Pi – Not Just an Ordinary Number
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Irrational Numbers Unit 4 Lesson 6 IRRATIONAL NUMBERS
Irrational Numbers Unit 4 Lesson 6 IRRATIONAL NUMBERS Students will be able to: Understand the meanings of Irrational Numbers Key Vocabulary: • Irrational Numbers • Examples of Rational Numbers and Irrational Numbers • Decimal expansion of Irrational Numbers • Steps for representing Irrational Numbers on number line IRRATIONAL NUMBERS A rational number is a number that can be expressed as a ratio or we can say that written as a fraction. Every whole number is a rational number, because any whole number can be written as a fraction. Numbers that are not rational are called irrational numbers. An Irrational Number is a real number that cannot be written as a simple fraction or we can say cannot be written as a ratio of two integers. The set of real numbers consists of the union of the rational and irrational numbers. If a whole number is not a perfect square, then its square root is irrational. For example, 2 is not a perfect square, and √2 is irrational. EXAMPLES OF RATIONAL NUMBERS AND IRRATIONAL NUMBERS Examples of Rational Number The number 7 is a rational number because it can be written as the 7 fraction . 1 The number 0.1111111….(1 is repeating) is also rational number 1 because it can be written as fraction . 9 EXAMPLES OF RATIONAL NUMBERS AND IRRATIONAL NUMBERS Examples of Irrational Numbers The square root of 2 is an irrational number because it cannot be written as a fraction √2 = 1.4142135…… Pi(휋) is also an irrational number. π = 3.1415926535897932384626433832795 (and more...) 22 The approx. value of = 3.1428571428571.. -
Euler's Square Root Laws for Negative Numbers
Ursinus College Digital Commons @ Ursinus College Transforming Instruction in Undergraduate Complex Numbers Mathematics via Primary Historical Sources (TRIUMPHS) Winter 2020 Euler's Square Root Laws for Negative Numbers Dave Ruch Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_complex Part of the Curriculum and Instruction Commons, Educational Methods Commons, Higher Education Commons, and the Science and Mathematics Education Commons Click here to let us know how access to this document benefits ou.y Euler’sSquare Root Laws for Negative Numbers David Ruch December 17, 2019 1 Introduction We learn in elementary algebra that the square root product law pa pb = pab (1) · is valid for any positive real numbers a, b. For example, p2 p3 = p6. An important question · for the study of complex variables is this: will this product law be valid when a and b are complex numbers? The great Leonard Euler discussed some aspects of this question in his 1770 book Elements of Algebra, which was written as a textbook [Euler, 1770]. However, some of his statements drew criticism [Martinez, 2007], as we shall see in the next section. 2 Euler’sIntroduction to Imaginary Numbers In the following passage with excerpts from Sections 139—148of Chapter XIII, entitled Of Impossible or Imaginary Quantities, Euler meant the quantity a to be a positive number. 1111111111111111111111111111111111111111 The squares of numbers, negative as well as positive, are always positive. ...To extract the root of a negative number, a great diffi culty arises; since there is no assignable number, the square of which would be a negative quantity. Suppose, for example, that we wished to extract the root of 4; we here require such as number as, when multiplied by itself, would produce 4; now, this number is neither +2 nor 2, because the square both of 2 and of 2 is +4, and not 4. -
Metrical Diophantine Approximation for Quaternions
This is a repository copy of Metrical Diophantine approximation for quaternions. White Rose Research Online URL for this paper: https://eprints.whiterose.ac.uk/90637/ Version: Accepted Version Article: Dodson, Maurice and Everitt, Brent orcid.org/0000-0002-0395-338X (2014) Metrical Diophantine approximation for quaternions. Mathematical Proceedings of the Cambridge Philosophical Society. pp. 513-542. ISSN 1469-8064 https://doi.org/10.1017/S0305004114000462 Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Under consideration for publication in Math. Proc. Camb. Phil. Soc. 1 Metrical Diophantine approximation for quaternions By MAURICE DODSON† and BRENT EVERITT‡ Department of Mathematics, University of York, York, YO 10 5DD, UK (Received ; revised) Dedicated to J. W. S. Cassels. Abstract Analogues of the classical theorems of Khintchine, Jarn´ık and Jarn´ık-Besicovitch in the metrical theory of Diophantine approximation are established for quaternions by applying results on the measure of general ‘lim sup’ sets. -
Section 3.6 Complex Zeros
210 Chapter 3 Section 3.6 Complex Zeros When finding the zeros of polynomials, at some point you're faced with the problem x 2 −= 1. While there are clearly no real numbers that are solutions to this equation, leaving things there has a certain feel of incompleteness. To address that, we will need utilize the imaginary unit, i. Imaginary Number i The most basic complex number is i, defined to be i = −1 , commonly called an imaginary number . Any real multiple of i is also an imaginary number. Example 1 Simplify − 9 . We can separate − 9 as 9 −1. We can take the square root of 9, and write the square root of -1 as i. − 9 = 9 −1 = 3i A complex number is the sum of a real number and an imaginary number. Complex Number A complex number is a number z = a + bi , where a and b are real numbers a is the real part of the complex number b is the imaginary part of the complex number i = −1 Arithmetic on Complex Numbers Before we dive into the more complicated uses of complex numbers, let’s make sure we remember the basic arithmetic involved. To add or subtract complex numbers, we simply add the like terms, combining the real parts and combining the imaginary parts. 3.6 Complex Zeros 211 Example 3 Add 3 − 4i and 2 + 5i . Adding 3( − i)4 + 2( + i)5 , we add the real parts and the imaginary parts 3 + 2 − 4i + 5i 5 + i Try it Now 1. Subtract 2 + 5i from 3 − 4i . -
