Algebra & Geometry: An Introduction to University Mathematics

425 Pages·2021·7.032 MB·English

by  Mark V. Lawson

Algebra & Geometry

Algebra & Geometry
An Introduction to
University Mathematics
Second Edition
Mark V. Lawson
Heriot-Watt University
Edinburgh, UK
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. If any copyright material has not been acknowledged please write and let us know so
we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or
hereafter invented, including photocopying, microfilming, and recording, or in any information stor-
age or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, access www.copyright.com
or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923,
978-750-8400. For works that are not available on CCC please contact mpkbookspermissions@tandf.
co.uk
Trademark notice: Product or corporate names may be trademarks or registered trademarks and are
used only for identification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data
Names: Lawson, Mark V., author.
Title: Algebra & geometry : an introduction to university mathematics /
Mark V. 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. | Geometry. | Mathematics.
Classification: LCC QA152.3 .L39 2021 (print) | LCC QA152.3 (ebook) | DDC
512.9--dc23
LC record available at https://lccn.loc.gov/2021000938
LC ebook record available at https://lccn.loc.gov/2021000939
ISBN: 9780367565084 (hbk)
ISBN: 9780367563035 (pbk)
ISBN: 9781003098072 (ebk)
Access the Support Material: www.routledge.com/9780367563035
Thissecondeditionisdedicatedtomysisters,
JulietRoseandJacquelineSusan,
whowillonlyreadthisdedication.

Contents
Preface to the Second Edition xi
Preface to the First Edition xiii
Prolegomena xvii
SECTION I IDEAS
CHAPTER 1 (cid:4) The Nature of Mathematics 3
1.1 MATHEMATICS IN HISTORY 3
1.2 MATHEMATICS TODAY 6
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 (cid:4) Proofs 11
2.1 MATHEMATICAL TRUTH 11
2.2 FUNDAMENTAL ASSUMPTIONS OF LOGIC 12
2.3 FIVE EASY PROOFS 12
2.4 AXIOMS 24
2.5 UN PETIT PEU DE PHILOSOPHIE 26
2.6 MATHEMATICAL CREATIVITY 27
2.7 PROVING SOMETHING FALSE 28
2.8 TERMINOLOGY 28
2.9 ADVICE ON PROOFS 29
vii
viii (cid:4) Contents
CHAPTER 3 (cid:4) Foundations 31
3.1 SETS 31
3.2 BOOLEAN OPERATIONS 40
3.3 RELATIONS 45
3.4 FUNCTIONS 50
3.5 EQUIVALENCE RELATIONS 62
3.6 ORDER RELATIONS 69
3.7 QUANTIFIERS 71
3.8 PROOF BY INDUCTION 73
3.9 COUNTING 75
3.10 INFINITE NUMBERS 84
CHAPTER 4 (cid: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
4.4 BINOMIAL THEOREM 114
4.5 BOOLEAN ALGEBRAS 116
4.6 CHARACTERIZING REAL NUMBERS 123
SECTION II THEORIES
CHAPTER 5 (cid:4) Number Theory 131
5.1 REMAINDER THEOREM 131
5.2 GREATEST COMMON DIVISORS 138
5.3 FUNDAMENTAL THEOREM OF ARITHMETIC 146
5.4 MODULAR ARITHMETIC 154
5.5 CONTINUED FRACTIONS 162
CHAPTER 6 (cid:4) Complex Numbers 171
6.1 COMPLEX NUMBER ARITHMETIC 171
6.2 COMPLEX NUMBER GEOMETRY 178
Contents (cid:4) ix
6.3 EULER’S FORMULA FOR COMPLEX NUMBERS 182
6.4 MAKING SENSE OF COMPLEX NUMBERS 183
CHAPTER 7 (cid:4) Polynomials 187
7.1 TERMINOLOGY 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
7.7 IRREDUCIBLE POLYNOMIALS 203
7.8 PARTIAL FRACTIONS 205
7.9 RADICAL SOLUTIONS 213
7.10 ALGEBRAIC AND TRANSCENDENTAL NUMBERS 223
7.11 MODULAR ARITHMETIC WITH POLYNOMIALS 224
CHAPTER 8 (cid:4) Matrices 227
8.1 MATRIX ARITHMETIC 227
8.2 MATRIX ALGEBRA 239
8.3 SOLVING SYSTEMS OF LINEAR EQUATIONS 246
8.4 DETERMINANTS 256
8.5 INVERTIBLE MATRICES 264
8.6 DIAGONALIZATION 280
8.7 BLANKINSHIP’S ALGORITHM 293
CHAPTER 9 (cid:4) Vectors 299
9.1 VECTORS GEOMETRICALLY 299
9.2 VECTORS ALGEBRAICALLY 312
9.3 GEOMETRIC MEANING OF DETERMINANTS 315
9.4 GEOMETRY WITH VECTORS 317
9.5 LINEAR FUNCTIONS 326
9.6 ALGEBRAIC MEANING OF DETERMINANTS 331
9.7 QUATERNIONS 334