depends on solutions corresponding to that number and the origin. y1; y2 only. absolute value function The function f(x) = jxj, defined for all real values x. abscissa In the plane Cartesian coordinate system, the name of the x-axis. absolute maximum and minimum (1) For func- tions of one variable. A number M in the range of a given function f(x), such that f(x) ≤ M for all values of x in the domain of f is the absolute maxi- mum. Similarly, a number m in the range of a given function f(x), such that f(x) ≥ m for all values of x in the domain of f is the absolute minimum. If the function f is given on a closed interval [a; b], P1 then to find absolute maximum and minimum values, absolutely convergent series A series n=1 an the following steps to be taken: is absolutely convergent if the series formed with the P1 (a) Find all critical points (points, where the deriva- absolute values of its terms n=1 janj converges. If tive f 0 is either zero or does not exist) of the function a series is absolutely convergent, then any rearrange- on the open interval (a; b) ment of that series is also convergent and the sum is (b) Find the values of f at critical points and end- the same as the original series. points a and b. acceleration In physics, the rate of change of (c) Compare all these values to determine the largest the velocity of a moving object. Mathematically, and smallest. They will be the absolute maximum if the distance traveled by the object is given by and minimum values respectively. See also local max- the function s(t), then acceleration is the second imum and minimum. derivative of this function: a(t) = s00(t). (2) For functions of several variables definitions similar to the above are also valid. The critical acute angle An angle that measures between 0 points are also used to find these maximum and and 90◦ in degree measure or between 0 and π=2 in minimum points. radian measure. 2 dimensions: the same number of rows and columns. If A and B are two m × n-matrices, then their sum is defined to be the matrix C which has as elements the sums of elements of the given matrices in each position. Example: 0 1 1 −2 1 0 −2 2 3 1 @ −1 0 4 A + @ 1 −3 −4 A 5 −2 0 −4 0 1 0 −1 3 1 1 addends In arithmetic, two or more numbers to be = @ 0 −3 0 A : added. Similarly, in algebra, two or more expressions 1 −2 1 to be added. Subtraction of matrices is defined similarly. addition One of the four basic operations in arith- addition and subtraction formulas In metic and algebra.
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