The Geometry of René Descartes
BOOK FIRST The Geometry of René Descartes BOOK I PROBLEMS THE CONSTRUCTION OF WHICH REQUIRES ONLY STRAIGHT LINES AND CIRCLES ANY problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction.1 Just as arithmetic consists of only four or five operations, namely, addition, subtraction, multiplication, division and the extraction of roots, which may be considered a kind of division, so in geometry, to find required lines it is merely necessary to add or subtract ther lines; or else, taking one line which I shall call unity in order to relate it as closely as possible to numbers,2 and which can in general be chosen arbitrarily, and having given two other lines, to find a fourth line which shall be to one of the given lines as the other is to unity (which is the same as multiplication) ; or, again, to find a fourth line which is to one of the given lines as unity is to the other (which is equivalent to division) ; or, finally, to find one, two, or several mean proportionals between unity and some other line (which is the same as extracting the square root, cube root, etc., of the given line.3 And I shall not hesitate to introduce these arithmetical terms into geometry, for the sake of greater clearness. For example, let AB be taken as unity, and let it be required to multiply BD by BC. I have only to join the points A and C, and draw DE parallel to CA ; then BE is the product of BD and BC. -
Calculating Numerical Roots = = 1.37480222744
#10 in Fundamentals of Applied Math Series Calculating Numerical Roots Richard J. Nelson Introduction – What is a root? Do you recognize this number(1)? 1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37694 80731 76679 73799 07324 78462 10703 88503 87534 32764 15727 35013 84623 09122 97024 92483 60558 50737 21264 41214 97099 93583 14132 22665 92750 55927 55799 95050 11527 82060 57147 01095 59971 60597 02745 34596 86201 47285 17418 64088 91986 09552 32923 04843 08714 32145 08397 62603 62799 52514 07989 68725 33965 46331 80882 96406 20615 25835 Fig. 1 - HP35s √ key. 23950 54745 75028 77599 61729 83557 52203 37531 85701 13543 74603 40849 … If you have had any math classes you have probably run into this number and you should recognize it as the square root of two, . The square root of a number, n, is that value when multiplied by itself equals n. 2 x 2 = 4 and = 2. Calculating the square root of the number is the inverse operation of squaring that number = n. The "√" symbol is called the "radical" symbol or "check mark." MS Word has the radical symbol, √, but we often use it with a line across the top. This is called the "vinculum" circa 12th century. The expression " (2)" is read as "root n", "radical n", or "the square root of n". The vinculum extends over the number to make it inclusive e.g. Most HP calculators have the square root key as a primary key as shown in Fig. 1 because it is used so much. Roots and Powers Numbers may be raised to any power (y, multiplied by itself x times) and the inverse operation, taking the root, may be expressed as shown below. -
How to Show That Various Numbers Either Can Or Cannot Be Constructed Using Only a Straightedge and Compass
How to show that various numbers either can or cannot be constructed using only a straightedge and compass Nick Janetos June 3, 2010 1 Introduction It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square on one side... -Indiana House Bill No. 246, 1897 Three problems of classical Greek geometry are to do the following using only a compass and a straightedge: 1. To "square the circle": Given a circle, to construct a square of the same area, 2. To "trisect an angle": Given an angle, to construct another angle 1/3 of the original angle, 3. To "double the cube": Given a cube, to construct a cube with twice the area. Unfortunately, it is not possible to complete any of these tasks until additional tools (such as a marked ruler) are provided. In section 2 we will examine the process of constructing numbers using a compass and straightedge. We will then express constructions in algebraic terms. In section 3 we will derive several results about transcendental numbers. There are two goals: One, to show that the numbers e and π are transcendental, and two, to show that the three classical geometry problems are unsolvable. The two goals, of course, will turn out to be related. 2 Constructions in the plane The discussion in this section comes from [8], with some parts expanded and others removed. The classical Greeks were clear on what constitutes a construction. Given some set of points, new points can be defined at the intersection of lines with other lines, or lines with circles, or circles with circles. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
Sequence Rules
