Irrational numbers
by Niven, Ivan
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 techniques. The last third of the monograph treats n…
Read More ↓Title Statements
- Full title :Irrational numbers / Ivan Niven.
Summary
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 techniques. The last third of the monograph treats normal and transcendental numbers, including the transcendence of p and its generalization in the Lindermann 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 book is almost wholly self-contained. 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 of notes at the end of each chapter citing the main sources used by the author, and suggesting further readings.
Read More ↓Edition Statements
- Publication Date:©1985
Publication Statements
- Publisher(s):Mathematical Association of America
- Place of Publication:Washington, District of Columbia
- Language:eng
- Publication Date:©1985
Notes
"Paperbound edition issued March 2005."
The Physical Object
- Description:1 online resource (177 pages) : : illustrations, tables
Edition Identifiers
- ISBN:9781614440116, 1614440115, 0883850389, 9780883850381
- oclc:ocn880452919
- LC Classification:QA247.5
- Dewey Decimal:512.81
Table of Contents
Front Cover -- Irrational Numbers -- Copyright Page -- Contents -- Chapter I. Rationals and Irrationals -- 1. The preponderance of irrationals -- 2. Countability -- 3. Dense sets -- 4. Decimal expansions -- Chapter II. Simple Irrationalities -- 1. Introduction -- 2. The trigonometric functions and π -- 3. The hyperbolic, exponential, and logarithmic functions -- Chapter III. Certain Algebraic Numbers -- 1. Introduction -- 2. Further background material -- 3. The factorization of xn � 1 -- 4. Certain trigonometric values -- 5. Extension to the tangent
Chapter IV. The Approximation of Irrationals by Rationals1. The problem -- 2. A generalization -- 3. Linearly dependent sets -- Chapter 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 -- Chapter VI. Further Diophantine Approximations -- 1. A basic result -- 2. Best possible approximations -- 3. Uniform distributions -- 4. A proof by Fourier analysis
Chapter VII. Algebraic and Transcendental Numbers1. Closure properties of algebraic numbers -- 2. A property…
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