DOCSLIB.ORG

An Introduction to Number Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
An Introduction to Number Theory An Introduction to Number Theory J. J. P. Veerman June 27, 2021 © 2020 J. J. P. Veerman This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) You are free to: Share—copy and redistribute the material in any medium or format Adapt—remix, transform, and build upon the material for any purpose, even commercially. The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms: Attribution—You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions—You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Non-Commercial—You may not use the material for commercial purposes. List of Figures p 1 Eratosthenes’ sieve up to n = 30. All multiples of a less than 31 are cancelled. The remainder are the primes less than n = 31. 4 2 A directed path g passing through all points of Z2. 12 R x 3 On the left, the function 2 lnt dt in blue, p(x) in red, and R x x=lnx in green. On the right, we have 2 lnt dt − x=lnx in blue, p(x) − x=lnx in red. 29 ¥ R ¥ −s 4 Proof that ∑n=1 f (n) is greater than 1 x dx if f is positive and decreasing. 34 ¥ 5 Proof that ∑n=1 f (n) (shaded in blue and green) minus f (1) R ¥ −s (shaded in blue) is less than 1 x dx if f is positive and decreasing to 0. 37 6 The origin is marked by “×”. The red dots are visible from ×; between any blue dot and × there is a red dot. The picture shows exactly one quarter of {−4;··· ;4g2n(0;0) ⊂ Z2. 38 7 The general solution of the inhomogeneous equation (~r;~x) = c in R2. 45 8 A one parameter family ft of maps from the circle to itself. For every t 2 [0;1] the map ft is constructed by truncating the map x ! 2x mod 1 as indicated in this figure. 65 3 4 List of Figures 9 Three branches of the Gauss map. 90 10 The line y = wx and (in red) successive iterates of the rotation Rw . Closest returns in this figure are q in f2;3;5;8g. 99 11 The geometry of successive closest returns. 99 12 Drawing y = w1x and successive approximations (an+1 is taken to be 3). The green arrows correspond to en−1, en, and en+1. 100 13 Black: thread from origin with golden mean slope; red: pulling the thread down from the origin; green: pulling the thread up from the origin. 103 14 Plots of the points (n;n) in polar coordinates, for n ranging from 1 to 50, 180, 330, and 3000, respectively. 106 15 Plots of the prime points (p; p) (p prime) in polar coordinates with p ranging between 2 and 3000, and between 2 and 30000, respectively. 107 16 The Gaussian integers are the lattice points in the complex plane; both real and imaginary parts are integers. For an arbitrary point z 2 C — marked by x in the figure, a nearby integer is k1 + ik2 where k1 is the closest integer to Re(z) and k2 the closest integer to Im(z). In this case that is 2 + 3i. 121 p 17 A depiction of Z[ −6] in the complex plane;p real parts are integers and imaginary parts are multiples of 6. 122 18 The part to the right of the intersection with ` : y = x + 1 (dashed) of bad path (in red) is reflected. The reflected part in indicated in green. The path becomes a monotone path from (0;0) to (n − 1;n + 1). 124 19 The Gaussian primes described in Proposition 7.30. There are approximately 950 within a radius 40 of the origin (left figure) and about 3300 within a radius 80 (right figure). 127 20 Possible values of rg−1 in the proof of Proposition 7.26. 129 21 A comparison between approximating the Lebesgue integral (left) and the Riemann integral (right). 138 22 The pushforward of a measure n. 139 − + 23 The functions m(Xc ) and m(Xc ). 140 List of Figures 5 24 This map is not uniquely ergodic 142 25 The first two stages of the construction of the singular measure np. The shaded parts are taken out. 145 26 The first two stages of the construction of the middle third Cantor set. 148 27 The inverse image of a small interval dy is T −1(dy) 154 p 28 w is irrational and q is a convergent of w. Then x + qw modulo 1 is close to x. Thus adding qw modulo 1 amounts to a translation by a small distance. Note: “om” in the figure stands for w. 157 1 1 29 `(I) is between 3 and 2 of `(J). So there are two disjoint images −1 of I under Rw that fall in J. 158 30 An example of the system described in Corollary 9.9. 