HISTORY OF MATHEMATICS: SPECIAL TOPICS OF ELEMENTARY MATHEMATICS

By V. MIEND

TABLE OF CONTENTS THE ANCIENT GREEKS PYTHAGORAS INTRODUCTION TO PYTHAGOREAN IDEAS PYTHAGOREAN TRIPLES EUCLID INTRODUCTION TO EUCLID THE IDEAS OF EUCLID ARCHIMEDES THE GENIUS OF ARCHIMEDES ARCHIMEDES’S CALCULATION OF THE AREA OF A CIRCLE ZENO’S PARADOX AND THE CONCEPT OF LIMIT THE CONTEXT OF THE PARADOX? THE LIFE OF ZENO OF ELEA CONSIDERATION OF THE PARADOXES DECIMAL NOTATION AND LIMITS IN?NITE SUMS AND LIMITS FINITE GEOMETRIC SERIES SOME USEFUL NOTATION CONCLUDING REMARKS THE MYSTICAL MATHEMATICS OF HYPATIA INTRODUCTION TO HYPATIA WHAT IS A CONIC SECTION? THE ARABS AND THE DEVELOPMENT OF ALGEBRA INTRODUCTORY REMARKS THE DEVELOPMENT OF ALGEBRA AL-KHOWˆARIZMˆI AND THE BASICS OF ALGEBRA THE LIFE OF AL-KHWARIZMI THE IDEAS OF AL-KHWARIZMI OMAR KHAYYAM AND THE RESOLUTION OF THE CUBIC THE GEOMETRY OF THE ARABS THE GENERALIZED PYTHAGOREAN THEOREM INSCRIBING A SQUARE IN AN ISOSCELES TRIANGLE A LITTLE ARAB NUMBER THEORY CARDANO, ABEL, GALOIS, AND THE SOLVING OF EQUATIONS INTRODUCTION THE STORY OF CARDANO FIRST-ORDER EQUATIONS RUDIMENTS OF SECOND-ORDER EQUATIONS COMPLETING THE SQUARE THE SOLUTION OF A QUADRATIC EQUATION THE CUBIC EQUATION A PARTICULAR EQUATION THE GENERAL CASE FOURTH DEGREE EQUATIONS AND BEYOND THE BRIEF AND TRAGIC LIVES OF ABEL AND GALOIS THE WORK OF ABEL AND GALOIS IN CONTEXT REN´E DESCARTES AND THE IDEA OF COORDINATES THE LIFE OF REN´E DESCARTES THE REAL NUMBER LINE THE CARTESIAN PLANE CARTESIAN COORDINATES AND EUCLIDEAN GEOMETRY COORDINATES IN THREE-DIMENSIONAL SPACE THE INVENTION OF DI?ERENTIAL CALCULUS THE LIFE OF FERMAT FERMAT’S METHOD MORE ADVANCED IDEAS OF CALCULUS: THE DERIVATIVE AND THE TANGENT LINE FERMAT’S LEMMA AND MAXIMUM/MINIMUM PROBLEMS COMPLEX NUMBERS AND POLYNOMIALS A NEW NUMBER SYSTEM PROGENITORS OF THE COMPLEX NUMBER SYSTEM CARDANO EULER ARGAND CAUCHY RIEMANN COMPLEX NUMBER BASICS THE FUNDAMENTAL THEOREM OF ALGEBRA FINDING THE ROOTS OF A POLYNOMIAL SOPHIE GERMAIN AND FERMAT’S LAST PROBLEM BIRTH OF AN INSPIRED AND UNLIKELY CHILD SOPHIE GERMAIN’S WORK ON FERMAT’S PROBLEM 10 CAUCHY AND THE FOUNDATIONS OF ANALYSIS 101 INTRODUCTION 102 WHY DO WE NEED THE REAL NUMBERS? 103 HOW TO CONSTRUCT THE REAL NUMBERS 104 PROPERTIES OF THE REAL NUMBER SYSTEM 1041 BOUNDED SEQUENCES 1042 MAXIMA AND MINIMA 1043 THE INTERMEDIATE VALUE PROPERTY 11 THE PRIME NUMBERS 111 THE SIEVE OF ERATOSTHENES 112 THE IN?NITUDE OF THE PRIMES 113 MORE PRIME THOUGHTS 12 DIRICHLET AND HOW TO COUNT 121 THE LIFE OF DIRICHLET 122 THE PIGEONHOLE PRINCIPLE 123 OTHER TYPES OF COUNTING 13 RIEMANN AND THE GEOMETRY OF SURFACES 130 INTRODUCTION 131 HOW TO MEASURE THE LENGTH OF A CURVE 132 RIEMANN’S METHOD FOR MEASURING ARC LENGTH 133 THE HYPERBOLIC DISC 14 GEORG CANTOR AND THE ORDERS OF IN?NITY 141 INTRODUCTORY REMARKS 142 WHAT IS A NUMBER? 