eBook ISBN:  9781614440116 
Product Code:  CAR/11.E 
List Price:  $55.00 
MAA Member Price:  $41.25 
AMS Member Price:  $41.25 
eBook ISBN:  9781614440116 
Product Code:  CAR/11.E 
List Price:  $55.00 
MAA Member Price:  $41.25 
AMS Member Price:  $41.25 

Book DetailsThe Carus Mathematical MonographsVolume: 11; 1985; 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 GelfondSchneider 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 selfcontained. The results needed from analysis and algebra are central and wellknown 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.

Table of Contents

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 GelfondSchneider theorem


Additional Material

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 topicsinnumber 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


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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 GelfondSchneider 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 selfcontained. The results needed from analysis and algebra are central and wellknown 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 GelfondSchneider 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 topicsinnumber 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