160 31 Illustration of the fact that for a concave function f , we have f (wx + (1 − w)y) ≥ w f (x) + (1 − w) f (y) (Jensen’s inequality). 168 32 Plot of the function ln(x)ln(1 + x) 169 33 Left, a curve. Then two simple, closed curves with opposite orientation. The curve on the right is a union of two simple, closed curves. 175 34 A singular point z0 where f is bounded does not contribute to H g f dz. 176 35 The curve g goes around z exactly once in counter-clockwise direction. If d is small enough, z + d also lies inside g. 179 36 The curve w goes around z0 exactly once in counter-clockwise direction. 181 37 g is analytic in DR := fRez ≥ −dRg \ fjzj ≤ Rg (shaded). The red is p p curve is given by C+(s) = Re with s 2 (− 2 ; 2 ). The green curve is p 3p is given by C+(s) = Re with s 2 ( 2 ; 2 ). The blue L− consists of 2 small circular segments plus the segment connecting their left endpoints at a distance d to the left of the the imaginary axis. 182 38 The complex plane with eit , −e−it and e−it on the unit circle. cost is the average of eit and e−it and isint as the average of eit and −e−it . 186 6 List of Figures 39 The functions gi and hi of exercise 10.20 for i 2 f2;8;15;30g. 190 40 The contour C is the concatenation of c (celeste), b1 (blue), r1 (red), g (green), p (purple), −g, r2, and b2. The path r is a semi-circle of radius R. The path p is a small circle of radius r. See exercise 10.21. 190 41 Integration over the shaded triangle of area 1=2 in equation 11.9. 199 42 The functions q(x)=x (green), y(x)=x (red), and p(x)lnx=x (blue) for x 2 [1;1000]. All converge to 1 as x tends to infinity. The x-axis is horizontal. 208 Contents List of Figures 3 Part 1. Introduction to Number Theory Chapter 1. A Quick Tour of Number Theory 3 x1.1. Divisors and Congruences 4 x1.2. Rational and Irrational Numbers 5 x1.3. Algebraic and Transcendental Numbers 7 x1.4. Countable and Uncountable Sets 9 x1.5. Exercises 13 Chapter 2. The Fundamental Theorem of Arithmetic 19 x2.1. Bezout’s´ Lemma 20 x2.2. Corollaries of Bezout’s´ Lemma 21 x2.3. The Fundamental Theorem of Arithmetic 23 x2.4. Corollaries of the Fundamental Theorem of Arithmetic 25 x2.5. The Riemann Hypothesis 27 x2.6. Exercises 30 Chapter 3. Linear Diophantine Equations 39 x3.1. The Euclidean Algorithm 39 7 8 Contents x3.2. A Particular Solution of ax + by = c 41 x3.3. Solution of the Homogeneous equation ax + by = 0 43 x3.4. The General Solution of ax + by = c 44 x3.5. Recursive Solution of x and y in the Diophantine Equation 46 x3.6. Exercises 47 Chapter 4. Number Theoretic Functions 55 x4.1. Multiplicative Functions 55 x4.2. Additive Functions 58 x4.3. Mobius¨ inversion 58 x4.4. Euler’s Phi or Totient Function 60 x4.5. Dirichlet and Lambert Series 62 x4.6. Exercises 65 Chapter 5. Modular Arithmetic and Primes 73 x5.1. Modular Arithmetic 73 x5.2. Euler’s Theorem 74 x5.3. Fermat’s Little Theorem and Primality Testing 76 x5.4. Fermat and Mersenne Primes 79 x5.5. Division in Zb 81 x5.6. Exercises 83 Chapter 6. Continued Fractions 89 x6.1. The Gauss Map 89 x6.2. Continued Fractions 90 x6.3. Computing with Continued Fractions 95 x6.4. The Geometric Theory of Continued Fractions 97 x6.5. Closest Returns 99 x6.6. Exercises 101 Part 2. Topics in Number Theory Chapter 7. Algebraic Integers 111 x7.1. Rings and Fields 111 Contents 9 x7.2. Primes and Integral Domains 113 x7.3. Norms 115 x7.4. Euclidean Domains 118 x7.5. Example and Counter-Example 120 x7.6. Exercises 123 Chapter 8. Ergodic Theory 133 x8.1. The Trouble with Measure Theory 133 x8.2. Measure and Integration 135 x8.3. The Birkhoff Ergodic Theorem 139 x8.4.
