Appendix I: on the Early History of Pi
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The Quest for Pi David H. Bailey, Jonathan M. Borwein, Peter B
The Quest for Pi David H. Bailey, Jonathan M. Borwein, Peter B. Borwein and Simon Plouffe June 25, 1996 Ref: Mathematical Intelligencer, vol. 19, no. 1 (Jan. 1997), pg. 50–57 Abstract This article gives a brief history of the analysis and computation of the mathematical constant π =3.14159 ..., including a number of the formulas that have been used to compute π through the ages. Recent developments in this area are then discussed in some detail, including the recent computation of π to over six billion decimal digits using high-order convergent algorithms, and a newly discovered scheme that permits arbitrary individual hexadecimal digits of π to be computed. D. Bailey: NASA Ames Research Center, Mail Stop T27A-1, Moffett Field, CA 94035-1000. E-mail: [email protected]. J. Borwein: Dept. of Mathematics and Statistics, Simon Fraser University Burnaby, BC V5A 1S6 Canada. Email: [email protected]. This work was supported by NSERC and the Shrum Endowment at Simon Fraser University. P. Borwein: Dept. of Mathematics and Statistics, Simon Fraser University Burnaby, BC V5A 1S6 Canada. Email: [email protected]. S. Plouffe: Dept. of Mathematics and Statistics, Simon Fraser University Burnaby, BC V5A 1S6 Canada. Email: [email protected]. 1 Introduction The fascinating history of the constant we now know as π spans several millennia, almost from the beginning of recorded history up to the present day. In many ways this history parallels the advancement of science and technology in general, and of mathematics and computer technology in particular. An overview of this history is presented here in sections one and two. -
Lesson 3: Rectangles Inscribed in Circles
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M5 GEOMETRY Lesson 3: Rectangles Inscribed in Circles Student Outcomes . Inscribe a rectangle in a circle. Understand the symmetries of inscribed rectangles across a diameter. Lesson Notes Have students use a compass and straightedge to locate the center of the circle provided. If necessary, remind students of their work in Module 1 on constructing a perpendicular to a segment and of their work in Lesson 1 in this module on Thales’ theorem. Standards addressed with this lesson are G-C.A.2 and G-C.A.3. Students should be made aware that figures are not drawn to scale. Classwork Scaffolding: Opening Exercise (9 minutes) Display steps to construct a perpendicular line at a point. Students follow the steps provided and use a compass and straightedge to find the center of a circle. This exercise reminds students about constructions previously . Draw a segment through the studied that are needed in this lesson and later in this module. point, and, using a compass, mark a point equidistant on Opening Exercise each side of the point. Using only a compass and straightedge, find the location of the center of the circle below. Label the endpoints of the Follow the steps provided. segment 퐴 and 퐵. Draw chord 푨푩̅̅̅̅. Draw circle 퐴 with center 퐴 . Construct a chord perpendicular to 푨푩̅̅̅̅ at and radius ̅퐴퐵̅̅̅. endpoint 푩. Draw circle 퐵 with center 퐵 . Mark the point of intersection of the perpendicular chord and the circle as point and radius ̅퐵퐴̅̅̅. 푪. Label the points of intersection . -
Squaring the Circle a Case Study in the History of Mathematics the Problem
Squaring the Circle A Case Study in the History of Mathematics The Problem Using only a compass and straightedge, construct for any given circle, a square with the same area as the circle. The general problem of constructing a square with the same area as a given figure is known as the Quadrature of that figure. So, we seek a quadrature of the circle. The Answer It has been known since 1822 that the quadrature of a circle with straightedge and compass is impossible. Notes: First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side √A clearly has the same area. Secondly, we are not saying that a quadrature of a circle is impossible, since it is possible, but not under the restriction of using only a straightedge and compass. Precursors It has been written, in many places, that the quadrature problem appears in one of the earliest extant mathematical sources, the Rhind Papyrus (~ 1650 B.C.). This is not really an accurate statement. If one means by the “quadrature of the circle” simply a quadrature by any means, then one is just asking for the determination of the area of a circle. This problem does appear in the Rhind Papyrus, but I consider it as just a precursor to the construction problem we are examining. The Rhind Papyrus The papyrus was found in Thebes (Luxor) in the ruins of a small building near the Ramesseum.1 It was purchased in 1858 in Egypt by the Scottish Egyptologist A. -
