In Memoriam WILLIAM HENRY LAWSON 24th April 1915, Ypres CHARLES PEIRCE ECCLESHALL 6th June 1915, Gallipoli Contents Preface xi Prolegomena xv SECTION I IDEAS CHAPTER 1 The nature of mathematics 3 1.1 MATHEMATICS IN HISTORY 3 1.2 MATHEMATICS TODAY 6 1.3 THE SCOPE OF MATHEMATICS 7 1.4 WHAT THEY (PROBABLY) DIDN’T TELL YOU IN SCHOOL 8 1.5 FURTHER READING 9 CHAPTER 2 Proofs 11 2.1 MATHEMATICAL TRUTH 11 2.2 FUNDAMENTAL ASSUMPTIONS OF LOGIC 12 2.3 FIVE EASY PROOFS 12 2.4 AXIOMS 24 2.5 UN PETIT PEU DE PHILOSOPHIE 26 2.6 MATHEMATICAL CREATIVITY 27 2.7 PROVING SOMETHING FALSE 28 2.8 TERMINOLOGY 28 2.9 ADVICE ON PROOFS 29 CHAPTER 3 Foundations 31 3.1 SETS 31 vii viii Contents 3.2 BOOLEAN OPERATIONS 40 3.3 RELATIONS 45 3.4 FUNCTIONS 49 3.5 EQUIVALENCE RELATIONS 61 3.6 ORDER RELATIONS 67 3.7 QUANTIFIERS 69 3.8 PROOF BY INDUCTION 71 3.9 COUNTING 74 3.10 INFINITE NUMBERS 81 CHAPTER 4 Algebra redux 89 4.1 RULES OF THE GAME 90 4.2 ALGEBRAIC AXIOMS FOR REAL NUMBERS 98 4.3 SOLVING QUADRATIC EQUATIONS 104 4.4 BINOMIAL THEOREM 108 4.5 BOOLEAN ALGEBRAS 111 4.6 CHARACTERIZING REAL NUMBERS 118 SECTION II THEORIES CHAPTER 5 Number theory 125 5.1 REMAINDER THEOREM 125 5.2 GREATEST COMMON DIVISORS 132 5.3 FUNDAMENTAL THEOREM OF ARITHMETIC 140 5.4 MODULAR ARITHMETIC 148 5.5 CONTINUED FRACTIONS 155 CHAPTER 6 Complex numbers 165 6.1 COMPLEX NUMBER ARITHMETIC 165 6.2 COMPLEX NUMBER GEOMETRY 172 6.3 EULER’S FORMULA 176 6.4 MAKING SENSE OF COMPLEX NUMBERS 177 CHAPTER 7 Polynomials 179 Contents ix 7.1 TERMINOLOGY 179 7.2 REMAINDER THEOREM 180 7.3 ROOTS OF POLYNOMIALS 183 7.4 FUNDAMENTAL THEOREM OF ALGEBRA 183 7.5 ARBITRARY ROOTS OF COMPLEX NUMBERS 188 7.6 GREATEST COMMON DIVISORS OF POLYNOMIALS 193 7.7 IRREDUCIBLE POLYNOMIALS 194 7.8 PARTIAL FRACTIONS 196 7.9 RADICAL SOLUTIONS 204 7.10 ALGEBRAIC AND TRANSCENDENTAL NUMBERS 214 7.11 MODULAR ARITHMETIC WITH POLYNOMIALS 215 CHAPTER 8 Matrices 219 8.1 MATRIX ARITHMETIC 219 8.2 MATRIX ALGEBRA 231 8.3 SOLVING SYSTEMS OF LINEAR EQUATIONS 238 8.4 DETERMINANTS 247 8.5 INVERTIBLE MATRICES 253 8.6 DIAGONALIZATION 267 8.7 BLANKINSHIP’S ALGORITHM 279 CHAPTER 9 Vectors 283 9.1 VECTORS GEOMETRICALLY 283 9.2 VECTORS ALGEBRAICALLY 296 9.3 GEOMETRIC MEANING OF DETERMINANTS 299 9.4 GEOMETRY WITH VECTORS 301 9.5 LINEAR FUNCTIONS 310 9.6 ALGEBRAIC MEANING OF DETERMINANTS 315 9.7 QUATERNIONS 318 CHAPTER 10 The principal axes theorem 323 10.1 ORTHOGONAL MATRICES 323 10.2 ORTHOGONAL DIAGONALIZATION 331 10.3 CONICS AND QUADRICS 336 x Contents Epilegomena 343 Bibliography 347 Index 355 Preface The aim of this book is to providea bridge between school and university mathemat- ics centred on algebra and geometry. Apart from pro forma proofs by induction at school, mathematics students usually meet the concept of proof for the first time at university. Thus, an important part of this book is an introduction to proof. My own experience is that aside from a few basic ideas, proofs are best learnt by doing and this is the approach I have adopted here. In addition, I have also tried to counter the view of mathematics as nothing more than a collection of methods by emphasizing ideas and their historical origins throughout.Context is important and leads to greater understanding. Mathematics does not divide into watertight compartments. A book on algebra and geometry must therefore also make connections with applications and other parts of mathematics. I have used the examples to introduce applications of algebra to topics such as cryptography and error-correcting codes and to illus- trate connections with calculus. In addition, scattered throughout the book, you will find boxes in smaller type which can be read or omitted according to taste. Some of the boxes describe more complex proofs or results, but many are asides on more advanced material. You do not need to read any of the boxes to understand the book. The book is organized around three topics: linear equations, polynomial equa- tions and quadratic forms. This choice was informed by consulting a range of older textbooks,in particular [26, 27, 56, 122, 71, 84, 15, 96, 126, 4], as well as some more modern ones [7, 25, 33, 40, 44, 47, 53, 94, 110, 116], and augmented by a survey of the first-year mathematics modules on offer in a number of British and Irish uni- versities. The older textbooks have been a revelation. For example, Chrystal’s books [26, 27], now Edwardian antiques, are full of good sense and good mathematics. They can be read with profit today. The two volumes [27] are freely available online. Exercises. One of my undergraduatelecturers used to divide exercises into