Load more

Recommended publications
  • 9N1.5 Determine the Square Root of Positive Rational Numbers That Are Perfect Squares
    9N1.5 Determine the square root of positive rational numbers that are perfect squares. Square Roots and Perfect Squares The square root of a given number is a value that when multiplied by itself results in the given number. For ​ ​ example, the square root of 9 is 3 because 3 × 3 = 9 . Example Use a diagram to determine the square root of 121. Solution Step 1 Draw a square with an area of 121 units. Step 2 Count the number of units along one side of the square. √121 = 11 units √121 = 11 units The square root of 121 is 11. This can also be written as √121 = 11. A number that has a whole number as its square root is called a perfect square. Perfect squares have the ​ ​ unique characteristic of having an odd number of factors. Example Given the numbers 81, 24, 102, 144, identify the perfect squares by ordering their factors from smallest to largest. Solution The square root of each perfect square is bolded. Factors of 81: 1, 3, 9, 27, 81 ​ ​ Since there are an odd number of factors, 81 is a perfect square. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Since there are an even number of factors, 24 is not a perfect square. Factors of 102: 1, 2, 3, 6, 17, 34, 51, 102 Since there are an even number of factors, 102 is not a perfect square. Factors of 144: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 ​ ​ Since there are an odd number of factors, 144 is a perfect square.
    [Show full text]
  • The Pseudoprimes to 25 • 109
    MATHEMATICS OF COMPUTATION, VOLUME 35, NUMBER 151 JULY 1980, PAGES 1003-1026 The Pseudoprimes to 25 • 109 By Carl Pomerance, J. L. Selfridge and Samuel S. Wagstaff, Jr. Abstract. The odd composite n < 25 • 10 such that 2n_1 = 1 (mod n) have been determined and their distribution tabulated. We investigate the properties of three special types of pseudoprimes: Euler pseudoprimes, strong pseudoprimes, and Car- michael numbers. The theoretical upper bound and the heuristic lower bound due to Erdös for the counting function of the Carmichael numbers are both sharpened. Several new quick tests for primality are proposed, including some which combine pseudoprimes with Lucas sequences. 1. Introduction. According to Fermat's "Little Theorem", if p is prime and (a, p) = 1, then ap~1 = 1 (mod p). This theorem provides a "test" for primality which is very often correct: Given a large odd integer p, choose some a satisfying 1 <a <p - 1 and compute ap~1 (mod p). If ap~1 pi (mod p), then p is certainly composite. If ap~l = 1 (mod p), then p is probably prime. Odd composite numbers n for which (1) a"_1 = l (mod«) are called pseudoprimes to base a (psp(a)). (For simplicity, a can be any positive in- teger in this definition. We could let a be negative with little additional work. In the last 15 years, some authors have used pseudoprime (base a) to mean any number n > 1 satisfying (1), whether composite or prime.) It is well known that for each base a, there are infinitely many pseudoprimes to base a.
    [Show full text]
  • Grade 7/8 Math Circles the Scale of Numbers Introduction
    Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 7/8 Math Circles November 21/22/23, 2017 The Scale of Numbers Introduction Last week we quickly took a look at scientific notation, which is one way we can write down really big numbers. We can also use scientific notation to write very small numbers. 1 × 103 = 1; 000 1 × 102 = 100 1 × 101 = 10 1 × 100 = 1 1 × 10−1 = 0:1 1 × 10−2 = 0:01 1 × 10−3 = 0:001 As you can see above, every time the value of the exponent decreases, the number gets smaller by a factor of 10. This pattern continues even into negative exponent values! Another way of picturing negative exponents is as a division by a positive exponent. 1 10−6 = = 0:000001 106 In this lesson we will be looking at some famous, interesting, or important small numbers, and begin slowly working our way up to the biggest numbers ever used in mathematics! Obviously we can come up with any arbitrary number that is either extremely small or extremely large, but the purpose of this lesson is to only look at numbers with some kind of mathematical or scientific significance. 1 Extremely Small Numbers 1. Zero • Zero or `0' is the number that represents nothingness. It is the number with the smallest magnitude. • Zero only began being used as a number around the year 500. Before this, ancient mathematicians struggled with the concept of `nothing' being `something'. 2. Planck's Constant This is the smallest number that we will be looking at today other than zero.
