Table of Contents

Chapter I. Properties of the Apparatus 1. Introduction 2. Convergents 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 11. Introduction 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 16. Average values Index