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MATHEMATICS Algebra, Geometry, Combinatorics MATHEMATICS Algebra, geometry, combinatorics Dr Mark V Lawson October 24, 2014 ii Contents 1 The nature of mathematics 1 1.1 What are algebra, geometry and combinatorics? . .1 1.1.1 Algebra . .1 1.1.2 Geometry . .5 1.1.3 Combinatorics . .7 1.2 The scope of mathematics . .8 1.3 Pure versus applied mathematics . .9 1.4 The antiquity of mathematics . 11 1.5 The modernity of mathematics . 12 1.6 The legacy of the Greeks . 14 1.7 The legacy of the Romans . 15 1.8 What they didn't tell you in school . 15 1.9 Further reading and links . 16 2 Proofs 19 2.1 How do we know what we think is true is true? . 20 2.2 Three fundamental assumptions of logic . 22 2.3 Examples of proofs . 23 2.3.1 Proof 1 . 23 2.3.2 Proof 2 . 26 2.3.3 Proof 3 . 28 2.3.4 Proof 4 . 29 2.3.5 Proof 5 . 31 2.4 Axioms . 37 2.5 Mathematics and the real world . 41 2.6 Proving something false . 41 2.7 Key points . 42 2.8 Mathematical creativity . 43 i ii CONTENTS 2.9 Set theory: the language of mathematics . 43 2.10 Proof by induction . 52 3 High-school algebra revisited 57 3.1 The rules of the game . 57 3.1.1 The axioms . 57 3.1.2 Indices . 63 3.1.3 Sigma notation . 66 3.1.4 Infinite sums . 68 3.2 Solving quadratic equations . 70 3.3 Order . 76 3.4 The real numbers . 77 4 Number theory 81 4.1 The remainder theorem . 81 4.2 Greatest common divisors . 91 4.3 The fundamental theorem of arithmetic . 97 4.4 Modular arithmetic . 108 4.4.1 Congruences . 109 4.4.2 Wilson's theorem . 112 4.5 Continued fractions . 113 4.5.1 Fractions of fractions . 113 4.5.2 Rabbits and pentagons . 116 5 Complex numbers 123 5.1 Complex number arithmetic . 123 5.2 The fundamental theorem of algebra . 131 5.2.1 The remainder theorem . 132 5.2.2 Roots of polynomials . 134 5.2.3 The fundamental theorem of algebra . 136 5.3 Complex number geometry . 141 5.3.1 sin and cos . 141 5.3.2 The complex plane . 141 5.3.3 Arbitrary roots of complex numbers . 145 5.3.4 Euler's formula . 148 5.4 Making sense of complex numbers . 150 5.5 Radical solutions . 151 5.5.1 Cubic equations . 151 CONTENTS iii 5.5.2 Quartic equations . 154 5.5.3 Symmetries and particles . 156 5.6 Gaussian integers and factorizing primes . 157 6 Rational functions 159 6.1 Numerical partial fractions . 159 6.2 Analogies . 162 6.3 Partial fractions . 163 6.4 Integrating rational functions . 167 7 Matrices I: linear equations 171 7.1 Matrix arithmetic . 171 7.1.1 Basic matrix definitions . 171 7.1.2 Addition, subtraction, scalar multiplication and the transpose . 173 7.1.3 Matrix multiplication . 175 7.1.4 Special matrices . 179 7.1.5 Linear equations . 181 7.1.6 Conics and quadrics . 182 7.1.7 Graphs . 183 7.2 Matrix algebra . 186 7.2.1 Properties of matrix addition . 186 7.2.2 Properties of matrix multiplication . 187 7.2.3 Properties of scalar multiplication . 188 7.2.4 Properties of the transpose . 189 7.2.5 Some proofs . 189 7.3 Solving systems of linear equations . 195 7.3.1 Some theory . 196 7.3.2 Gaussian elimination . 198 7.4 Blankinship's algorithm . 206 8 Matrices II: inverses 209 8.1 What is an inverse? . 209 8.2 Determinants . 213 8.3 When is a matrix invertible? . 217 8.4 Computing inverses . 223 8.5 The Cayley-Hamilton theorem . 227 8.6 Complex numbers via matrices . 230 iv CONTENTS 9 Vectors 231 9.1 Vector algebra . 232 9.1.1 Addition and scalar multiplication of vectors . 232 9.1.2 Inner, scalar or dot products . 238 9.1.3 Vector or cross products . 240 9.1.4 Scalar triple products . 243 9.2 Vector arithmetic . 245 9.2.1 i's, j's and k's....................... 245 9.3 Geometry with vectors . 249 9.3.1 Position vectors . 249 9.3.2 Linear combinations . 250 9.3.3 Lines . 251 9.3.4 Planes . 255 9.3.5 Determinants . 258 9.4 Summary of vectors . 263 9.5 *Two vector proofs* . 266 9.6 Quaternions . 268 Chapter 1 The nature of mathematics This chapter is a guide to the mathematics described in this book. 1.1 What are algebra, geometry and combi- natorics? 1.1.1 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 1 2 CHAPTER 1. THE NATURE OF MATHEMATICS 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 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 which enable you to solve the equations. I don't mean that no formulae have yet been discovered, I mean that someone has proved that such a formula is impossible, that someone being the young French mathematician Evariste Galois (1811{1832), the James Dean of mathematics. Galois's work meant the end of the view that algebra was about finding formulae to solve equations. We shall not study Galois's work in this book but it has had a huge impact on algebra. It is one of the reasons why the algebra you study later in your university careers will look very different from the algebra you studied at school. In fact, one of my goals in writing this book is to help you navigate this transition. 1.1. WHAT ARE ALGEBRA, GEOMETRY AND COMBINATORICS? 3 I have talked about solving equations where there is one unknown but there is no reason to stop there. We can also study equations where there are any finite number of unknowns and those unknowns occur to any powers. The best place to start is where we have any number of unknowns but each unknown can occur only to the first power and no products of unknowns are allowed. This means we are studying linear equations like x + 2y + 3z = 4: Our goal is to find all the values of x, y and z that satisfy this equation. Thus the solutions are ordered triples (x; y; z). For example, both (0; 2; 0) and (2; 1; 0) are solutions whereas (1; 1; 1) is not a solution. It is unusual to have just one linear equation to solve. Usually we have two or more such as x + 2y + 3z = 4 and x + y + z = 0: We then need to find all the triples (x; y; z) that satisfy both equations simultaneously. In fact, as you should check, all the triples (λ − 4; 4 − 2λ, λ) where λ is any number satisfy both equations. For this reason, we often speak about simultaneous linear equations. It turns out that solving systems of linear equations never becomes difficult however many unknowns there are. The modern way of studying systems of linear equations uses matrix theory. That leaves studying equations where there are at least 2 unknowns and where there are no constraints on the powers of the unknowns and the extent to which they may be multiplied together. This is much more complicated. If you only allow squares such as x2 or products of at most two unknowns, such as xy, then there are relatively simple methods for solving them.
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