Real Numbers and Fascinating Fractions

Based on N.M. Beskin's work with the title Fascinating Fractions , translated by V.I.Kisin, MIR Publishers, Moscow, 1986

 Preface 2.2.1 Euclid's Algorithm 4.1.2 Principle of Nested Segments V Approximation of Real Numbers I Two Historical Puzzles 2.2.2 Examples of applications of Euclid's algorithm 4.1.3 The Set of Rational Numbers 5.1 Approximation by Convergents 1.1 Archimedes' Puzzle 2.2.3 Summary 4.1.4 Existence of Non-rational Points on the Number Line 5.1.1 High-quality Approximation 1.1.1 Archimedes' Number III Convergents 4.1.5 Non-terminating Decimal Fractions 5.1.2. The Main Property of Convergents 1.1.2 Approximation 3.1 Concept of Convergents 4.1.6 Irrational Numbers 5.1.3. Convergents have the Highest Quality 1.1.3 Error of approximation 3.1.1 Preliminary Definition of Convergents 4.1.7 Real Numbers VI Solutions 1.1.4 Quality of approximation 3.1.2 How to generate convergents 4.1.8 Representation of Real Numbers on the Number Line 6.1 The Mystery of Archimedes' Number 1.2 The Puzzle of Pope Gregory XIII 3.1.3 Final Definition of Convergents 4.1.9 Condition of Rationality of Non-terminating Decimals 6.1.1 The Key to all Puzzles 1.2.1 The Mathematical Problem of the Calendar 3.1.4 Evaluation of Convergents 4.2 Non-terminating Continued Fractions 6.1.2. The Secret of Archimedes' Number 1.2.2 The Julian and Gregorian Calendars 3.1.5 Complete quotients 4.2.1 Numerical Value of a Non-terminating Continued Fraction 6.2 The Solution of the Calendar Problem II Formation of Continued Fractions 3.2 Properties of Convergents 4.2.2 Representation of Irrationals by Non-terminating Continued Fractions 6.2.1 The Use of Continued Fractions 2.1 Expansion of a Real Number into a Continued Fraction 3.2.1 Difference Between Two Neighbouring Convergents 4.2.3 The Single-valuedness of the Representation of a Real Number by a Continued Fraction 6.2.2 How to Choose a Calendar 2.1.1 Algorithm of Expansion into a Continued Fraction 3.2.2 Comparison of Neighbouring Convergents 4.3 The nature of Numbers Given by Continued Fractions 6.2.3 The Secret of Pope Gregory XIII 2.1.2 Notation for Continued Fractions 3.2.3 Irreducibility of convergent 4.3.1. Classification of Irrationals 2.1.3 Expansion of Negative Numbers into Continued Fractions IV Non-terminating Continued Fractions 4.3.2 Quadratic Irrationals 2.1.4 Examples of Non-terminating Expansion 4.1 Real Numbers 4.3.3. Euler's Theorem 2.2 Euclid's Algorithm 4.1.1 The Gulf Between the Finite and the Infinite 4.3.4. Lagrange's Theorem

INDEX