Irrationality and Transcendence in Number Theory.

243 Pages·2022·14.35 MB·English

by  David Angell

Irrationality and
Transcendence
in Number Theory

Irrationality and
Transcendence
in Number Theory
David Angell
University of New South Wales, Australia
Cover image: Helena Brusic The Imagination Agency
First edition published 2022
by CRC Press
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and by CRC Press
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© 2022 David Angell
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Library of Congress Cataloging‑in‑Publication Data
Names: Angell, David (Mathematics), author.
Title: Irrationality and transcendence in number theory / David Angell,
University of New South Wales, Australia.
Description: First edition. | Boca Raton : C&H/CRC Press, 2022. | Includes
bibliographical references and index.
Identifiers: LCCN 2021037176 (print) | LCCN 2021037177 (ebook) | ISBN
9780367628376 (hardback) | ISBN 9780367628758 (paperback) | ISBN
9781003111207 (ebook)
Subjects: LCSH: Irrational numbers. | Transcendental numbers. | Number
theory.
Classification: LCC QA247.5 .A54 2022 (print) | LCC QA247.5 (ebook) | DDC
512.7/3--dc23
LC record available at https://lccn.loc.gov/2021037176
LC ebook record available at https://lccn.loc.gov/2021037177
ISBN: 978-0-367-62837-6 (hbk)
ISBN: 978-0-367-62875-8 (pbk)
ISBN: 978-1-003-11120-7 (ebk)
DOI: 10.1201/9781003111207
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
In memory of
Dr David K. Crooke
who would have loved to read this book.

Contents
Foreword xi
Preface xiii
Author xvii
Chapter 1 (cid:4) INTRODUCTION 1
1.1 Irrational surds 2
1.2 Irrational decimals 6
1.3 Irrationality oftheexponential constant 8
1.4 Other results,andsome openquestions 9
Exercises 10
Appendix:Someelementary number theory 14
Chapter 2 (cid:4) HERMITE’S METHOD 17
2.1 Irrationality ofer 18
2.2 Irrationality ofπ 22
2.3 Irrational valuesoftrigonometric functions 23
Exercises 26
Appendix:Someresults ofelementary calculus 28
vii
viii (cid:4) Contents
Chapter 3 (cid:4) ALGEBRAIC AND TRANSCENDENTAL
NUMBERS 31
3.1 Definitions andbasic properties 31
3.1.1 Proving polynomials irreducible 33
3.1.2 Closure properties of algebraic numbers 37
3.2 Existence oftranscendental numbers 40
3.3 Approximation ofrealnumbers byrationals 42
3.4 Irrationality ofζ(3):asketch 55
Exercises 57
Appendix1:Countable anduncountable sets 62
Appendix2:TheMeanValueTheorem 63
Appendix3:ThePrimeNumber Theorem 63
Chapter 4 (cid:4) CONTINUED FRACTIONS 65
4.1 Definition andbasicproperties 66
4.2 Continued fractions ofirrational numbers 70
4.3 Approximation properties ofconvergents 76
4.4 Twoimportant approximation problems 81
4.4.1 How many days should we count in a
calendar year? 81
4.4.2 How many semitones should there be
in an octave? 84
4.5 A“computational” testfor rationality 86
4.6 Further approximation properties ofconvergents 87
4.7 Computing thecontinued fractionofanalgebraic irrational 93
4.8 Thecontinued fraction ofe 95
Exercises 100
Appendix1:Aproperty ofpositive fractions 105
Appendix2:Simultaneous equations withintegral coefficients 106
Contents (cid:4) ix
Appendix3:Cardinality ofsetsofsequences 106
Appendix4:Basicmusical terminology 107
Chapter 5 (cid:4) HERMITE’S METHOD FOR TRANSCENDENCE 109
5.1 Transcendence ofe 110
5.2 Transcendence ofπ 114
5.2.1 Symmetric polynomials 114
5.2.2 The transcendence proof 117
5.3 Somemoreirrationality proofs 121
5.4 Transcendence ofeα 127
5.5 Other results 139
Exercises 141
Appendix1:Rootsandcoefficients ofpolynomials 143
Appendix2:Somerealandcomplex analysis 143
Appendix3:Ordering complex numbers 145
Chapter 6 (cid:4) AUTOMATA AND TRANSCENDENCE 147
6.1 Deterministic finiteautomata 148
6.2 Mahler’s transcendence proof 150
6.3 Amoregeneraltranscendence result 156
6.4 Atranscendence prooffor theThuesequence 162
6.5 Automata andfunctional equations 165
6.6 Conclusion 168
Exercises 169
Appendix1:Alphabets, languages andDFAs 171
Appendix2:Someresultsofcomplex analysis 171
A2.1 Taylor series and analytic functions 171
A2.2 Limit points of roots of an analytic function 172