Numbers: Rational and Irrational

Description

In this monograph, Ivan Niven, provides a masterful exposition of some central results on irrational, transcendental, and normal numbers. He gives a complete treatment by elementary methods of the irrationality of the exponential, logarithmic, and trigonometric functions with rational arguments. The approximation of irrational numbers by rationals, up to such results as the best possible approximation of Hurwitz, is also given with elementary technique. The last third of the monograph treats normal and transcendental numbers, including the transcendence of and its generalization in the Lindemann theorem, and the Gelfond-Schneider theorem. Most of the material in the first two-thirds of the book presupposes only calculus and beginning number theory. The results needed from analysis and algebra are central, and well-known theorems, and complete references to standard works are given to help the beginner. The chapters are for the most part independent. There is a set if bites at the end of each chapter citing

This book belongs to the series of monographs published by the School Mathematics Study Group. The ideas of the first five chapters are accessible to high school students. Chapters 1 and 2 deal with the integers and rationals, properties of primes, identification of the rationals with the repeating decimals, etc. Chapters 3, 4, and 5 produce examples of irrational numbers and proofs of their irrationality. Concepts of algebraic and transcendental numbers are introduced. The three famous construction problems of classical Greek geometry are analyzed.
Chapter 6 deals with approximation of irrationals by rationals, for example, approximations to within 1/n21/n^2. In Chapter 7 Liouville's method is followed to establish the existence of transcendentals. Cantor's existence proof is also given.
This book is ideally suited for the purposes for which it was written.

Reviewed by C. Brumfiel