## MATH ODYSSEY 2000

By Clement Falbo

Stipe Publishing Co. / 10-12 Chester Street

Champlain Illinois 61820

```  Preface                                            P-1

INTRODUCTION:MATH ODYSSEY 2000--The Metaphor        I-1

I .  PUZZLES, PARADOXES, AND LOGIC

1.1 Puzzle Problems                                               1
1.3 Aristotelian Logic                                            7
1.4 Rational and Irrational Numbers                              12
1.5 Two Methods of Proof                                         18

I I. AXIOMATIC SYSTEMS

2.1 Euclid and the Axiomatic Method in Geometry                  23
2.2 Non-Euclidean Geometries                                     29
2.3 A Small Algebraic System --Introduction to Groups            32

I I I. SOME MEN AND WOMEN OF MATHEMATICS

3.1  Srinivasa Ramanujan (1887-1920)                             39
3.2  Sonya Kovalevsky (1850-1891)                                42
3.3  David Hilbert(1862-1943)                                    44
3.4  Maria Gaetana Agnesi(1718-1799)                             47
3.5  Carl Fredrich Gauss(1777-1855)                              50
3.6  Ada Byron Lovelace (1815-1852)                              52
3.7  Evariste Galois (1811-1832)                                 56
3.8  Sophie Germain (1776-1831)                                  59
3.9  Leonhard Euler (1707-1783)                                  61
3.10  Emmy Noether (1882-1935)                                   65

I V.  INFINITY OF THE COUNTING NUMBERS

4.1 Archimedes and Finite Number Sets                             68
4.2 Introduction to Infinity                                      72
4.3 Prime Numbers                                                 75
4.4 The Goldbach Conjecture                                       79
4.5 Countable Infinity and the Efficient
Rearrangement of the Counting Numbers                         82
4.6 The Rational Numbers are Countable                            85

V. HIGHER INFINITIES

5.1 Adding up an Infinity of Numbers                              88
5.2 The Length vs The Cardinality of a Set                        94
5.3 Sets of "measure zero"                                       102
5.4 The irrational numbers are uncountable                       106
5.5 The Continuum Hypothesis -- The Power Set                    109
5.6 Continuum Hypothesis--Independence Proved by Godel & Cohen 113

V I. CRISES & BREAKTHROUGHS

6.1 Imaginary Numbers -- Discoveries and Disbelief              115
6.2 Complex Numbers                                             119
6.3 Four-color Problem & Computer proofs                        126
6.4 Fibonacci Sequence                                          131
6.5 The Golden Ratio                                            135
6.6 The Pythagorean Theorem and Fermat's Last Theorem           137

V I I.  UNCERTAINTY

7.1 Foundation of Arithmetic. Frege and Russell's paradox       143
7.2 Formal Axiomatic Systems                                    146
7.3 Godel's Proof                                               152

Appendix 1  Teaching & Learning                           157
Appendix 2   Hints for Solving  Problems                 164
Appendix 3   Answers to Selected Problems                186
Appendix 4   Review of Algebra                             208
Index                                                          216
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