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. -
Squaring the Circle in Elliptic Geometry
Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 2 Article 1 Squaring the Circle in Elliptic Geometry Noah Davis Aquinas College Kyle Jansens Aquinas College, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Davis, Noah and Jansens, Kyle (2017) "Squaring the Circle in Elliptic Geometry," Rose-Hulman Undergraduate Mathematics Journal: Vol. 18 : Iss. 2 , Article 1. Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss2/1 Rose- Hulman Undergraduate Mathematics Journal squaring the circle in elliptic geometry Noah Davis a Kyle Jansensb Volume 18, No. 2, Fall 2017 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 a [email protected] Aquinas College b scholar.rose-hulman.edu/rhumj Aquinas College Rose-Hulman Undergraduate Mathematics Journal Volume 18, No. 2, Fall 2017 squaring the circle in elliptic geometry Noah Davis Kyle Jansens Abstract. Constructing a regular quadrilateral (square) and circle of equal area was proved impossible in Euclidean geometry in 1882. Hyperbolic geometry, however, allows this construction. In this article, we complete the story, providing and proving a construction for squaring the circle in elliptic geometry. We also find the same additional requirements as the hyperbolic case: only certain angle sizes work for the squares and only certain radius sizes work for the circles; and the square and circle constructions do not rely on each other. Acknowledgements: We thank the Mohler-Thompson Program for supporting our work in summer 2014. Page 2 RHIT Undergrad. Math. J., Vol. 18, No. 2 1 Introduction In the Rose-Hulman Undergraduate Math Journal, 15 1 2014, Noah Davis demonstrated the construction of a hyperbolic circle and hyperbolic square in the Poincar´edisk [1]. -
Three Points on the Plane
THREE POINTS ON THE PLANE ALEXANDER GIVENTAL UC BERKELEY In this lecture, we examine some properties of a geometric figure formed by three points on the plane. Any three points not lying on the same line determine a triangle. Namely, the points are the vertices of the triangle, and the segments connecting them are the sides of it. We expect the reader to be familiar with a few basic notions and facts of geometry of triangles. References like “[K], §140” will point to specific sections in: Kiselev’s Geometry. Book I: Planimetry, Sumizdat, 2006, 248 pages. Everything we assume known, as well as the initial part of the present exposition, is contained in the first half of this textbook. Circumcenter. One says a circle is circumscribed about a triangle (or shorter, is the circumcircle of it), if the circle passes through all vertices of the triangle. Theorem 1. About every triangle, a circle can be circum- scribed, and such a circle is unique. Let △ABC be a triangle with vertices A, B, C (Figure 1). A point O equidistant from1 all the three vertices is the center of a circumcircle of radius OA = OB = OC. Thus we need to show that a point equidistant from the vertices exists and is unique. Indeed, the geometric locus 2 of points equidistant from A and B is the perpendicular bisector to the segment AB, i.e. it is the line (denoted by MN in Figure 1) perpendicular to the segment AB at its midpoint (see [K], §56). Likewise, the geometric locus of points equidistant from B and C is the perpendicular bisector P Q to the segment BC. -
Evaluating Fourier Transforms with MATLAB
