Numbers Rational and Irrational

✍ Scribed by Niven, Ivan Morton

• Publisher: Mathematical Association of America (MAA)
• Year: 1961;2014
• Tongue: English
• Leaves: 149
• Series: Anneli Lax New Mathematical Library 1
• Category: Library

✦ Synopsis

A superb development that starts with the natural numbers and carries the reader through the rationals and their decimal representations to algebraic numbers and then to the real numbers. Along the way, you will see characterizations of the rationals and of certain special (Liouville) transcendental numbers.pThis material is basic to all of algebra and analysis. Professor Niven's book may be read with profit by interested high school students as well as by college students and others who want to know more about the basic aspects of pure mathematics. Most readers will find the early chapters well within their grasp while ambitious readers will profit by the more advanced material to be found in later chapters.

✦ Table of Contents

Front Cover......Page 1
Numbers: Rational and Irrational......Page 4
Copyright Page......Page 5
CONTENTS......Page 8
Introduction......Page 12
Chapter 1. Natural Numbers aud Integers......Page 18
1.1 Primes......Page 19
1.2 Unique Factorization......Page 20
1.3 Integers......Page 22
1.4 Even and Odd Integers......Page 24
1.5 Closure Properties......Page 27
1.6 A Remark on the Nature of Proof......Page 28
2.1 Definition of Rational Numbers......Page 30
2.2 Terminating and Non-terminating Decimals......Page 32
2.3 The Many Ways of Stating and Proving Propositions......Page 35
2.4 Periodic Decimals......Page 39
2.5 Terminating Decimals Written as Periodic Decimals......Page 43
2.6 A Summary......Page 45
3.1 The Geometric Viewpoint......Page 47
3.2 Decimal Representations......Page 48
3.3 The Irrationality of √2......Page 51
3.4 The Irrationality of √3......Page 52
3.5 The Irrationality of v6 and v2 +√3......Page 53
3.6 The Words We Use......Page 54
3.7 An Application to Geometry......Page 55
3.8 A summary......Page 60
4.1 Closure Properties......Page 61
4.2 Polynomial Equations......Page 63
4.3 Rational Roots of Polynomial Equations......Page 66
4.4 Further Examples......Page 71
4.5 A Summary......Page 73
5.1 Irrational Values of Trigonometric Functions......Page 74
5.2 A Chain Device......Page 77
5.3 Irrational Values of Common Logarithms......Page 78
5.4 Transcendental Numbers......Page 80
5.5 Three Famous Construction Problems......Page 82
5.6 Further Analysis of 3√2......Page 87
5.7 A Summary......Page 88
6.1 Inequalities......Page 90
6.2 Approximation by Integers......Page 93
6.3 Approximation by Rationals......Page 95
6.4 Better Approximations......Page 98
6.5 Approximations to within1/n2......Page 103
6.6 Limitations on Approximations......Page 108
6.7 A Summary......Page 111
Chapter 7. The Existence of Transcendental Numbers......Page 112
7.1 Some Algebraic Preliminaries......Page 113
7.2 An Approximation to α......Page 116
7.3 The Plan of the Proof......Page 117
7.4 Properties of Polynomials......Page 118
7.5 The Transcendence of α......Page 120
7.6 A Summary......Page 122
Appendix A Proof That There Are Infinitely Many Prime Numbers......Page 124
Appendix B Proof of the Fundamental Theorem of Arithmetic......Page 126
Appendix C Cantor’s Proof of the Existence of Transcendental Numbers......Page 131
Appendix D Trigonometric Numbers......Page 138
Answers and Suggestions to Selected Problems......Page 142
Index......Page 148

✦ Subjects

Science;Mathematics;Textbooks