Irrational Numbers

✍ Scribed by Ivan Niven

• Publisher: Mathematical Assn of Amer
• Year: 1956
• Tongue: English
• Leaves: 176
• Edition: Second Printing
• Category: Library

✦ Synopsis

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

✦ Table of Contents

Cover
Series
Title page
Date-line
Preface
CONTENTS
I. Rationals and Irrationals
1. The preponderance of irrationals
2. Countability
3. Dense sets
4. Decimal expansions
II. Simple Irrationalities
1. Introduction
2. The trigonometric functions and $\pi$
3. The hyperbolic, exponential, and logarithmic functions
III. Certain Algebraic Numbers
1. Introduction
2. Further background material
3. The factorization of $x^n—1$
4. Certain trigonometric values
5. Extension to the tangent
IV. The Approximation of Irrationals by Rationals
1. The problem
2. A generalization
3. Linearly dependent sets
V. Continued Fractions
1. The Euclidean algorithm
2. Uniqueness
3. Infinite continued fractions
4. Infinite continued fraction expansions
5. The convergents as approximations
6. Periodic continued fractions
VI. Further Diophantine Approximations
1. A basic result
2. Best possible approximations
3. Uniform distributions
4. A proof by Fourier analysis
VII. Algebraic and Transcendental Numbers
1. Closure properties of algebraic numbers
2. A property of algebraic integers
3. Transcendental numbers
4. The order of approximation
VIII. Normal Numbers
1. Definition of a normal number
2. The measure of the set of normal numbers
3. Equivalent definitions
4. A normal number exhibited
IX. The Generalized Lindemann Theorem
1. Statement of the theorem
2. Preliminaries
3. Proof of the theorem
4. Applications of the theorem
5. Squaring the circle
X. The Gelfond-Schneider Theorem
1. Hilbert's seventh problem
2. Background material
3. Two lemmas
4. Proof of the Gelfond-Schneider theorem
List of Notation
Glossary
Reference Books
Index of Topics
Index of Names