Episodic History Of Mathematics Pdf

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An Episodic History of Mathematics Mathematical Culture through Problem Solving by Steven G. Krantz September 23, 2006 To Marvin J. Greenberg, an inspiring teacher. iii Preface Together with philosophy, mathematics is the oldest academic discipline known to mankind. Today mathematics is a huge and complex enterprise, far beyond the ken of any one individual. Those of us who choose to study the subject can only choose a piece of it, and in the end must specialize rather drastically in order to make any contribution to the evolution of ideas. An important development of twenty-first century life is that mathematical and analytical thinking have permeated all aspects of our world. We all need to understand the spread of diseases, the likelihood that we will contract SARS or hepatitis. We all must deal with financial matters. Finally, we all must deal with computers and databases and the Internet. Mathematics is an integral part of the theory and the operating systems that make all these computer systems work. Theoretical mathematics is used to design automobile bodies, to plan reconstructive surgery procedures, and to analyze prison riots. The modern citizen who is unaware of mathematical thought is lacking a large part of the equipment of life. Thus it is worthwhile to have a book that will introduce the student to some of the genesis of mathematical ideas. While we cannot get into the nuts and bolts of Andrew Wiles's solution of Fermat's Last Theorem, we can instead describe some of the stream of thought that created the problem and led to its solution. While we cannot describe all the sophisticated mathematics that goes into the theory behind black holes and modern cosmology, we can instead indicate some of Bernhard Riemann's ideas about the geometry of space. While we cannot describe in specific detail the mathematical research that professors at the University of Paris are performing today, we can instead indicate the development of ideas that has led to that work. Certainly the modern school teacher, who above all else serves as a role model for his/her students, must be conversant with mathematical thought. As a matter of course, the teacher will use mathematical examples and make mathematical allusions just as examples of reasoning. Certainly the grade school teacher will seek a book that is broadly accessible, and that speaks to the level and interests of K-6 students. A book with this audience in mind should serve a good purpose. iv Mathematical history is exciting and rewarding, and it is a significant slice of the intellectual pie. A good education consists of learning different methods of discourse, and certainly mathematics is one of the most well-developed and important modes of discourse that we have. The purpose of this book, then, is to acquaint the student with mathematical language and mathematical life by means of a number of historically important mathematical vignettes. And, as has already been noted, the book will also serve to help the prospective school teacher to become inured in some of the important ideas of mathematics—both classical and modern. The focus in this text is on doing—getting involved with the mathematics and solving problems. This book is unabashedly mathematical: The history is primarily a device for feeding the reader some doses of mathematical meat. In the course of reading this book, the neophyte will become involved with mathematics by working on the same problems that Zeno and Pythagoras and Descartes and Fermat and Riemann worked on. This is a book to be read with pencil and paper in hand, and a calculator or computer close by. The student will want to experiment, to try things, to become a part of the mathematical process. This history is also an opportunity to have some fun. Most of the mathematicians treated here were complex individuals who led colorful lives. They are interesting to us as people as well as scientists. There are wonderful stories and anecdotes to relate about Pythagoras and Galois and Cantor and Poincaré, and we do not hesitate to indulge ourselves in a little whimsy and gossip. This device helps to bring the subject to life, and will