    [Show full text]
  • Input for Carnival of Math: Number 115, October 2014
    Input for Carnival of Math: Number 115, October 2014 I visited Singapore in 1996 and the people were very kind to me. So I though this might be a little payback for their kindness. Good Luck. David Brooks The “Mathematical Association of America” (http://maanumberaday.blogspot.com/2009/11/115.html ) notes that: 115 = 5 x 23. 115 = 23 x (2 + 3). 115 has a unique representation as a sum of three squares: 3 2 + 5 2 + 9 2 = 115. 115 is the smallest three-digit integer, abc , such that ( abc )/( a*b*c) is prime : 115/5 = 23. STS-115 was a space shuttle mission to the International Space Station flown by the space shuttle Atlantis on Sept. 9, 2006. The “Online Encyclopedia of Integer Sequences” (http://www.oeis.org) notes that 115 is a tridecagonal (or 13-gonal) number. Also, 115 is the number of rooted trees with 8 vertices (or nodes). If you do a search for 115 on the OEIS website you will find out that there are 7,041 integer sequences that contain the number 115. The website “Positive Integers” (http://www.positiveintegers.org/115) notes that 115 is a palindromic and repdigit number when written in base 22 (5522). The website “Number Gossip” (http://www.numbergossip.com) notes that: 115 is the smallest three-digit integer, abc, such that (abc)/(a*b*c) is prime. It also notes that 115 is a composite, deficient, lucky, odd odious and square-free number. The website “Numbers Aplenty” (http://www.numbersaplenty.com/115) notes that: It has 4 divisors, whose sum is σ = 144.
    [Show full text]
  • Zero-One Laws
    1 THE LOGIC IN COMPUTER SCIENCE COLUMN by 2 Yuri GUREVICH Electrical Engineering and Computer Science UniversityofMichigan, Ann Arb or, MI 48109-2122, USA [email protected] Zero-One Laws Quisani: I heard you talking on nite mo del theory the other day.Itisinteresting indeed that all those famous theorems ab out rst-order logic fail in the case when only nite structures are allowed. I can understand that for those, likeyou, educated in the tradition of mathematical logic it seems very imp ortantto ndoutwhichof the classical theorems can b e rescued. But nite structures are to o imp ortant all by themselves. There's got to b e deep nite mo del theory that has nothing to do with in nite structures. Author: \Nothing to do" sounds a little extremist to me. Sometimes in nite ob jects are go o d approximations of nite ob jects. Q: I do not understand this. Usually, nite ob jects approximate in nite ones. A: It may happ en that the in nite case is cleaner and easier to deal with. For example, a long nite sum may b e replaced with a simpler integral. Returning to your question, there is indeed meaningful, inherently nite, mo del theory. One exciting issue is zero- one laws. Consider, say, undirected graphs and let be a propertyofsuch graphs. For example, may b e connectivity. What fraction of n-vertex graphs have the prop erty ? It turns out that, for many natural prop erties , this fraction converges to 0 or 1asn grows to in nity. If the fraction converges to 1, the prop erty is called almost sure.
    [Show full text]
  • A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000)
    A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000) 1. INTRODUCTION In 1982, Albert Wilansky, a mathematics professor at Lehigh University wrote a short article in the Two-Year College Mathematics Journal [6]. In that article he identified a new subset of the composite numbers. He defined a Smith number to be a composite number where the sum of the digits in its prime factorization is equal to the digit sum of the number. The set was named in honor of Wi!anskyJs brother-in-law, Dr. Harold Smith, whose telephone number 493-7775 when written as a single number 4,937,775 possessed this interesting characteristic. Adding the digits in the number and the digits of its prime factors 3, 5, 5 and 65,837 resulted in identical sums of42. Wilansky provided two other examples of numbers with this characteristic: 9,985 and 6,036. Since that time, many things have been discovered about Smith numbers including the fact that there are infinitely many Smith numbers [4]. The largest Smith numbers were produced by Samuel Yates. Using a large repunit and large palindromic prime, Yates was able to produce Smith numbers having ten million digits and thirteen million digits. Using the same large repunit and a new large palindromic prime, the author is able to find a Smith number with over thirty-two million digits. 2. NOTATIONS AND BASIC FACTS For any positive integer w, we let S(ri) denote the sum of the digits of n.
