This is a somewhat schizophrenic book on the arithmetical theory of continued fractions, with the first half covering what everybody knows and the second half what almost nobody knows. The present volume is an unaltered reprint of the 1964 English translation of the 1961 Russian third edition.
The first two chapters are a thorough and fairly conventional development of the theory of continued fractions, very similar to what is in Hardy & Wright’s An Introduction to the Theory of Numbers and in Niven & Zuckerman & Montgomery’s An Introduction to the Theory of Numbers. The latter two books have examples and give applications (in particular solving Pell’s equation), while the present book has neither examples nor applications.
The third chapter, which is about half of the book, covers some rarely-seen material in statistical (measure-theoretic) properties of the partial quotients. These theorems typically give an extreme or average value for some function of the partial quotients that is satisfied by all real numbers except a set of measure zero. The highlights of this theory are also in Hardy & Wright, but Khinchin gives a much more detailed and thorough exposition.
Bottom line: If you want a book that deals only with continued fractions, this is a good choice, but most readers would be better served by one of the general number theory books that has more examples and integrates the subject better with the rest of number theory.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.
|Properties of the Apparatus|
|3.||Infinite continued fractions|
|4.||Continued fractions with natural elements|
|Chapter II.||The Representation of Numbers by Continued Fractions|
|5.||Continued fractions as an apparatus for representing real numbers|
|6.||Convergents as best approximations|
|7.||The order of approximation|
|8.||General approximation theorems|
|9.||The approximation of algebraic irrational numbers and Liouville's transcendental numbers|
|10.||Quadratic irrational numbers and periodic continued fractions|
|Chapter III.||The Measure Theory of Continued Fractions|
|12.||The elements as functions of the number represented|
|13.||Measure-theoretic evaluation of the increase in the elements|
|14.||Measure-theoretic evaluation of the increase in the denominators of the convergents. The fundamental theorem of the measure theory of approximation|
|15.||Gauss's problem and Kuz'min's theorem|