retain reader interest. It should be clearly understood that this is in no sense a thoroughgoing history of mathematics, in the sense of the wonderful treatises of Boyer/Merzbach [BOM] or Katz [KAT] or Smith [SMI]. It is instead a collection of snapshots of aspects of the world of mathematics, together with some cultural information to put the mathematics into perspective. The reader will pick up history on the fly, while actually doing mathematics— developing mathematical ideas, working out problems, formulating questions. And we are not shy about the things we ask the reader to do. This book will be accessible to students with a wide variety of backgrounds v and interests. But it will give the student some exposure to calculus, to number theory, to mathematical induction, cardinal numbers, cartesian geometry, transcendental numbers, complex numbers, Riemannian geometry, and several other exciting parts of the mathematical enterprise. Because it is our intention to introduce the student to what mathematicians think and what mathematicians value, we actually prove a number of important facts: (i) the existence of irrational numbers, (ii) the existence of transcendental numbers, (iii) Fermat's little theorem, (iv) the completeness of the real number system, (v) the fundamental theorem of algebra, and (vi) Dirichlet's theorem. The reader of this text will come away with a hands-on feeling for what mathematics is about and what mathematicians do. This book is intended to be pithy and brisk. Chapters are short, and it will be easy for the student to browse around the book and select topics of interest to dip into. Each chapter will have an exercise set, and the text itself will be peppered with items labeled "For You to Try". This device gives the student the opportunity to test his/her understanding of a new idea at the moment of impact. It will be both rewarding and reassuring. And it should keep interest piqued. In fact the problems in the exercise sets are of two kinds. Many of them are for the individual student to work out on his/her own. But many are labeled for class discussion. They will make excellent group projects or, as appropriate, term papers. It is a pleasure to thank my editor, Richard Bonacci, for enlisting me to write this book and for providing decisive advice and encouragement along the way. Certainly the reviewers that he engaged in the writing process provided copious and detailed advice that have turned this into a more accurate and useful teaching tool. I am grateful to all. The instructor teaching from this book will find grist for a number of interesting mathematical projects. Term papers, and even honors projects, will be a natural outgrowth of this text. The book can be used for a course in mathematical culture (for non-majors), for a course in the history of mathematics, for a course of mathematics for teacher preparation, or for a course in problem-solving. We hope that it will help to bridge the huge and demoralizing gap between the technical world and the humanistic world. For certainly the most important thing that we do in our society is to communicate. My wish is to communicate mathematics. SGK St. Louis, MO Table of Contents Preface 1 The Ancient Greeks 1.1 Pythagoras . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Introduction to Pythagorean Ideas . . . . . . . 1.1.2 Pythagorean Triples . . . . . . . . . . . . . . . 1.2 Euclid . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Introduction to Euclid . . . . . . . . . . . . . . 1.2.2 The Ideas of Euclid . . . . . . . . . . . . . . . . 1.3 Archimedes . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 The Genius of Archimedes . . . . . . . . . . . . 1.3.2 Archimedes's Calculation of the Area of a Circle . . . . . . . . . 1 1 1 7 10 10 14 21 21 24 . . . . . . . . 43 43 44 51 56 57 59 63 64 3 The Mystical Mathematics of Hypatia 3.1 Introduction to Hypatia . . . . . . . . . . . . . . . . . . 3.2 What is a Conic Section? . . . . . . . . . . . . . . . . . . 69 69 78 2 Zeno's Paradox and the Concept 2.1 The Context of the Paradox? . 2.2 The Life of Zeno of Elea . . . . 2.3 Consideration of the Paradoxes 2.4 Decimal Notation and Limits . 2.5 Infinite Sums and Limits . . . . 2.6 Finite Geometric Series . . . . . 