Irrationality and Transcendence in Number Theory

"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.”

­—MAA Reviews

“This thoughtfully organised book is very well written—meticulous care has been taken to make the material accessible to students new to the topics.”

—The Mathematical Gazette

Irrationality and Transcendence in Number Theory, Second Edition 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

New to the Second Edition

• A theorem on sums of irrational square roots. As well as the result being of interest in itself, the proof serves as an early example of ideas which will be of great importance in Chapter 5
• A theorem of Rado concerning irrationality of values of functions which are solutions to systems of differential equations
• Pell’s equation is treated twice using different techniques
• An introduction to the measure theory of continued fractions, leading to certain astounding regularities in the continued fractions of real numbers
• 40% more problems