The Essence of Mathematics Through Elementary Problems ALEXANDRE BOROVIK AND TONY GARDINER To access digital resources including: blog posts videos online appendices and to purchase copies of this book in: hardback paperback ebook editions Go to: https://www.openbookpublishers.com/product/979 Open Book Publishers is a non-profit independent initiative. We rely on sales and donations to continue publishing high-quality academic works. THE ESSENCE OF MATHEMATICS THROUGH ELEMENTARY PROBLEMS The Essence of Mathematics Through Elementary Problems Alexandre Borovik and Tony Gardiner http://www.openbookpublishers.com c 2019 Alexandre Borovik and Tony Gardiner This work is licensed under a Creative Commons Attribution 4.0 International license (CC BY 4.0). This license allows you to share, copy, distribute and transmit the work; to adapt the work and to make commercial use of the work providing attribution is made to the author (but not in any way that suggests that they endorse you or your use of the work). Attribution should include the following information: Alexandre Borovik and Tony Gardiner, The Essence of Mathematics through Elementary Problems. Cambridge, UK: Open Book Publishers, 2019. http://dx.doi.org/10.11647/OBP.0168 Further details about CC BY licenses are available at http://creativecommons.org/licenses/by/4.0/ All external links were active on 10/04/2019 and archived via the Internet Archive Wayback Machine: https://archive.org/web/ Every effort has been made to identify and contact copyright holders and any omission or error will be corrected if notification is made to the publisher. Digital material and resources associated with this volume are available at http://www.openbookpublishers.com/product/979#resources This is the third volume of the OBP Series in Mathematics: ISSN 2397-1126 (Print) ISSN 2397-1134 (Online) ISBN Paperback 9781783746996 ISBN Hardback: 9781783747009 ISBN Digital (PDF): 9781783747016 DOI: 10.11647/OBP.0168 Cover photo: Abstract Spiral Pattern (2015) by Samuel Zeller, https://unsplash.com/photos/j0g8taxHZa0 Cover design by Anna Gatti. Contents Preface v About this text xi I. Mental Skills 1 1.1 Mental arithmetic and algebra 2 1.1.1 Times tables. 2 1.1.2 Squares, cubes, and powers of 2. 2 1.1.3 Primes 5 1.1.4 Common factors and common multiples 5 1.1.5 The Euclidean algorithm 6 1.1.6 Fractions and ratio 7 1.1.7 Surds 9 1.2 Direct and inverse procedures 9 1.2.1 Factorisation 12 1.3 Structural arithmetic 12 1.4 Pythagoras' Theorem 13 1.4.1 Pythagoras' Theorem, trig for special angles, and CAST 14 1.4.2 Converses and Pythagoras' Theorem 16 1.4.3 Pythagorean triples 17 1.4.4 Sums of two squares 19 1.5 Visualisation 20 1.6 Trigonometry and radians 22 1.6.1 Sine Rule 22 1.6.2 Radians and spherical triangles 23 1.6.3 Polar form and sin(A+B) 27 1.7 Regular polygons and regular polyhedra 27 1.7.1 Regular polygons are cyclic 28 1.7.2 Regular polyhedra 28 1.8 Chapter 1: Comments and solutions 29 II. Arithmetic 51 2.1 Place value and decimals: basic structure 51 2.2 Order and factors 53 2.3 Standard written algorithms 53 2.4 Divisibility tests 54 2.5 Sequences 56 2.5.1 Triangular numbers 56 2.5.2 Fibonacci numbers 56 2.6 Commutative, associative and distributive laws 60 2.7 Infinite decimal expansions 61 2.8 The binary numeral system 64 2.9 The Prime Number Theorem 66 2.10 Chapter 2: Comments and solutions 69 III. Word Problems 91 3.1 Twenty problems which embody \3 ´ 1 “ 2" 93 3.2 Some classical examples 94 3.3 Speed and acceleration 95 3.4 Hidden connections 96 3.5 Chapter 3: Comments and solutions 97 IV. Algebra 111 4.1 Simultaneous linear equations and symmetry 112 4.2 Inequalities and modulus 115 4.2.1 Geometrical interpretation of modulus, of inequalities, and of modulus inequalities 115 4.2.2 Inequalities 117 4.3 Factors, roots, polynomials and surds 119 4.3.1 Standard factorisations 119 4.3.2 Quadratic equations 123 4.4 Complex numbers 126 4.5 Cubic equations 131 4.6 An extra 133 4.7 Chapter 4: Comments and solutions 134 V. Geometry 169 5.1 Comparing geometry and arithmetic 171 5.2 Euclidean geometry: a brief summary 173 5.3 Areas, lengths and angles 194 5.4 Regular and semi-regular tilings in the plane 196 5.5 Ruler and compasses constructions for regular polygons 199 5.6 Regular and semi-regular polyhedra 201 5.7 The Sine Rule and the Cosine Rule 206 5.8 Circular arcs and circular sectors 211 5.9 Convexity 217 5.10 Pythagoras' Theorem in three dimensions 217 5.11 Loci and conic sections 220 5.12 Cubes in higher dimensions 227 5.13 Chapter 5: Comments and solutions 230 VI. Infinity: recursion, induction, infinite descent 283 6.1 Proof by mathematical induction I 286 6.2 `Mathematical induction' and `scientific induction' 287 6.3 Proof by mathematical