Table Of Content

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

Description:

Irrationality and Transcendence in Number Theory tells the story of irrational numbers from their discovery in the days of Pythagoras to the ideas behind the work of Baker and Mahler on transcendence in the 20th century. It focuses on themes of irrationality, algebraic and transcendental numbers, continued fractions, approximation of real numbers by rationals, and relations between automata and transcendence. This book serves as a guide and introduction to number theory for advanced undergraduates and early postgraduates. Readers are led through the developments in number theory from ancient to modern times. The book includes a wide range of exercises, from routine problems to surprising and thought-provoking extension material. Features Uses techniques from widely diverse areas of mathematics, including number theory, calculus, set theory, complex analysis, linear algebra, and the theory of computation. Suitable as a primary textbook for advanced undergraduate courses in number theory, or as supplementary reading for interested postgraduates. Each chapter concludes with an appendix setting out the basic facts needed from each topic, so that the book is accessible to readers without any specific specialist background. Access a list of errata for this book here.