    [Show full text]
  • Representing Square Numbers Copyright © 2009 Nelson Education Ltd
    Representing Square 1.1 Numbers Student book page 4 You will need Use materials to represent square numbers. • counters A. Calculate the number of counters in this square array. ϭ • a calculator number of number of number of counters counters in counters in in a row a column the array 25 is called a square number because you can arrange 25 counters into a 5-by-5 square. B. Use counters and the grid below to make square arrays. Complete the table. Number of counters in: Each row or column Square array 525 4 9 4 1 Is the number of counters in each square array a square number? How do you know? 8 Lesson 1.1: Representing Square Numbers Copyright © 2009 Nelson Education Ltd. NNEL-MATANSWER-08-0702-001-L01.inddEL-MATANSWER-08-0702-001-L01.indd 8 99/15/08/15/08 55:06:27:06:27 PPMM C. What is the area of the shaded square on the grid? Area ϭ s ϫ s s units ϫ units square units s When you multiply a whole number by itself, the result is a square number. Is 6 a whole number? So, is 36 a square number? D. Determine whether 49 is a square number. Sketch a square with a side length of 7 units. Area ϭ units ϫ units square units Is 49 the product of a whole number multiplied by itself? So, is 49 a square number? The “square” of a number is that number times itself. For example, the square of 8 is 8 ϫ 8 ϭ .
    [Show full text]
  • 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.
    [Show full text]
  • Composite Numbers That Give Valid RSA Key Pairs for Any Coprime P
    information Article Composite Numbers That Give Valid RSA Key Pairs for Any Coprime p Barry Fagin ID Department of Computer Science, US Air Force Academy, Colorado Springs, CO 80840, USA; [email protected]; Tel.: +1-719-339-4514 Received: 13 August 2018; Accepted: 25 August 2018; Published: 28 August 2018 Abstract: RSA key pairs are normally generated from two large primes p and q. We consider what happens if they are generated from two integers s and r, where r is prime, but unbeknownst to the user, s is not. Under most circumstances, the correctness of encryption and decryption depends on the choice of the public and private exponents e and d. In some cases, specific (s, r) pairs can be found for which encryption and decryption will be correct for any (e, d) exponent pair. Certain s exist, however, for which encryption and decryption are correct for any odd prime r - s. We give necessary and sufficient conditions for s with this property. Keywords: cryptography; abstract algebra; RSA; computer science education; cryptography education MSC: [2010] 11Axx 11T71 1. Notation and Background Consider the RSA public-key cryptosystem and its operations of encryption and decryption [1]. Let (p, q) be primes, n = p ∗ q, f(n) = (p − 1)(q − 1) denote Euler’s totient function and (e, d) the ∗ Z encryption/decryption exponent pair chosen such that ed ≡ 1. Let n = Un be the group of units f(n) Z mod n, and let a 2 Un. Encryption and decryption operations are given by: (ae)d ≡ (aed) ≡ (a1) ≡ a mod n We consider the case of RSA encryption and decryption where at least one of (p, q) is a composite number s.