five-finger exercises and lollipops. I have done the same in this book. The exercises, of which there are about 250, are listed at the end of the section of the chapter to which they refer. If they are not marked with a star ( ), they are five-finger exercises and can be solved simply by reading the section. Those∗ marked with a star are not necessarily hard, but are also not merely routine applications of what you have read. They are there to make you think and to be enjoyable. For further practice in solving problems, the Schaum’s Outline Series of books are an excellent resource and cheap second- hand copies are easy to find. xi xii Preface Prerequisites. If the following topics are familiar then you probably have the back- ground needed to read this book: basic Euclidean and analytic geometry in two and three dimensions; the trigonometric, exponential and logarithm functions; the arith- metic of polynomials and the roots of the quadratic; experience in algebraic manipu- lation. Organization. The book is divided into two sections. Section I consists of Chapters 1to 4. Chapters 1 and 2 set the tone for the whole book and in particular attempt to • explain what proofs are and why they are important. Chapter 3 is a reference chapter of which only Section 3.1, Section 3.8 and the • first few pages of Section 3.4 need be read first. Everything else can be read when needed or when the fancy takes you. Chapter 4 is an essential prerequisite for reading Section II. It is partly revision • but mainly an introduction to properties that are met with time and again in studying algebra and are likely to be unfamiliar. Section II consists of Chapters 5 to 10. This is the mathematical core of the book and the chapters have been written to be read in order. Chapters 5, 6 and 7 are linked thematically by the remainder theorem and Euclid’s algorithm, whereas Chapters 8, 9 and 10 form an introduction to linear algebra. I have organized each chapter so that the more advanced material occurs towards the end. The three themes I had constantly in mind whilst writing these chapters were: 1. The solution of different kinds of algebraic equation. 2. The nature of the solutions. 3. The interplay between geometry and algebra. Wise words from antiquity. Mathematics is, and always has been, difficult. The commentator Proclus in the fifth century records a story about the mathematician Euclid. He was asked by Ptolomy, the ruler of Egypt, if there was not some easier way of learning mathematics than by reading Euclid’s big book on geometry, known as the Elements. Euclid’s reply was correct in every respect but did not contribute to the popularity of mathematicians. There was, he said, no royal road to geometry. In other words: no shortcuts, not even for god-kings. Despite that, I hope my book will make the road a little easier. Acknowledgements. I would like to thank my former colleagues in Wales, Tim Porter and Ronnie Brown, whose Mathematics in context module has influenced my thinking on presenting mathematics. The bibliography contains a list of every book or paper I read in connection with the writing of this one. Of these, I referred to Archbold [4] the most and regard it as an unsung classic. My own copy originally belonged to Ruth Coyte and was passed onto me by her family. This is my chance to Preface xiii thank them for all their kindnesses over the years. The book originated in a course I taught at Heriot-Watt University inherited from my colleagues Richard Szabo and Nick Gilbert. Although the text has been rethought and rewritten, some of the ex- ercises go back to them, and I have had numerous discussions over the years with both of them about what and how we should be teaching. Thanks are particularly due to Lyonell Boulton, Robin Knops and Phil Scott for reading selected chapters, and to Bernard Bainson, John Fountain, Jamie Gabbay, Victoria Gould, Des John- ston and Bruce MacDougall for individual comments. Sonya Gale advised on Greek. At CRC Press, thank you to Sunil Nair and Alexander Edwards for encouraging me to write the book, amongst other things, Amber Conley, Robin Loyd-Starkes, Katy E. Smith and an anonymous copy-editor for producing the book, and Shashi Kumar for technical support. I have benefited at Heriot-Watt University from the technical support of Iain McCrone and Steve Mowbray over many years with some fine-tuning by Dugald Duncan. The TeX-LaTeX Stack Exchange has been an invaluable source of good advice. The pictures were created using Till Tantau’s TikZ and Alain Matthes’ tkz-euclide which are warmly recommended.