1421 AN UNCOUNTABLE SET 1422 COUNTABLE AND UNCOUNTABLE 143 THE EXISTENCE OF TRANSCENDENTAL NUMBERS 15 THE NUMBER SYSTEMS 151 THE NATURAL NUMBERS 1511 INTRODUCTORY REMARKS 1512 CONSTRUCTION OF THE NATURAL NUMBERS 1513 AXIOMATIC TREATMENT OF THE NATURAL NUMBERS 152 THE INTEGERS 1521 LACK OF CLOSURE IN THE NATURAL NUMBERS 1522 THE INTEGERS AS A SET OF EQUIVALENCE CLASSES 1523 EXAMPLES OF INTEGER ARITHMETIC 1524 ARITHMETIC PROPERTIES OF THE INTEGERS 153 THE RATIONAL NUMBERS 1531 LACK OF CLOSURE IN THE INTEGERS 1532 THE RATIONAL NUMBERS AS A SET OF EQUIVALENCE CLASSES 1533 EXAMPLES OF RATIONAL ARITHMETIC 1534 SUBTRACTION AND DIVISION OF RATIONAL NUMBERS 154 THE REAL NUMBERS 1541 LACK OF CLOSURE IN THE RATIONAL NUMBERS 1542 AXIOMATIC TREATMENT OF THE REAL NUMBERS 155 THE COMPLEX NUMBERS 1551 INTUITIVE VIEW OF THE COMPLEX NUMBERS 1552 DE?NITION OF THE COMPLEX NUMBERS 1553 THE DISTINGUISHED COMPLEX NUMBERS 1 AND I 1554 ALGEBRAIC CLOSURE OF THE COMPLEX NUMBERS 16 HENRI POINCAR´E, CHILD PRODIGY 161 INTRODUCTORY REMARKS 162 RUBBER SHEET GEOMETRY 163 THE IDEA OF HOMOTOPY 164 THE BROUWER FIXED POINT THEOREM 165 THE GENERALIZED HAM SANDWICH THEOREM 1651 CLASSICAL HAM SANDWICHES 1652 GENERALIZED HAM SANDWICHES 17 SONYA KOVALEVSKAYA AND MECHANICS 171 THE LIFE OF SONYA KOVALEVSKAYA 172 THE SCIENTI?C WORK OF SONYA KOVALEVSKAYA 1721 PARTIAL DI?ERENTIAL EQUATIONS 1722 A FEW WORDS ABOUT POWER SERIES 1723 THE MECHANICS OF A SPINNING GYROSCOPE AND THE IN?UENCE OF GRAVITY 1724 THE RINGS OF SATURN 1725 THE LAM´E EQUATIONS 1726 BRUNS’S THEOREM 173 AFTERWARD ON SONYA KOVALEVSKAYA 18 EMMY NOETHER AND ALGEBRA 181 THE LIFE OF EMMY NOETHER 182 EMMY NOETHER AND ABSTRACT ALGEBRA: GROUPS 183 EMMY NOETHER AND ABSTRACT ALGEBRA: RINGS 1831 THE IDEA OF AN IDEAL 19 METHODS OF PROOF 191 AXIOMATICS 1911 UNDE?NABLES 1912 DE?NITIONS 1913 AXIOMS 1914 THEOREMS, MODUSPONENDOPONENS, AND MODUSTOL- LENS 192 PROOF BY INDUCTION 1921 MATHEMATICAL INDUCTION 1922 EXAMPLES OF INDUCTIVE PROOF 193 PROOF BY CONTRADICTION 1931 EXAMPLES OF PROOF BY CONTRADICTION 194 DIRECT PROOF 1941 EXAMPLES OF DIRECT PROOF 195 OTHER METHODS OF PROOF 1951 EXAMPLES OF COUNTING ARGUMENTS 20 ALAN TURING AND CRYPTOGRAPHY 200 BACKGROUND ON ALAN TURING 201 THE TURING MACHINE 2011 AN EXAMPLE OF A TURING MACHINE 202 MORE ON THE LIFE OF ALAN TURING 203 WHAT IS CRYPTOGRAPHY? 204 ENCRYPTION BY WAY OF A?NE TRANSFORMATIONS 2041 DIVISION IN MODULAR ARITHMETIC 2042 INSTANCES OF THE A?NE TRANSFORMATION ENCRYPTION 205 DIGRAPH TRANSFORMATIONS