## Continued Fractions – Khinchin

In this post, we will see the book Continued Fractions by A. Ya. Khinchin.

The late Alexander J. Khinchin was born in Russia in 1894. One of the founders of the Soviet school of probability theory, Khinchin was made a full professor at Moscow University in 1922 and held that position until his death. His teaching skill is discernible in the clear and straightforward presentation of his subject. Designed for use as an expository text in the university curriculum, the book is basically of an elementary nature, the author confining his attention to continued fractions with positive-integral elements. The essentials needed for applications in probability theory, mechanics, and, especially, number theory are given and the real number system is constructed from continued fractions. The last chapter is somewhat more advanced and deals with the metric, or probability, theory of continued fractions. This first English translation is based on the third edition of the text which was issued in 1961.

The book was translated from Russian by Scripta Technica and was published in 1964.

You can get the book here.

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# Contents

## Chapter I. Properties of the Apparatus 1

1. Introduction 1
2. Convergents 3
3. Infinite continued fractions 8
4. Continued fractions with natural elements 12

## Chapter II. The Representation of Numbers by Continued Fractions 16

5. Continued fractions as an apparatus for representing real numbers 16
6. Convergents as best approximations 20
7. The order of approximation 28
8. General approximation theorems 34
9. The approximation of algebraic irrational numbers and Liouville’s transcendental numbers 45
10. Quadratic irrational numbers and periodic continued fractions 47

## Chapter III. The Measure Theory of Continued Fractions 51

11. Introduction 51
12. The elements as functions of the number represented 52
13. Measure-theoretic evaluation of the increase in the elements 60
14. Measure-theoretic evaluation of the increase in the denominators of the convergents. The fundamental theorem of the measure theory of approximation 65
15. Gauss’s problem and Kuz’min’s theorem 71
16. Average values 86
95 Index