AN EPISODIC HISTORY OF MATHEMATICS
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 10 1 INTRODUCTION 10 2 WHY DO WE NEED THE REAL NUMBERS? 10 3 HOW TO CONSTRUCT THE REAL NUMBERS 10 4 PROPERTIES OF THE REAL NUMBER SYSTEM 10 4 1 BOUNDED SEQUENCES 10 4 2 MAXIMA AND MINIMA 10 4 3 THE INTERMEDIATE VALUE PROPERTY 11 THE PRIME NUMBERS 11 1 THE SIEVE OF ERATOSTHENES 11 2 THE IN?NITUDE OF THE PRIMES 11 3 MORE PRIME THOUGHTS 12 DIRICHLET AND HOW TO COUNT 12 1 THE LIFE OF DIRICHLET 12 2 THE PIGEONHOLE PRINCIPLE 12 3 OTHER TYPES OF COUNTING 13 RIEMANN AND THE GEOMETRY OF SURFACES 13 0 INTRODUCTION 13 1 HOW TO MEASURE THE LENGTH OF A CURVE 13 2 RIEMANN’S METHOD FOR MEASURING ARC LENGTH 13 3 THE HYPERBOLIC DISC 14 GEORG CANTOR AND THE ORDERS OF IN?NITY 14 1 INTRODUCTORY REMARKS 14 2 WHAT IS A NUMBER? 14 2 1 AN UNCOUNTABLE SET 14 2 2 COUNTABLE AND UNCOUNTABLE 14 3 THE EXISTENCE OF TRANSCENDENTAL NUMBERS 15 THE NUMBER SYSTEMS 15 1 THE NATURAL NUMBERS 15 1 1 INTRODUCTORY REMARKS 15 1 2 CONSTRUCTION OF THE NATURAL NUMBERS 15 1 3 AXIOMATIC TREATMENT OF THE NATURAL NUMBERS 15 2 THE INTEGERS 15 2 1 LACK OF CLOSURE IN THE NATURAL NUMBERS 15 2 2 THE INTEGERS AS A SET OF EQUIVALENCE CLASSES 15 2 3 EXAMPLES OF INTEGER ARITHMETIC 15 2 4 ARITHMETIC PROPERTIES OF THE INTEGERS 15 3 THE RATIONAL NUMBERS 15 3 1 LACK OF CLOSURE IN THE INTEGERS 15 3 2 THE RATIONAL NUMBERS AS A SET OF EQUIVALENCE CLASSES 15 3 3 EXAMPLES OF RATIONAL ARITHMETIC 15 3 4 SUBTRACTION AND DIVISION OF RATIONAL NUMBERS 15 4 THE REAL NUMBERS 15 4 1 LACK OF CLOSURE IN THE RATIONAL NUMBERS 15 4 2 AXIOMATIC TREATMENT OF THE REAL NUMBERS 15 5 THE COMPLEX NUMBERS 15 5 1 INTUITIVE VIEW OF THE COMPLEX NUMBERS 15 5 2 DE?NITION OF THE COMPLEX NUMBERS 15 5 3 THE DISTINGUISHED COMPLEX NUMBERS 1 AND I 15 5 4 ALGEBRAIC CLOSURE OF THE COMPLEX NUMBERS 16 HENRI POINCAR´E, CHILD PRODIGY 16 1 INTRODUCTORY REMARKS 16 2 RUBBER SHEET GEOMETRY 16 3 THE IDEA OF HOMOTOPY 16 4 THE BROUWER FIXED POINT THEOREM 16 5 THE GENERALIZED HAM SANDWICH THEOREM 16 5 1 CLASSICAL HAM SANDWICHES 16 5 2 GENERALIZED HAM SANDWICHES 17 SONYA KOVALEVSKAYA AND MECHANICS 17 1 THE LIFE OF SONYA KOVALEVSKAYA 17 2 THE SCIENTI?C WORK OF SONYA KOVALEVSKAYA 17 2 1 PARTIAL DI?ERENTIAL