ECE 460 – Introduction to Communication Systems MATLAB Tutorial #2 Evaluating Fourier Transforms with MATLAB In class we study the analytic approach for determining the Fourier transform of a continuous time signal. In this tutorial numerical methods are used for finding the Fourier transform of continuous time signals with MATLAB are presented. Using MATLAB to Plot the Fourier Transform of a Time Function The aperiodic pulse shown below: x(t) 1 t -2 2 has a Fourier transform: X ( jf ) = 4sinc(4π f ) This can be found using the Table of Fourier Transforms. We can use MATLAB to plot this transform. MATLAB has a built-in sinc function. However, the definition of the MATLAB sinc function is slightly different than the one used in class and on the Fourier transform table. In MATLAB: sin(π x) sinc(x) = π x Thus, in MATLAB we write the transform, X, using sinc(4f), since the π factor is built in to the function. The following MATLAB commands will plot this Fourier Transform: >> f=-5:.01:5; >> X=4*sinc(4*f); >> plot(f,X) In this case, the Fourier transform is a purely real function. Thus, we can plot it as shown above. In general, Fourier transforms are complex functions and we need to plot the amplitude and phase spectrum separately. This can be done using the following commands: >> plot(f,abs(X)) >> plot(f,angle(X)) Note that the angle is either zero or π. This reflects the positive and negative values of the transform function. Performing the Fourier Integral Numerically For the pulse presented above, the Fourier transform can be found easily using the table. -
On the Normality of Numbers
ON THE NORMALITY OF NUMBERS Adrian Belshaw B. Sc., University of British Columbia, 1973 M. A., Princeton University, 1976 A THESIS SUBMITTED 'IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Mathematics @ Adrian Belshaw 2005 SIMON FRASER UNIVERSITY Fall 2005 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. APPROVAL Name: Adrian Belshaw Degree: Master of Science Title of Thesis: On the Normality of Numbers Examining Committee: Dr. Ladislav Stacho Chair Dr. Peter Borwein Senior Supervisor Professor of Mathematics Simon Fraser University Dr. Stephen Choi Supervisor Assistant Professor of Mathematics Simon Fraser University Dr. Jason Bell Internal Examiner Assistant Professor of Mathematics Simon Fraser University Date Approved: December 5. 2005 SIMON FRASER ' u~~~snrllbrary DECLARATION OF PARTIAL COPYRIGHT LICENCE The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection, and, without changing the content, to translate the thesislproject or extended essays, if technically possible, to any medium or format for the purpose of preservation of the digital work. -
History of the Numerical Methods of Pi
History of the Numerical Methods of Pi By Andy Yopp MAT3010 Spring 2003 Appalachian State University This project is composed of a worksheet, a time line, and a reference sheet. This material is intended for an audience consisting of undergraduate college students with a background and interest in the subject of numerical methods. This material is also largely recreational. Although there probably aren't any practical applications for computing pi in an undergraduate curriculum, that is precisely why this material aims to interest the student to pursue their own studies of mathematics and computing. The time line is a compilation of several other time lines of pi. It is explained at the bottom of the time line and further on the worksheet reference page. The worksheet is a take-home assignment. It requires that the student know how to construct a hexagon within and without a circle, use iterative algorithms, and code in C. Therefore, they will need a variety of materials and a significant amount of time. The most interesting part of the project was just how well the numerical methods of pi have evolved. Currently, there are iterative algorithms available that are more precise than a graphing calculator with only a single iteration of the algorithm. This is a far cry from the rough estimate of 3 used by the early Babylonians. It is also a different entity altogether than what the circle-squarers have devised. The history of pi and its numerical methods go hand-in-hand through shameful trickery and glorious achievements. It is through the evolution of these methods which I will try to inspire students to begin their own research. -
MATLAB Examples Mathematics
