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Irrational Numbers

MAA Press: An Imprint of the American Mathematical Society
Available Formats:
Electronic ISBN: 978-1-61444-011-6
Product Code: CAR/11.E
List Price: $50.00 MAA Member Price:$37.50
AMS Member Price: $37.50 Click above image for expanded view Irrational Numbers MAA Press: An Imprint of the American Mathematical Society Available Formats:  Electronic ISBN: 978-1-61444-011-6 Product Code: CAR/11.E  List Price:$50.00 MAA Member Price: $37.50 AMS Member Price:$37.50
• Book Details

The Carus Mathematical Monographs
Volume: 111985; 164 pp

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.

• Chapters
• Chapter I. Rationals and irrationals
• Chapter II. Simple irrationalities
• Chapter III. Certain algebraic numbers
• Chapter IV. The approximation of irrationals by rationals
• Chapter V. Continued fractions
• Chapter VI. Further Diophantine approximations
• Chapter VII. Algebraic and transcendental numbers
• Chapter VIII. Normal numbers
• Chapter IX. The generalized Lindemann theorem
• Chapter X. The Gelfond-Schneider theorem

• Reviews

• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Volume: 111985; 164 pp

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.

• Chapters
• Chapter I. Rationals and irrationals
• Chapter II. Simple irrationalities
• Chapter III. Certain algebraic numbers
• Chapter IV. The approximation of irrationals by rationals
• Chapter V. Continued fractions
• Chapter VI. Further Diophantine approximations
• Chapter VII. Algebraic and transcendental numbers
• Chapter VIII. Normal numbers
• Chapter IX. The generalized Lindemann theorem
• Chapter X. The Gelfond-Schneider theorem
• 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
Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
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