EQUATIONS 17 2 2 A FEW WORDS ABOUT POWER SERIES 17 2 3 THE MECHANICS OF A SPINNING GYROSCOPE AND THE IN?UENCE OF GRAVITY 17 2 4 THE RINGS OF SATURN 17 2 5 THE LAM´E EQUATIONS 17 2 6 BRUNS’S THEOREM 17 3 AFTERWARD ON SONYA KOVALEVSKAYA 18 EMMY NOETHER AND ALGEBRA 18 1 THE LIFE OF EMMY NOETHER 18 2 EMMY NOETHER AND ABSTRACT ALGEBRA: GROUPS 18 3 EMMY NOETHER AND ABSTRACT ALGEBRA: RINGS 18 3 1 THE IDEA OF AN IDEAL 19 METHODS OF PROOF 19 1 AXIOMATICS 19 1 1 UNDE?NABLES 19 1 2 DE?NITIONS 19 1 3 AXIOMS 19 1 4 THEOREMS, MODUSPONENDOPONENS, AND MODUSTOL- LENS 19 2 PROOF BY INDUCTION 19 2 1 MATHEMATICAL INDUCTION 19 2 2 EXAMPLES OF INDUCTIVE PROOF 19 3 PROOF BY CONTRADICTION 19 3 1 EXAMPLES OF PROOF BY CONTRADICTION 19 4 DIRECT PROOF 19 4 1 EXAMPLES OF DIRECT PROOF 19 5 OTHER METHODS OF PROOF 19 5 1 EXAMPLES OF COUNTING ARGUMENTS 20 ALAN TURING AND CRYPTOGRAPHY 20 0 BACKGROUND ON ALAN TURING 20 1 THE TURING MACHINE 20 1 1 AN EXAMPLE OF A TURING MACHINE 20 2 MORE ON THE LIFE OF ALAN TURING 20 3 WHAT IS CRYPTOGRAPHY? 20 4 ENCRYPTION BY WAY OF A?NE TRANSFORMATIONS 20 4 1 DIVISION IN MODULAR ARITHMETIC 20 4 2 INSTANCES OF THE A?NE TRANSFORMATION ENCRYPTION 20 5 DIGRAPH TRANSFORMATIONS CHAPTER INTRODUCTION TO PYTHAGOREAN IDEAS PYTHAGORAS (569–500 B C E ) WAS BOTH A PERSON AND A SOCIETY (I E , THE PYTHAGOREANS) HE WAS ALSO A POLITICAL ?GURE AND A MYSTIC HE WAS SPECIAL IN HIS TIME BECAUSE, AMONG OTHER REASONS, HE INVOLVED WOMEN AS EQUALS IN HIS ACTIVITIES ONE CRITIC CHARACTERIZED THE MAN AS “ONE TENTH OF HIM GENIUS, NINE-TENTHS SHEER FUDGE ” PYTHAGORAS DIED, ACCORDING TO LEGEND, IN THE ?AMES OF HIS OWN SCHOOL ?RED BY POLITICAL AND RELIGIOUS BIGOTS WHO STIRRED UP THE MASSES TO PROTEST AGAINST THE ENLIGHTENMENT WHICH PYTHAGORAS SOUGHT TO BRING THEM AS WITH MANY ?GURES FROM ANCIENT TIMES, THERE IS LITTLE SPECI?C THAT WE KNOW ABOUT PYTHAGORAS’S LIFE WE KNOW A LITTLE ABOUT HIS IDEAS AND HIS SCHOOL, AND WE SKETCH SOME OF THESE HERE THE PYTHAGOREAN SOCIETY WAS INTENSELY MATHEMATICAL IN NATURE, BUT IT WAS ALSO QUASI-RELIGIOUS AMONG ITS TENETS (ACCORDING TO [RUS]) WERE: TO ABSTAIN FROM BEANS NOT TO PICK UP WHAT HAS FALLEN NOT TO TOUCH A WHITE COCK NOT TO BREAK BREAD NOT TO STEP OVER A CROSSBAR