Book 3

2 Background 1.2 Cosmological 1.2.11 Mathematics

Book 5

Book 4 - 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



Two Historical Puzzles

1.1 Archimedes' Puzzle
1.1.1 Archimedes' Number
1.1.2 Approximation
1.1.3 Error of approximation
1.1.4 Quality of approximation
1.2 The Puzzle of Pope Gregory XIII
1.2.1 The Mathematical Problem of the Calendar
1.2.2 The Julian and Gregorian Calendars

Formation of Continued Fractions

2.1 Expansion of a Real Number into a Continued Fraction
2.1.1 Algorithm of Expansion into a Continued Fraction
2.1.2 Notation for Continued Fractions
2.1.3 Expansion of Negative Numbers into Continued Fractions
2.1.4 Examples of Non-terminating Expansion
2.2 Euclid's Algorithm
2.2.1 Euclid's Algorithm
2.2.2 Examples of applications of Euclid's algorithm
2.2.3 Summary


3.1 Concept of Convergents
3.1.1 Preliminary Definition of Convergents
3.1.2 How to generate convergents
3.1.3 Final Definition of Convergents
3.1.4 Evaluation of Convergents
3.1.5 Complete quotients
3.2 Properties of Convergents
3.2.1 Difference Between Two Neighbouring Convergents
3.2.2 Comparison of Neighbouring Convergents
3.2.3 Irreducibility of convergent

Non-terminating Continued Fractions

4.1 Real Numbers
4.1.1 The Gulf Between the Finite and the Infinite
4.1.2 Principle of Nested Segments
4.1.3 The Set of Rational Numbers
4.1.4 Existence of Non-rational Points on the Number Line
4.1.5 Non-terminating Decimal Fractions
4.1.6 Irrational Numbers
4.1.7 Real Numbers
4.1.8 Representation of Real Numbers on the Number Line
4.1.9 Condition of Rationality of Non-terminating Decimals
4.2 Non-terminating Continued Fractions
4.2.1 Numerical Value of a Non-terminating Continued Fraction
4.2.2 Representation of Irrationals by Non-terminating Continued Fractions
4.2.3 The Single-valuedness of the Representation
 of a Real Number by a Continued Fraction
4.3 The nature of Numbers Given by Continued Fractions
4.3.1 Classification of Irrationals
4.3.2 Quadratic Irrationals
4.3.3 Euler's Theorem
4.3.4 Lagrange's Theorem

Approximation of Real Numbers

5.1 Approximation by Convergents
5.1.1 High-quality Approximation
5.1.2 The Main Property of Convergents
5.1.3 Convergents have the Highest Quality


6.1 The Mystery of Archimedes' Number
6.1.1 The Key to all Puzzles
6.1.2 The Secret of Archimedes' Number
6.2 The Solution of the Calendar Problem
6.2.1 The Use of Continued Fractions
6.2.2 How to Choose a Calendar
6.2.3 The Secret of Pope Gregory XIII