induction II 290 6.4 Infinite geometric series 297 6.5 Some classical inequalities 299 6.6 The harmonic series 304 6.7 Induction in geometry, combinatorics and number theory 311 6.8 Two problems 313 6.9 Infinite descent 314 6.10 Chapter 6: Comments and solutions 317 Simon Phillips Norton 1952{2019 In memoriam Preface Understanding mathematics cannot be transmitted by painless entertainment . actual contact with the content of living mathematics is necessary. The present book . is not a concession to the dangerous tendency toward dodging all exertion. Richard Courant (1888{1972) and Herbert Robbins (1915{2001) Preface to the first edition of What is mathematics? Interested students of mathematics, who seek insight into the \essence of the discipline", and who read more widely with a view to discovering what the subject is really about, may emerge with the justifiable impression of serious mathematics as an austere, but distant mountain range { accessible only to those who devote their lives to its exploration. And they may conclude that the beginner can only appreciate its rough outline through a haze of unbridgeable distance. The best popularisers sometimes manage to convey more than this { including hints of the human story behind recent developments, and the way different branches and results interact in unexpected ways; but the essence of mathematics still tends to remain elusive, and the picture they paint is inevitably a broad brush substitute for the detail of living mathematics. This collection takes a different approach. We start out by observing that mathematics is not a fixed entity { as one might unconsciously infer from the metaphor of an \austere mountain range". Mathematics is a mental universe, a work-in-progress in our collective imagination, which grows dramatically over time, and whose eventual extent would seem to be unconstrained { without any obvious limits. This boundlessness also works in reverse, when applied to small details: features which we thought we had understood are repeatedly filled in, or reinterpreted, in new ways to reveal finer and finer micro-structures. Hence whatever the essence of the discipline may be, it is clearly not something which can only be accessed through the complete exploration of some fixed corpus of knowledge. Rather the essential character of mathematics seems to be related to viii Essence of Mathematics • the kind of material that counts as mathematical, • the way this material is addressed, • the changes in perspective that occur as our understanding grows and deepens, and • the unexpected connections that regularly emerge between separate strands and layers. There are a number of books giving excellent general advice to prospective students about how university mathematics differs from school mathematics. In contrast, this collection { which we hope will be enjoyed by interested high school students and their teachers, by undergraduates and postgraduates, and by many others is more like a messy workshop than a polished exposition. Here the reader is asked to tackle a sequence of problems, to reflect on what they discover, and mostly to draw their own conclusions (though some key messages are explicitly discussed in the text, or in the solutions at the end of each chapter). This attempt to engage the reader as an active participant along the way is inevitably untidy { and may sometimes prove frustrating. In particular, whereas a polished exposition would break up the text with eye-catching diagrams, an untidy workshop will usually leave the reader to draw their own figures as an essential part of the struggle. This temporary untidiness and frustration is an integral part of \the essence" that we seek to capture { provided it leads to occasional glimpses of the power, and the elegance of mathematics. Young children and students of all ages regularly experience the power, the economy, the beauty, and the elegance of mathematics and of mathematical thinking on a small scale, through struggling with certain elementary results and problems (or groups of problems). For example, one of the problems we have included in Chapter 3 was mentioned explicitly in an interview1 with the leading Russian mathematician Vladimir Arnold (1937{2010): Interviewer: Please tell us a little bit about your early education. Were you already interested in mathematics as a child? Arnold: [. ] The first real mathematical experience I had was when our schoolteacher I.V. Morotzkin gave us the following problem [VA then formulated Problem 89 in Chapter 3]. I spent a whole day thinking on this oldie, and the solution (based on what are now called scaling arguments, dimensional analysis, or toric variety theory, depending on your taste) came as a revelation.