2.7 Some Useful Notation . . . . . . 2.8 Concluding Remarks . . . . . . of Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii . . . . . . . . . . . . . . . . viii 4 The Arabs and the Development of Algebra 4.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . 4.2 The Development of Algebra . . . . . . . . . . . . . . . . 4.2.1 Al-Khowârizmı̂ and the Basics of Algebra . . . . . 4.2.2 The Life of Al-Khwarizmi . . . . . . . . . . . . . 4.2.3 The Ideas of Al-Khwarizmi . . . . . . . . . . . . . 4.2.4 Omar Khayyam and the Resolution of the Cubic . 4.3 The Geometry of the Arabs . . . . . . . . . . . . . . . . 4.3.1 The Generalized Pythagorean Theorem . . . . . . 4.3.2 Inscribing a Square in an Isosceles Triangle . . . . 4.4 A Little Arab Number Theory . . . . . . . . . . . . . . . 93 93 94 94 95 100 105 108 108 112 114 5 Cardano, Abel, Galois, and the Solving of Equations 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Story of Cardano . . . . . . . . . . . . . . . . . . 5.3 First-Order Equations . . . . . . . . . . . . . . . . . . 5.4 Rudiments of Second-Order Equations . . . . . . . . . 5.5 Completing the Square . . . . . . . . . . . . . . . . . . 5.6 The Solution of a Quadratic Equation . . . . . . . . . . 5.7 The Cubic Equation . . . . . . . . . . . . . . . . . . . 5.7.1 A Particular Equation . . . . . . . . . . . . . . 5.7.2 The General Case . . . . . . . . . . . . . . . . . 5.8 Fourth Degree Equations and Beyond . . . . . . . . . . 5.8.1 The Brief and Tragic Lives of Abel and Galois . 5.9 The Work of Abel and Galois in Context . . . . . . . . . . . . . . . . . . . . 123 123 124 129 130 131 133 136 137 139 140 141 148 . . . . . . 151 151 152 156 158 165 169 6 René Descartes and the Idea of Coordinates 6.0 Introductory Remarks . . . . . . . . . . . . . . 6.1 The Life of René Descartes . . . . . . . . . . . . 6.2 The Real Number Line . . . . . . . . . . . . . . 6.3 The Cartesian Plane . . . . . . . . . . . . . . . 6.4 Cartesian Coordinates and Euclidean Geometry 6.5 Coordinates in Three-Dimensional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The Invention of Differential Calculus 177 7.1 The Life of Fermat . . . . . . . . . . . . . . . . . . . . . 177 7.2 Fermat's Method . . . . . . . . . . . . . . . . . . . . . . 180 ix 7.3 7.4 More Advanced Ideas of Calculus: The Derivative and the Tangent Line . . . . . . . . . . . . . . . . . . . . . . . . 183 Fermat's Lemma and Maximum/Minimum Problems . . 191 8 Complex Numbers and Polynomials 8.1 A New Number System . . . . . . . . . . . . 8.2 Progenitors of the Complex Number System 8.2.1 Cardano . . . . . . . . . . . . . . . . 8.2.2 Euler . . . . . . . . . . . . . . . . . . 8.2.3 Argand . . . . . . . . . . . . . . . . 8.2.4 Cauchy . . . . . . . . . . . . . . . . 8.2.5 Riemann . . . . . . . . . . . . . . . . 8.3 Complex Number Basics . . . . . . . . . . . 8.4 The Fundamental Theorem of Algebra . . . 8.5 Finding the Roots of a Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 205 205 206 206 210 212 212 213 219 226 9 Sophie Germain and Fermat's Last Problem 231 9.1 Birth of an Inspired and Unlikely Child . . . . . . . . . . 231 9.2 Sophie Germain's Work on Fermat's Problem . . . . . . 239 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 11.1 11.2 11.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 249 254 255 260 261 262 267 Prime Numbers 275 The Sieve of Eratosthenes . . . . . . . . . . . . . . . . . 275 The Infinitude of the Primes . . . . . . . . . . . . . . . . 278 More Prime Thoughts . . . . . . . . . . . . . . . . . . . 279 12 Dirichlet and How to Count 289 12.1 The Life of Dirichlet . . . . . . . . . . . . . . . . . . . . 289 12.2 The Pigeonhole Principle . . . . . . . . . . . . . . . . . . 292 x 12.3 Other Types of Counting . . . . . . . . . . . . . . . . . . 296 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 305 309 312 316 14 Georg Cantor and the Orders of Infinity 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 323 327 332 334 337 . . . . . . . . . . 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 Definition of the Complex Numbers . . . . . . . . 343 345 345 345 346 347 347 348 348 349 349 349 350 350 351 351 351 352 354 354 354

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