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 Nonrational Points
on the Number Line 

5.1.1 Highquality Approximation 
1.1.1 Archimedes'
Number 

III Convergents 

4.1.5 Nonterminating 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 Nonterminating 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 Nonterminating 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
Nonterminating 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 Nonterminating 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
Singlevaluedness 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 Nonterminating
Continued Fractions 

4.3.2
Quadratic Irrationals 


2.1.4 Examples of Nonterminating
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 

