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Irrationality and Transcendence in Number Theory by David Angell (2021, Hardcover)
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About this product
Product Identifiers
PublisherCRC Press LLC
ISBN-100367628376
ISBN-139780367628376
eBay Product ID (ePID)12050416844
Product Key Features
Number of Pages224 Pages
Publication NameIrrationality and Transcendence in Number Theory
LanguageEnglish
SubjectNumber Systems, General, Number Theory
Publication Year2021
TypeTextbook
Subject AreaMathematics
AuthorDavid Angell
FormatHardcover
Dimensions
Item Weight16.5 Oz
Item Length9.2 in
Item Width6.1 in
Additional Product Features
LCCN2021-037176
Reviews"Exceptionally informative, impressively organized and presented, Irrationality and Transcendence in Number Theory is an ideal selection as a curriculum textbook." - Midwest Books Review "This excellent book not only helps fill a substantial gap in the undergraduate mathematics literature, but it does so in a way that most students will, I think, find interesting, inviting and accessible. [. . .] This material is, of course, very nontrivial, but Angell goes to great lengths to make it accessible. He writes slowly and clearly and spends a lot of time motivating results. As previously noted, he also includes background Appendices in each chapter. There are other useful pedagogical features. Each chapter ends with an extensive collection of exercises, most of them non-routine; a 20-page section at the end of the book offers hints to these. The book also contains a five-page bibliography (one that, surprisingly, omits the Burger/Tubbs book mentioned earlier) that directs a reader to useful sources. The subject matter of this book is interesting and beautiful and deserves to be made accessible to well-prepared senior undergraduates. Angell has done an excellent job in helping to do so." - MAA Reviews
Dewey Edition23
IllustratedYes
Dewey Decimal512.7/3
Table Of Content1. 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. 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. 3. Algebraic & Transcendental Numbers. 3.1. Definitions and Basic Properties. 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. 4. Continued Fractions. Definition and Basic Properties. 4.2. Continued Fractions of Irrational Numbers. 4.3. Approximation Properties of Convergents. 4.4. Two important Approximation Problems. 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. 5. Hermite's Method for Transcendence. 5.1. Transcendence of e. 5.2. Transcendence of π. 5.3. Some more Irrationality Proofs. 5.4. Transcendence of ea .5.5. Other Results. Exercises. Appendix 1: Roots and Coefficients of Polynomials. Appendix 2: Some Real and Complex Analysis. Appendix 3: Ordering Complex Numbers. 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. Appendix 3: A Result on Linear Equations. 7. Lambert's Irrationality Proofs. 7.1. Generalised Continued Fractions. 7.2. Further Continued Fractions. Exercises. Appendix: Some Results from Elementary Algebra and Calculus. Hints for Exercises. Bibliography. Index.
SynopsisThis book 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. 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., This book 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, etc.
LC Classification NumberQA247.5.A54 2022
