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MATHEMATICS Algebra, Geometry, Combinatorics MATHEMATICS Algebra, geometry, combinatorics Dr Mark V Lawson November 17, 2014 ii Contents Preface v Introduction vii 1 The nature of mathematics 1 1.1 The scope of mathematics . .1 1.2 Pure versus applied mathematics . .3 1.3 The antiquity of mathematics . .4 1.4 The modernity of mathematics . .6 1.5 The legacy of the Greeks . .8 1.6 The legacy of the Romans . .8 1.7 What they didn't tell you in school . .9 1.8 Further reading and links . 10 2 Proofs 13 2.1 How do we know what we think is true is true? . 14 2.2 Three fundamental assumptions of logic . 16 2.3 Examples of proofs . 17 2.3.1 Proof 1 . 17 2.3.2 Proof 2 . 20 2.3.3 Proof 3 . 22 2.3.4 Proof 4 . 23 2.3.5 Proof 5 . 25 2.4 Axioms . 31 2.5 Mathematics and the real world . 35 2.6 Proving something false . 35 2.7 Key points . 36 2.8 Mathematical creativity . 37 i ii CONTENTS 2.9 Set theory: the language of mathematics . 37 2.10 Proof by induction . 46 3 High-school algebra revisited 51 3.1 The rules of the game . 51 3.1.1 The axioms . 51 3.1.2 Indices . 57 3.1.3 Sigma notation . 60 3.1.4 Infinite sums . 62 3.2 Solving quadratic equations . 64 3.3 *Order . 70 3.4 *The real numbers . 71 4 Number theory 75 4.1 The remainder theorem . 75 4.2 Greatest common divisors . 85 4.3 The fundamental theorem of arithmetic . 91 4.4 *Modular arithmetic . 102 4.4.1 Congruences . 103 4.4.2 Wilson's theorem . 106 4.5 *Continued fractions . 107 4.5.1 Fractions of fractions . 107 4.5.2 Rabbits and pentagons . 110 5 Complex numbers 117 5.1 Complex number arithmetic . 117 5.2 The fundamental theorem of algebra . 125 5.2.1 The remainder theorem . 126 5.2.2 Roots of polynomials . 128 5.2.3 The fundamental theorem of algebra . 130 5.3 Complex number geometry . 135 5.3.1 sin and cos . 135 5.3.2 The complex plane . 135 5.3.3 Arbitrary roots of complex numbers . 139 5.3.4 Euler's formula . 142 5.4 *Making sense of complex numbers . 144 5.5 *Radical solutions . 145 5.5.1 Cubic equations . 145 CONTENTS iii 5.5.2 Quartic equations . 148 5.5.3 Symmetries and particles . 150 5.6 *Gaussian integers and factorizing primes . 151 6 *Rational functions 153 6.1 Numerical partial fractions . 153 6.2 Analogies . 156 6.3 Partial fractions . 157 6.4 Integrating rational functions . 161 7 Matrices I: linear equations 165 7.1 Matrix arithmetic . 165 7.1.1 Basic matrix definitions . 165 7.1.2 Addition, subtraction, scalar multiplication and the transpose . 167 7.1.3 Matrix multiplication . 169 7.1.4 Special matrices . 173 7.1.5 Linear equations . 175 7.1.6 Conics and quadrics . 176 7.1.7 Graphs . 177 7.2 Matrix algebra . 180 7.2.1 Properties of matrix addition . 180 7.2.2 Properties of matrix multiplication . 181 7.2.3 Properties of scalar multiplication . 182 7.2.4 Properties of the transpose . 183 7.2.5 Some proofs . 183 7.3 Solving systems of linear equations . 189 7.3.1 Some theory . 190 7.3.2 Gaussian elimination . 192 7.4 Blankinship's algorithm . 200 8 Matrices II: inverses 203 8.1 What is an inverse? . 203 8.2 Determinants . 207 8.3 When is a matrix invertible? . 212 8.4 Computing inverses . 218 8.5 The Cayley-Hamilton theorem . 222 8.6 Determinants redux . 228 iv CONTENTS 8.7 *Complex numbers via matrices . 228 9 Vectors 231 9.1 Vectors geometrically . 232 9.1.1 Addition and scalar multiplication of vectors . 232 9.1.2 Inner products . 237 9.1.3 Vector products . 240 9.2 Vectors algebraically . 246 9.2.1 The geometric meaning of determinants . 249 9.3 Geometry with vectors . 253 9.3.1 Position vectors . 253 9.3.2 Linear combinations . 253 9.3.3 Lines . 254 9.3.4 Planes . 258 9.3.5 The geometric meaning of linear equations . 261 9.4 *Quaternions . 262 10 Combinatorics 265 10.1 More set theory . 265 10.1.1 Operations on sets . 265 10.1.2 Partitions . 270 10.2 Ways of counting . 271 10.2.1 Counting principles . 271 10.2.2 The power set . 273 10.2.3 Counting arrangements: permutations . 274 10.2.4 Counting choices: combinations . 275 10.2.5 Examples of counting . 277 10.3 The binomial theorem . 279 10.4 *Infinite numbers . 