    [Show full text]
  • Sequences of Primes Obtained by the Method of Concatenation
    SEQUENCES OF PRIMES OBTAINED BY THE METHOD OF CONCATENATION (COLLECTED PAPERS) Copyright 2016 by Marius Coman Education Publishing 1313 Chesapeake Avenue Columbus, Ohio 43212 USA Tel. (614) 485-0721 Peer-Reviewers: Dr. A. A. Salama, Faculty of Science, Port Said University, Egypt. Said Broumi, Univ. of Hassan II Mohammedia, Casablanca, Morocco. Pabitra Kumar Maji, Math Department, K. N. University, WB, India. S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia. Mohamed Eisa, Dept. of Computer Science, Port Said Univ., Egypt. EAN: 9781599734668 ISBN: 978-1-59973-466-8 1 INTRODUCTION The definition of “concatenation” in mathematics is, according to Wikipedia, “the joining of two numbers by their numerals. That is, the concatenation of 69 and 420 is 69420”. Though the method of concatenation is widely considered as a part of so called “recreational mathematics”, in fact this method can often lead to very “serious” results, and even more than that, to really amazing results. This is the purpose of this book: to show that this method, unfairly neglected, can be a powerful tool in number theory. In particular, as revealed by the title, I used the method of concatenation in this book to obtain possible infinite sequences of primes. Part One of this book, “Primes in Smarandache concatenated sequences and Smarandache-Coman sequences”, contains 12 papers on various sequences of primes that are distinguished among the terms of the well known Smarandache concatenated sequences (as, for instance, the prime terms in Smarandache concatenated odd
    [Show full text]
  • Diophantine Approximation and Transcendental Numbers
    Diophantine Approximation and Transcendental Numbers Connor Goldstick June 6, 2017 Contents 1 Introduction 1 2 Continued Fractions 2 3 Rational Approximations 3 4 Transcendental Numbers 6 5 Irrationality Measure 7 6 The continued fraction for e 8 6.1 e's Irrationality measure . 10 7 Conclusion 11 A The Lambert Continued Fraction 11 1 Introduction Suppose we have an irrational number, α, that we want to approximate with a rational number, p=q. This question of approximating an irrational number is the primary concern of Diophantine approximation. In other words, we want jα−p=qj < . However, this method of trying to approximate α is boring, as it is possible to get an arbitrary amount of precision by making q large. To remedy this problem, it makes sense to vary with q. The problem we are really trying to solve is finding p and q such that p 1 1 jα − j < which is equivalent to jqα − pj < (1) q q2 q Solutions to this problem have applications in both number theory and in more applied fields. For example, in signal processing many of the quickest approximation algorithms are given by solutions to Diophantine approximation problems. Diophantine approximations also give many important results like the continued fraction expansion of e. One of the most interesting aspects of Diophantine approximations are its relationship with transcendental 1 numbers (a number that cannot be expressed of the root of a polynomial with rational coefficients). One of the key characteristics of a transcendental number is that it is easy to approximate with rational numbers. This paper is separated into two categories.
    [Show full text]
  • Number Pattern Hui Fang Huang Su Nova Southeastern University, [email protected]
    Transformations Volume 2 Article 5 Issue 2 Winter 2016 12-27-2016 Number Pattern Hui Fang Huang Su Nova Southeastern University, [email protected] Denise Gates Janice Haramis Farrah Bell Claude Manigat See next page for additional authors Follow this and additional works at: https://nsuworks.nova.edu/transformations Part of the Curriculum and Instruction Commons, Science and Mathematics Education Commons, Special Education and Teaching Commons, and the Teacher Education and Professional Development Commons Recommended Citation Su, Hui Fang Huang; Gates, Denise; Haramis, Janice; Bell, Farrah; Manigat, Claude; Hierpe, Kristin; and Da Silva, Lourivaldo (2016) "Number Pattern," Transformations: Vol. 2 : Iss. 2 , Article 5. Available at: https://nsuworks.nova.edu/transformations/vol2/iss2/5 This Article is brought to you for free and open access by the Abraham S. Fischler College of Education at NSUWorks. It has been accepted for inclusion in Transformations by an authorized editor of NSUWorks. For more information, please contact [email protected]. Number Pattern Cover Page Footnote This article is the result of the MAT students' collaborative research work in the Pre-Algebra course. The research was under the direction of their professor, Dr. Hui Fang Su. The ap per was organized by Team Leader Denise Gates. Authors Hui Fang Huang Su, Denise Gates, Janice Haramis, Farrah Bell, Claude Manigat, Kristin Hierpe, and Lourivaldo Da Silva This article is available in Transformations: https://nsuworks.nova.edu/transformations/vol2/iss2/5 Number Patterns Abstract In this manuscript, we study the purpose of number patterns, a brief history of number patterns, and classroom uses for number patterns and magic squares.
    [Show full text]