Irrationality and Transcendence in Number Theory
- Length: 240 pages
- Edition: 1
- Language: English
- Publisher: Chapman and Hall/CRC
- Publication Date: 2022-01-27
- ISBN-10: 0367628376
- ISBN-13: 9780367628376
- Sales Rank: #1350572 (See Top 100 Books)
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.
Cover Half Title Title Page Copyright Page Dedication Contents Foreword Preface Author CHAPTER 1: INTRODUCTION 1.1. Irrational surds 1.2. Irrational decimals 1.3. Irrationality of the exponential constant 1.4. Other results, and some open questions Exercises Appendix: Some elementary number theory CHAPTER 2: HERMITE’S METHOD 2.1. Irrationality of er 2.2. Irrationality of π 2.3. Irrational values of trigonometric functions Exercises Appendix: Some results of elementary calculus CHAPTER 3: ALGEBRAIC AND TRANSCENDENTAL NUMBERS 3.1. Definitions and basic properties 3.1.1. Proving polynomials irreducible 3.1.2. Closure properties of algebraic numbers 3.2. Existence of transcendental numbers 3.3. Approximation of real numbers by rationals 3.4. Irrationality of (3) : a sketch Exercises Appendix 1: Countable and uncountable sets Appendix 2: The Mean Value Theorem Appendix 3: The Prime Number Theorem CHAPTER 4: CONTINUED FRACTIONS 4.1. Definition and basic properties 4.2. Continued fractions of irrational numbers 4.3. Approximation properties of convergents 4.4. Two important approximation problems 4.4.1. How many days should we count in a calendar year? 4.4.2. How many semitones should there be in an octave? 4.5. A “computational” test for rationality 4.6. Further approximation properties of convergents 4.7. Computing the continued fraction of an algebraic irrational 4.8. The continued fraction of e Exercises Appendix 1: A property of positive fractions Appendix 2: Simultaneous equations with integral coefficients Appendix 3: Cardinality of sets of sequences Appendix 4: Basic musical terminology CHAPTER 5: HERMITE’S METHOD FOR TRANSCENDENCE 5.1. Transcendence of e 5.2. Transcendence of π 5.2.1. Symmetric polynomials 5.2.2. The transcendence proof 5.3. Some more irrationality proofs 5.4. Transcendence of eα 5.5. Other results Exercises Appendix 1: Roots and coefficients of polynomials Appendix 2: Some real and complex analysis Appendix 3: Ordering complex numbers CHAPTER 6: AUTOMATA AND TRANSCENDENCE 6.1. Deterministic finite automata 6.2. Mahler’s transcendence proof 6.3. A more general transcendence result 6.4. A transcendence proof for the Thue sequence 6.5. Automata and functional equations 6.6. Conclusion Exercises Appendix 1: Alphabets, languages and DFAs Appendix 2: Some results of complex analysis A2.1. Taylor series and analytic functions A2.2. Limit points of roots of an analytic function A2.3. Estimation of power series A2.4. Algebraic and transcendental functions Appendix 3: A result on linear equations CHAPTER 7: LAMBERT’S IRRATIONALITY PROOFS 7.1. Generalised continued fractions 7.1.1. Irrationality of tanh r 7.2. Further continued fractions 7.2.1. Irrationality of tan r Exercises Appendix: Some results from elementary algebra and calculus Hints for exercises Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Bibliography Index
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