2.1.11.3 Book 3 |
Book 5 2.1.11.5 |
Based on N.M. Beskin's work with the title Fascinating Fractions , translated by V.I.Kisin, MIR Publishers, Moscow, 1986
1.0 |
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 |
2.0 |
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 |
3.0 |
Convergents |
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 |
4.0 |
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 |
5.0 |
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 |
6.0 |
Solutions |
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 |