SEQUENCE RULES A connected series of five of the same colored chip either up or THE JACKS down, across or diagonally on the playing surface. There are 8 Jacks in the card deck. The 4 Jacks with TWO EYES are wild. To play a two-eyed Jack, place it on your discard pile and place NOTE: There are printed chips in the four corners of the game board. one of your marker chips on any open space on the game board. The All players must use them as though their color marker chip is in 4 jacks with ONE EYE are anti-wild. To play a one-eyed Jack, place the corner. When using a corner, only four of your marker chips are it on your discard pile and remove one marker chip from the game needed to complete a Sequence. More than one player may use the board belonging to your opponent. That completes your turn. You same corner as part of a Sequence. cannot place one of your marker chips on that same space during this turn. You cannot remove a marker chip that is already part of a OBJECT OF THE GAME: completed SEQUENCE. Once a SEQUENCE is achieved by a player For 2 players or 2 teams: One player or team must score TWO SE- or a team, it cannot be broken. You may play either one of the Jacks QUENCES before their opponents. whenever they work best for your strategy, during your turn. For 3 players or 3 teams: One player or team must score ONE SE- DEAD CARD QUENCE before their opponents. -
Finding Pi Project
Name: ________________________ Finding Pi - Activity Objective: You may already know that pi (π) is a number that is approximately equal to 3.14. But do you know where the number comes from? Let's measure some round objects and find out. Materials: • 6 circular objects Some examples include a bicycle wheel, kiddie pool, trash can lid, DVD, steering wheel, or clock face. Be sure each object you choose is shaped like a perfect circle. • metric tape measure Be sure your tape measure has centimeters on it. • calculator It will save you some time because dividing with decimals can be tricky. • “Finding Pi - Table” worksheet It may be attached to this page, or on the back. What to do: Step 1: Choose one of your circular objects. Write the name of the object on the “Finding Pi” table. Step 2: With the centimeter side of your tape measure, accurately measure the distance around the outside of the circle (the circumference). Record your measurement on the table. Step 3: Next, measure the distance across the middle of the object (the diameter). Record your measurement on the table. Step 4: Use your calculator to divide the circumference by the diameter. Write the answer on the table. If you measured carefully, the answer should be about 3.14, or π. Repeat steps 1 through 4 for each object. Super Teacher Worksheets - www.superteacherworksheets.com Name: ________________________ “Finding Pi” Table Measure circular objects and complete the table below. If your measurements are accurate, you should be able to calculate the number pi (3.14). Is your answer name of circumference diameter circumference ÷ approximately circular object measurement (cm) measurement (cm) diameter equal to π? 1. -
UNDERSTANDING MATHEMATICS to UNDERSTAND PLATO -THEAETEUS (147D-148B Salomon Ofman
UNDERSTANDING MATHEMATICS TO UNDERSTAND PLATO -THEAETEUS (147d-148b Salomon Ofman To cite this version: Salomon Ofman. UNDERSTANDING MATHEMATICS TO UNDERSTAND PLATO -THEAETEUS (147d-148b. Lato Sensu, revue de la Société de philosophie des sciences, Société de philosophie des sciences, 2014, 1 (1). hal-01305361 HAL Id: hal-01305361 https://hal.archives-ouvertes.fr/hal-01305361 Submitted on 20 Apr 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNDERSTANDING MATHEMATICS TO UNDERSTAND PLATO - THEAETEUS (147d-148b) COMPRENDRE LES MATHÉMATIQUES POUR COMPRENDRE PLATON - THÉÉTÈTE (147d-148b) Salomon OFMAN Institut mathématique de Jussieu-PRG Histoire des Sciences mathématiques 4 Place Jussieu 75005 Paris [email protected] Abstract. This paper is an updated translation of an article published in French in the Journal Lato Sensu (I, 2014, p. 70-80). We study here the so- called ‘Mathematical part’ of Plato’s Theaetetus. Its subject concerns the incommensurability of certain magnitudes, in modern terms the question of the rationality or irrationality of the square roots of integers. As the most ancient text on the subject, and on Greek mathematics and mathematicians as well, its historical importance is enormous. -
0.999… = 1 an Infinitesimal Explanation Bryan Dawson
0 1 2 0.9999999999999999 0.999… = 1 An Infinitesimal Explanation Bryan Dawson know the proofs, but I still don’t What exactly does that mean? Just as real num- believe it.” Those words were uttered bers have decimal expansions, with one digit for each to me by a very good undergraduate integer power of 10, so do hyperreal numbers. But the mathematics major regarding hyperreals contain “infinite integers,” so there are digits This fact is possibly the most-argued- representing not just (the 237th digit past “Iabout result of arithmetic, one that can evoke great the decimal point) and (the 12,598th digit), passion. But why? but also (the Yth digit past the decimal point), According to Robert Ely [2] (see also Tall and where is a negative infinite hyperreal integer. Vinner [4]), the answer for some students lies in their We have four 0s followed by a 1 in intuition about the infinitely small: While they may the fifth decimal place, and also where understand that the difference between and 1 is represents zeros, followed by a 1 in the Yth less than any positive real number, they still perceive a decimal place. (Since we’ll see later that not all infinite nonzero but infinitely small difference—an infinitesimal hyperreal integers are equal, a more precise, but also difference—between the two. And it’s not just uglier, notation would be students; most professional mathematicians have not or formally studied infinitesimals and their larger setting, the hyperreal numbers, and as a result sometimes Confused? Perhaps a little background information wonder .