TABLE OF CONTENTS

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 1.2 Pooled Data Paradoxes 4 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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