Irrationality and Transcendence in Number Theory

2022·6.07 MB·English

by  David Angell

Cover Page Page: i
Half-Title Page Page: i
Title Page Page: iii
Copyright Page Page: iv
Dedication Page Page: v
Contents Page: vii
Foreword Page: xi
Preface Page: xiii
Author Page: xvii
Chapter 1 ◾ Introduction Page: 1
1.1 Irrational surds Page: 2
1.2 Irrational decimals Page: 6
1.3 Irrationality of the exponential constant Page: 8
1.4 Other results, and some open questions Page: 9
Exercises Page: 10
Appendix: Some elementary number theory Page: 14
Chapter 2 ◾ Hermite's Method Page: 17
2.1 Irrationality of er Page: 18
2.2 Irrationality of π Page: 22
2.3 Irrational values of trigonometric functions Page: 23
Exercises Page: 25
Appendix: Some results of elementary calculus Page: 28
Chapter 3 ◾ Algebraic and Transcendental Numbers Page: 31
3.1 Definitions and basic properties Page: 31
3.1.1 Proving polynomials irreducible Page: 33
3.1.2 Closure properties of algebraic numbers Page: 37
3.2 Existence of transcendental numbers Page: 40
3.3 Approximation of real numbers by rationals Page: 42
3.4 Irrationality of ζ(3): a sketch Page: 54
Exercises Page: 57
Appendix 1: Countable and uncountable sets Page: 61
Appendix 2: The Mean Value Theorem Page: 63
Appendix 3: The Prime Number Theorem Page: 63
Chapter 4 ◾ Continued Fractions Page: 65
4.1 Definition and basic properties Page: 66
4.2 Continued fractions of irrational numbers Page: 69
4.3 Approximation properties of convergents Page: 76
4.4 Two important approximation problems Page: 80
4.4.1 How many days should we count in a calendar year? Page: 81
4.4.2 How many semitones should there be in an octave? Page: 83
4.5 A “computational” test for rationality Page: 85
4.6 Further approximation properties of convergents Page: 87
4.7 Computing the continued fraction of an algebraic irrational Page: 92
4.8 The continued fraction of e Page: 94
Exercises Page: 99
Appendix 1: A property of positive fractions Page: 105
Appendix 2: Simultaneous equations with integral coefficients Page: 105
Appendix 3: Cardinality of sets of sequences Page: 106
Appendix 4: Basic musical terminology Page: 107
Chapter 5 ◾ Hermite's Method for Transcendence Page: 109
5.1 Transcendence of e Page: 110
5.2 Transcendence of π Page: 113
5.2.1 Symmetric polynomials Page: 114
5.2.2 The transcendence proof Page: 117
5.3 Some more irrationality proofs Page: 121
5.4 Transcendence of eα Page: 127
5.5 Other results Page: 139
Exercises Page: 141
Appendix 1: Roots and coefficients of polynomials Page: 143
Appendix 2: Some real and complex analysis Page: 143
Appendix 3: Ordering complex numbers Page: 145
Chapter 6 ◾ Automata and Transcendence Page: 147
6.1 Deterministic finite automata Page: 148
6.2 Mahler's transcendence proof Page: 150
6.3 A more general transcendence result Page: 156
6.4 A transcendence proof for the Thue sequence Page: 162
6.5 Automata and functional equations Page: 164
6.6 Conclusion Page: 168
Exercises Page: 168
Appendix 1: Alphabets, languages and DFAs Page: 170
Appendix 2: Some results of complex analysis Page: 171
A2.1 Taylor series and analytic functions Page: 171
A2.2 Limit points of roots of an analytic function Page: 171
A2.3 Estimation of power series Page: 172
A2.4 Algebraic and transcendental functions Page: 173
Appendix 3: A result on linear equations Page: 175
Chapter 7 ◾ Lambert's Irrationality Proofs Page: 177
7.1 Generalised continued fractions Page: 179
7.1.1 Irrationality of tanhr Page: 182
7.2 Further continued fractions Page: 184
7.2.1 Irrationality of tanr Page: 188
Exercises Page: 190
Appendix: Some results from elementary algebra and calculus Page: 192
Hints for exercises Page: 193
Chapter 1 Page: 193
Chapter 2 Page: 195
Chapter 3 Page: 196
Chapter 4 Page: 201
Chapter 5 Page: 207
Chapter 6 Page: 209
Chapter 7 Page: 212
Bibliography Page: 215
Index Page: 221