# Irrational Numbers

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*Ivan Niven*

MAA Press: An Imprint of the American Mathematical Society

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 reading.

#### Reviews & Endorsements

The book is fantastic and remains valuable even fifty years after its first appearance. It certainly qualifies (still) as a wonderful choice for a topics-in-number theory seminar or a tutorial or reading course. Individual chapters of “Irrational Numbers” already go a long way in this regard all by themselves.

-- Michael Berg, MAA Reviews

# Table of Contents

## Irrational Numbers

- Front Cover Cover11
- Irrational Numbers i2
- Copyright Page ii3
- Contents xi12
- Chapter I. Rationals and Irrationals 114
- Chapter II. Simple Irrationalities 1528
- Chapter III. Certain Algebraic Numbers 2841
- Chapter IV. The Approximation of Irrationals by Rationals 4255
- Chapter V. Continued Fractions 5164
- Chapter VI. Further Diophantine Approximations 6881
- Chapter VII. Algebraic and Transcendental Numbers 8396
- Chapter VIII. Normal Numbers 94107
- Chapter IX. The Generalized Lindemann Theorem 117130
- Chapter X. The Gelfond-Schneider Theorem 134147
- List of Notation 151164
- Glossary 153166
- Reference Books 157170
- Index of Topics 159172
- Index of Names 163176
- Back cover 165178