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An Introduction to Contemporary Mathematics

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An Introduction to Contemporary Mathematics An Introduction to Contemporary Mathematics John Hutchinson (suggestions and comments to: [email protected]) March 21, 2010 c 2006 John Hutchinson Mathematical Sciences Institute College of Science Australian National University A text for the ANU secondary college course \An Introduction to Contemporary Mathematics" I wish to dedicate this text: • to the memory of my father George Hutchinson and to my mother Ellen Hutchinson for their moral and financial support over many years of my interest in mathematics; • to my mentor Kevin Friel for being such an inspirational high school teacher of mathematics; • and to my partner and wife Malise Arnstein for her unflagging support and encouragement, despite her insight from the begin- ning that this project was going to take far more time than I ever anticipated. Contents Introduction iii For Whom are these Notes? . iii What is Mathematics? . iii Philosophy of this Course . iii These Notes and The Heart of Mathematics . iv What is Covered in this Course? . iv Studying Mathematics . v Acknowledgements . v Quotations vi 1 Fun and Games 1 2 Numbers and Cryptography 2 2.1 Counting .............................. 6 2.2 The Fibonacci Sequence ...................... 13 2.3 Prime Numbers ........................... 24 2.4 Modular Arithmetic ........................ 36 2.5 RSA Public Key Cryptography . 52 2.6 Irrational Numbers ......................... 70 2.7 The Real Number System ..................... 75 3 Infinity 86 3.1 Comparing Sets ........................... 89 3.2 Countably Infinite Sets ....................... 94 3.3 Different Sizes of Infinity . 103 3.4 An Infinite Hierarchy of Infinities . 113 3.5 Geometry and Infinity . 122 4 Chaos and Fractals 132 4.1 A Gallery of Fractals . 138 i ii Contents 4.2 Iterative Dynamical Systems . 143 4.3 Fractals By Repeated Replacement . 151 4.4 Iterated Function Systems . 161 4.5 Simple Processes Can Lead to Chaos . 182 4.6 Julia Sets and Mandelbrot Sets . 204 4.7 Dimensions Which Are Not Integers . 213 5 Geometry and Topology 216 5.1 Euclidean Geometry and Pythagoras's Theorem . 220 5.2 Platonic Solids and Euler's Formula . 225 5.3 Visualising the Fourth Dimension . 239 5.4 Topology, Isotopy and Homeomorphisms . 248 5.5 One Sided Surfaces and Non Orientable Surfaces . 255 5.6 Classifying Surfaces . 264 Introduction For Whom are these Notes? These notes, together with the book The Heart of Mathematics [HM] by Burger and Starbird, are the texts for the ANU College Mathematics Minor for Years 11 and 12 students. If you are doing this course you will have a strong interest in mathematics, and probably be in the top 5% or so of students academically. What is Mathematics? Mathematics is the study of pattern and structure. Mathematics is funda- mental to the physical and biological sciences, engineering and information technology, to economics and increasingly to the social sciences. The patterns and structures we study in mathematics are universal. It is perhaps possible to imagine a universe in which the biology and physics are dif- ferent, it is much more difficult to imagine a universe in which the mathematics is different. Philosophy of this Course The goal is to introduce you to contemporary mainstream 20th and 21st century mathematics. This is not an easy task. Mathematics is like a giant scaffolding. You need to build the superstructure before you can ascend for the view. The calculus and algebra you will learn in college is an essential part of this scaffolding and is fundamental for your further mathematics, but most of it was discovered in the 18th century. We will take a few short cuts and only use calculus later in this course. We will investigate some very exciting and useful modern mathematics and get a feeling for \what mathematics is all about". The mathematics you will see in this course is usually not seen until higher level courses in second or third year at University. Of course, you will not cover the mathematics in the same depth or general- ity as you will if you pursue mathematics as a part of your University studies (as I hope most of you will do). The way we will proceed is by studying carefully chosen parts and representative examples from various areas of mathematics which illustrate important and general key concepts. In the process you will iii iv Introduction gain a real understanding and feeling for the beauty, utility and breadth of mathematics. These Notes and The Heart of Mathematics [HM] is an excellent book. It is one of a small number of texts intended to give you, the reader, a feeling for the theory and applications of contemporary mathematics at an early stage in your mathematical studies. However, [HM] is directed at a