The theory of continued fractions is vast. This booklet covers 
only its fundamentals, but it contains everything that may be 
useful for a layman interested in mathematics.

One more addition to the Little Mathematics Library series. In this post we see Fascinating Fractions by N. M. Beskin.

This booklet is intended for high-school students interested
in mathematics. It is concerned with approximating real
numbers by rational ones, which is one of the most captivating
topics in arithmetic.

…
Continued fractions represent one of the most perfect
creations of 17-18th century mathematicians: Huygens, Euler,
Lagrange, and Legendre. The properties of these fractions
are really striking.

The following should be borne in mind when reading this
booklet. Topics easily understandable are presented in normal print,
while those more difficult are given in small print. Proofs of some theorems given in small print may be omitted safely. These theorems will necessarily be taken for granted. However, mathematics is not just reading for entertainment. A future mathematician as well as a physicist or an engineer has to acquire skill in dealing with  mathematical constructions and proofs. So take a pencil and a sheet of  paper and study carefully the topics given in small print. You may succeed in simplifying some proofs or finding better ones.

The book was translated from the Russian by V. I. Kisin and was first published by Mir Publishers in 1986.

DJVU | OCR | Bookmarked |  Cover | 827 KB (Zip 823) | 88 pp

All credits to the original uploader, thanks to KS for the link.

You can get the book here.

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Contents

Preface 7
Chapter 1. Two Historical Puzzles
1.1. Archimedes’ Puzzle 9
1.1.1. Archimedes’ Number 9
1.1.2. Approximation 10
1.1.3. Error of Approximation 12
1.1.4. Quality of Approximation 13
1.2. The Puzzle of Pope Gregory XIII 15
1.2.1. The Mathematical Problem of the Calendar … 15
1.2.2. Julian and Gregorian Calendars 17

Chapter 2. Formation of Continued Fractions
2.1. Expansion of a Real Number into a Continued Fraction 19
2.1.1. Algorithm of Expansion into a Continued Fraction 19
2.1.2. Notation for Continued Fractions 21
2.1.3. Expansion of Negative Numbers into Continued Fractions 21
2.1.4. Examples of Nonterminating Expansion 22
2.2. Euclid’s Algorithm 24
2.2.1. Euclid’s Algorithm 24
2.2.2. Examples of Application of Euclid’s Algorithm 26
2.2.3. Summary 27

Chapter 3. Convergents
3.1. The Concept of Convergents 29
3.1.1. Preliminary Definition of Convergents 29
3.1.2. How to Generate Convergents 30
3.1.3. The Final Definition of Convergents 33
3.1.4. Evaluation of Convergents 34
3.1.5. Complete Quotients 34
3.2. The Properties of Convergents 36
3.2.1. The Difference Between Two Neighbouring Convergents 36
3.2.2. Comparison of Neighbouring Convergents  37
3.2.3. Irreducibility of Convergents 39

Chapter 4. Nonterminating Continued Fractions
4.1. Real Numbers 40
4.1.1. The Gulf Between the Finite and the Infinite . . 40
4.1.2. Principle of Nested Segments 41
4.1.3. The Set of Rational Numbers 44
4.1.4. The Existence of Nonrational Points on the  Number Line 45
4.1.5. Nonterminating Decimal Fractions 46
4.1.6. Irrational Numbers 48
4.1.7. Real Numbers 49
4.1.8. Representing Real Numbers on the Number Line 50
4.1.9. The Condition of Rationality of Nonterminating Decimals 52
4.2. Nonterminating Continued Fractions 52
4.2.1. Numerical Value of a Nonterminating Continued Fraction 52
4.2.2. Representation of Irrationals by Nonterminating Continued Fractions 54
4.2.3. The Single-Valuedness of the Representation of a Real Number by a Continued  Fraction 55
4.3. The Nature of Numbers Given by Continued Fractions 58
4.3.1. Classification of Irrationals 58
4.3.2. Quadratic Irrationals 60
4.3.3. Euler’s Theorem 66
4.3.4. Lagrange Theorem 69

Chapter 5. Approximation of Real Numbers
5.1. Approximation by Convergents 72
5.1.1. High-Quality Approximation 72
5.1.2. The Main Property of Convergents 72
5.1.3. Convergents Have the Highest Quality 76

Chapter 6. Solutions
6.1. The Mystery of Archimedes’ Number 81
6.1.1. The Key to All Puzzles 81
6.1.2. The Secret of Archimedes’ Number 81
6.2. The Solution to the Calendar Problem 83
6.2.1. The Use of Continued Fractions 83
6.2.2. How to Choose a Calendar 84
6.2.3. The Secret of Pope Gregory XIII 86
Bibliography 88