Algebra & Geometry www.com Algebra & Geometry An Introduction to University Mathematics Second Edition Mark V. Lawson Heriot-Watt University Edinburgh, UK www.com Second edition published 2021 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2021 Mark V. Lawson First edition published by CRC Press 2016 CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and pub- lisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained.
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Lawson, Heriot-Watt University Edinburgh, UK. Other titles: Algebra and geometry Description: Second edition. | Boca Raton : Chapman & Hall/CRC Press, 2021. | Includes bibliographical references and index.
Identifiers: LCCN 2021000938 (print) | LCCN 2021000939 (ebook) | ISBN 9780367565084 (hardback) | ISBN 9780367563035 (paperback) | ISBN 9781003098072 (ebook) Subjects: LCSH: Algebra. Classification: LCC QA152.9--dc23 LC record available at https://lccn.gov/2021000938 LC ebook record available at https://lccn.gov/2021000939 ISBN: 9780367565084 (hbk) ISBN: 9780367563035 (pbk) ISBN: 9781003098072 (ebk) Access the Support Material: www.com This second edition is dedicated to my sisters, Juliet Rose and Jacqueline Susan, who will only read this dedication.com Contents Preface to the Second Edition xi Preface to the First Edition xiii Prolegomena xvii SECTION I IDEAS CHAPTER 1 The Nature of Mathematics 3 1.1 MATHEMATICS IN HISTORY 3 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 2.2 FUNDAMENTAL ASSUMPTIONS OF LOGIC 12 2.3 FIVE EASY PROOFS 12 2.5 UN PETIT PEU DE PHILOSOPHIE 26 2.7 PROVING SOMETHING FALSE 28 2.9 ADVICE ON PROOFS 29 vii www.com viii Contents CHAPTER 3 Foundations 31 3.8 PROOF BY INDUCTION 73 3.10 INFINITE NUMBERS 84 CHAPTER 4 Algebra Redux 93 4.1 RULES OF THE GAME 94 4.2 ALGEBRAIC AXIOMS FOR REAL NUMBERS 103 4.3 SOLVING QUADRATIC EQUATIONS 110 4.6 CHARACTERIZING REAL NUMBERS 123 SECTION II THEORIES CHAPTER 5 Number Theory 131 5.2 GREATEST COMMON DIVISORS 138 5.3 FUNDAMENTAL THEOREM OF ARITHMETIC 146 5.5 CONTINUED FRACTIONS 162 CHAPTER 6 Complex Numbers 171 6.1 COMPLEX NUMBER ARITHMETIC 171 6.2 COMPLEX NUMBER GEOMETRY 178 www.com Contents ix 6.3 EULER’S FORMULA FOR COMPLEX NUMBERS 182 6.4 MAKING SENSE OF COMPLEX NUMBERS 183 CHAPTER 7 Polynomials 187 7.2 THE REMAINDER THEOREM 188 7.3 ROOTS OF POLYNOMIALS 191 7.4 FUNDAMENTAL THEOREM OF ALGEBRA 192 7.5 ARBITRARY ROOTS OF COMPLEX NUMBERS 197 7.6 GREATEST COMMON DIVISORS OF POLYNOMIALS 201 7.10 ALGEBRAIC AND TRANSCENDENTAL NUMBERS 223 7.11 MODULAR ARITHMETIC WITH POLYNOMIALS 224 CHAPTER 8 Matrices 227 8.3 SOLVING SYSTEMS OF LINEAR EQUATIONS 246 8.7 BLANKINSHIP’S ALGORITHM 293 CHAPTER 9 Vectors 299 9.3 GEOMETRIC MEANING OF DETERMINANTS 315 9.4 GEOMETRY WITH VECTORS 317 9.6 ALGEBRAIC MEANING OF DETERMINANTS 331 9.com x Contents CHAPTER 10 The Principal Axes Theorem 339 10.3 CONICS AND QUADRICS 352 CHAPTER 11 What are the Real Numbers? 359 11.1 THE PROPERTIES OF THE REAL NUMBERS 360 11.2 APPROXIMATING REAL NUMBERS BY RATIONAL NUMBERS 372 11.3 A CONSTRUCTION OF THE REAL NUMBERS 375 Epilegomena 383 Bibliography 387 Index 395 www.com Preface to the Second Edition I am grateful to everyone who helped realize this new edition: specifically, Callum Fraser, Mansi Kabra, Meeta Singh and the anonymous copyeditor. I have incorpo- rated all the typos that people were kind enough to send in–with thanks to Harith Faris, Benjamin Gardner, Jennie Hansen, Roger Luther, James J. Ward and Amelia Wilson-Lake–smoothed out the text in a few places and augmented it in others, and added a new chapter, Chapter 11, that explains how to construct the real numbers.
In the first edition of this book, I divided the material into two types: the bulk of the material of the book proper in the usual font, and then material in boxes in smaller font. The aim of the material in the boxes was, and is, to describe more advanced mathematics. However, I realized that there was a jump in level between the two types of material, so in this edition, I have added fifteen short ‘essays’ that are at the same level as the main text but which direct the reader to particular developments of the material. These essays range in length from a paragraph to a page.
You can read them or omit them as you choose. The exercises have remained essentially the same with only minor changes. How- ever, I have tried to give a more explicit alert to the nature of a question by usually placing a star beside it if it requires some thought. I shall post these as before at the following page also accessible via my homepage http://www.uk/~markl/Algebra-geometry.
Lawson Edinburgh, Vernal Equinox, 2021.com Preface to the First Edition The aim of this book is to provide a 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 [29, 30, 68, 148, 85, 103, 17, 118, 152, 5], as well as some more modern ones [9, 28, 38, 47, 52, 55, 65, 116, 134, 140], and augmented by a sur- vey of the first-year mathematics modules on offer in a number of British and Irish universities. The older textbooks have been a revelation. For example, Chrystal’s books [29, 30], now Edwardian antiques, are full of good sense and good mathemat- ics.
They can be read with profit today. The two volumes [30] are freely available online. One of my undergraduate lecturers 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, xiii www.com xiv Preface to the First Edition the Schaum’s Outline Series of books are an excellent resource and cheap second- hand copies are easy to find. 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. The book is divided into two parts. Part I consists of Chapters 1 to 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 Sections 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. Part 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. The nature of the solutions.
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. 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 www.com Preface to the First Edition xv Archbold [5] 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 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.