different group of students | undergraduate students in the United States with little mathematics background (e.g. no calculus) who might take no other mathematics courses in their studies. Despite its apparently informal style, [HM] develops a significant amount of interesting contemporary mathematics. The arguments are usually complete (and if not, this is indicated), correct and well motivated. They are often done by means of studying particular but important examples which cover the main ideas in the general case. However, you might find that the language is a little verbose at times (and you may or may not find the jokes tedious!). After first studying the arguments in [HM] you may then find the more precisely written mathematical arguments in these Notes more helpful in understanding \how it all hangs together". So here is a suggested procedure: 1. Look very briefly at these notes both to see what parts of [HM] you should study and to gain an overview. 2. Study (= read, think about, cogitate over) the relevant section in [HM]. 3. Then study the relevant section in these Notes. You may want to change the order, do what is best for you. In the Notes we: • Follow the same chapter and section numbering as in [HM] • Discuss and extend the material in [HM] and fill in some gaps • Often write out more succinct and general arguments • Indicate which parts of [HM] are to be studied and sometimes recommend questions to attempt • Include some more difficult and challenging questions What is Covered in this Course? There are four parts to the course. Each will take approximately 1.5 terms. You will study the first 2 parts in terms 2,3,4 of year 11 and the second 2 parts in terms 1,2,3 of year 12. Part 1 An introduction to number theory and its application to cryptography. Essentially Chapter 2 from [HM] and supplementary material from these Notes. The RSA cryptography we discuss is essential to internet security and the method was discovered in 1977. The 3 mathematicians involved started a company which they sold for about $600,000,000(US). Part 2 A Hierarchy of Infinities. Essentially Chapter 3 from [HM] and sup- plementary material from these Notes. What is infinity? Can one infinite Studying Mathematics v set be larger than another (Yes). If you remove 23 objects from an infi- nite set is the resulting set \smaller" (No). These ideas are interesting, but are they important or useful? (Yes). Part 3 Dynamical Processes, Chaos and Fractals. Modelling change by dy- namical processes, how chaos can arise out of simple processes, how frac- tal sets have fractional dimensions. Some of the ideas here on fractals were first developed by the present writer (iterated function systems) and other ideas (the chaos game) by another colleague now at the ANU, Michael Barnsley. Barnsley applied these ideas to image compression and was a founder of the company \Iterated Systems", at one stage valued at $200,000,000(US), later known as \Media Bin" and then acquired by \Interwoven". Part 4 Geometry and Topology. Parts of Chapters 4 and 5 from [HM] and sup- plementary material from these Notes. Platonic solids, visualising higher dimensions, topology, classifying surfaces, and more. This is beautiful mathematics and it is fundamental to our understanding of the universe in which we live | some current theories model our universe by 10 di- mensional curved geometry I suggest you also • read ix{xiv of [HM] in order to understand the philosophy of that book; • read xv{xxi of [HM] to gain an idea of the material you will be investi- gating over the next 2 years. Studying Mathematics This takes time and effort but it is very interesting material and intellectually rewarding. Do lots of Questions from [HM] and from these Notes, answer the questions here marked with a - and keep your solutions and comments in a folder. Material marked ? is not in [HM] and is more advanced. Some is a little more advanced and some is a lot more advanced. It is included to give you an idea of further connections. Don't worry if it does not make complete sense or you don't fully understand. Just relax and realise it is not examinable, except in those cases where your teacher specifically says so, in which case you will also be told how and to what extent it is examinable. Acknowledgements I would like to thank Richard Brent, Tim Brook, Clare Byrne, Jonathan Manton, Neil Montgomery, Phoebe Moore, Simon Olivero, Raiph McPherson, Jeremy Reading, Bob Scealy, Lisa Walker and Chris Wetherell, for comments and suggestions on various drafts of these notes. Quotations Philosophy is written in this grand book|I mean the universe| which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other mathematical figures, without which it is humanly impossible to understand a single word of it; without these one is wandering about in a dark labyrinth.
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