MATLAB Examples Mathematics Hans-Petter Halvorsen, M.Sc. Mathematics with MATLAB • MATLAB is a powerful tool for mathematical calculations. • Type “help elfun” (elementary math functions) in the Command window for more information about basic mathematical functions. Mathematics Topics • Basic Math Functions and Expressions � = 3�% + ) �% + �% + �+,(.) • Statistics – mean, median, standard deviation, minimum, maximum and variance • Trigonometric Functions sin() , cos() , tan() • Complex Numbers � = � + �� • Polynomials = =>< � � = �<� + �%� + ⋯ + �=� + �=@< Basic Math Functions Create a function that calculates the following mathematical expression: � = 3�% + ) �% + �% + �+,(.) We will test with different values for � and � We create the function: function z=calcexpression(x,y) z=3*x^2 + sqrt(x^2+y^2)+exp(log(x)); Testing the function gives: >> x=2; >> y=2; >> calcexpression(x,y) ans = 16.8284 Statistics Functions • MATLAB has lots of built-in functions for Statistics • Create a vector with random numbers between 0 and 100. Find the following statistics: mean, median, standard deviation, minimum, maximum and the variance. >> x=rand(100,1)*100; >> mean(x) >> median(x) >> std(x) >> mean(x) >> min(x) >> max(x) >> var(x) Trigonometric functions sin(�) cos(�) tan(�) Trigonometric functions It is quite easy to convert from radians to degrees or from degrees to radians. We have that: 2� ������� = 360 ������� This gives: 180 � ������� = �[�������] M � � �[�������] = �[�������] M 180 → Create two functions that convert from radians to degrees (r2d(x)) and from degrees to radians (d2r(x)) respectively. Test the functions to make sure that they work as expected. The functions are as follows: function d = r2d(r) d=r*180/pi; function r = d2r(d) r=d*pi/180; Testing the functions: >> r2d(2*pi) ans = 360 >> d2r(180) ans = 3.1416 Trigonometric functions Given right triangle: • Create a function that finds the angle A (in degrees) based on input arguments (a,c), (b,c) and (a,b) respectively. -
The Number of Solutions of Φ(X) = M
Annals of Mathematics, 150 (1999), 1–29 The number of solutions of φ(x) = m By Kevin Ford Abstract An old conjecture of Sierpi´nskiasserts that for every integer k > 2, there is a number m for which the equation φ(x) = m has exactly k solutions. Here φ is Euler’s totient function. In 1961, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an unconditional proof of Sierpi´nski’sconjecture. The proof uses many results from sieve theory, in particular the famous theorem of Chen. 1. Introduction Of fundamental importance in the theory of numbers is Euler’s totient function φ(n). Two famous unsolved problems concern the possible values of the function A(m), the number of solutions of φ(x) = m, also called the multiplicity of m. Carmichael’s Conjecture ([1],[2]) states that for every m, the equation φ(x) = m has either no solutions x or at least two solutions. In other words, no totient can have multiplicity 1. Although this remains an open problem, very large lower bounds for a possible counterexample are relatively easy to obtain, the latest being 101010 ([10], Th. 6). Recently, the author has also shown that either Carmichael’s Conjecture is true or a positive proportion of all values m of Euler’s function give A(m) = 1 ([10], Th. 2). In the 1950’s, Sierpi´nski conjectured that all multiplicities k > 2 are possible (see [9] and [16]), and in 1961 Schinzel [17] deduced this conjecture from his well-known Hypothesis H [18]. -
PRIME-PERFECT NUMBERS Paul Pollack [email protected] Carl
PRIME-PERFECT NUMBERS Paul Pollack Department of Mathematics, University of Illinois, Urbana, Illinois 61801, USA [email protected] Carl Pomerance Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755, USA [email protected] In memory of John Lewis Selfridge Abstract We discuss a relative of the perfect numbers for which it is possible to prove that there are infinitely many examples. Call a natural number n prime-perfect if n and σ(n) share the same set of distinct prime divisors. For example, all even perfect numbers are prime-perfect. We show that the count Nσ(x) of prime-perfect numbers in [1; x] satisfies estimates of the form c= log log log x 1 +o(1) exp((log x) ) ≤ Nσ(x) ≤ x 3 ; as x ! 1. We also discuss the analogous problem for the Euler function. Letting N'(x) denote the number of n ≤ x for which n and '(n) share the same set of prime factors, we show that as x ! 1, x1=2 x7=20 ≤ N (x) ≤ ; where L(x) = xlog log log x= log log x: ' L(x)1=4+o(1) We conclude by discussing some related problems posed by Harborth and Cohen. 1. Introduction P Let σ(n) := djn d be the sum of the proper divisors of n. A natural number n is called perfect if σ(n) = 2n and, more generally, multiply perfect if n j σ(n). The study of such numbers has an ancient pedigree (surveyed, e.g., in [5, Chapter 1] and [28, Chapter 1]), but many of the most interesting problems remain unsolved.