285 Preface Mathematics is the single most important cultural innovation after language. But if your recollections of school mathematics don't go much beyond solv- ing quadratic equations, then you would be forgiven for thinking this a wild claim. In fact, the modern world would be impossible without mathemat- ics. I don't mean just more difficult and inconvenient | I mean impossible. However, the mathematics that makes the world go round is hidden, usually embedded in the programs that turn inert silicon into useful technology. But the usefulness of mathematics is not the only reason to study it. Mathemat- ics is also a man-made universe that is endlessly fascinating. Where else, for example, could such ideas as being infinitely small or infinitely large be contemplated and used? The aim of this book is essentially a practical one: to provide a bridge between school and university mathematics, and a firm foundation for fur- ther study. But to do this, I have to talk about mathematics as well as do mathematics. The talking-about should help you with the doing | it is not supposed to be just waffle. This book is mainly self-contained but where there are connections with calculus I have mentioned them. Mathematics does not divide into water-tight compartments. For example, some of the deepest theorems about the prime numbers require the tools of calculus ap- plied to complex numbers, that is complex analysis, for their proof. Purity of method is artificial and misleading. Maths is difficult and 'twas always so. The commentator Proclus in the fifth century records a story about the mathematician Euclid. He was asked by the ruler of Egypt, Ptolomy, if there wasn't some easier way of learning maths than through Euclid's big book on geometry, known as the Elements. Euclid's reply was correct in every respect but didn't contribute to the pop- ularity of mathematicians. There was, he said, no royal road to geometry. In other words: no short-cuts, not even for god-kings. Despite that, I hope v vi PREFACE this book will make the road a little easier. Introduction: what are algebra, geometry and combinatorics? Algebra Algebra started as the study of equations. The simplest kinds of equations are ones like 3x − 1 = 0 where there is only one unknown x and that unknown occurs to the power 1. This means we have x alone and not, say, x1000. It is easy to solve this specific equation. Add 1 to both sides to get 3x = 1 and then divide both sides by 3 to get 1 x = : 3 This is the solution to my original equation and, to make sure, we check our answer by calculating 1 3 · − 1 3 and observing that we really do get 0 as required. Even this simple example raises an important point: to carry out these calculations, I had to know what rules the numbers and symbols obeyed. You probably applied these rules unconsciously, but in this book it will be important to know explicitly what they are. The method used for the specific example above can be applied to any equation of the form ax + b = 0 vii viii INTRODUCTION as long as a 6= 0. Here a; b are specific numbers, probably real numbers, and x is the real number I am trying to find. This equation is the most general example of a linear equation in one unknown. If x occurs to the power 2 then we get ax2 + bx + c = 0 where a 6= 0. This is an example of a quadratic equation in one unknown. You will have learnt a formula to solve such equations. But there is no reason to stop at 2. If x occurs to the power 3 we get a cubic equation in one unknown ax3 + bx2 + cx + d = 0 where a 6= 0. Solving such equations is much harder than solving quadratics but there is also an algebraic formula for the roots. But there is no reason to stop at cubics. We could look at equations in which x occurs to the power 4, quartics, and once again there is a formula for finding the roots. The highest power of x that occurs in such an equation is called its degree. These results might lead you to expect that there are always algebraic formulae for finding the roots of any polynomial equation whatever its degree. There aren't. For equations of degree 5, the quintics, and more, there are no algebraic formulae.
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