Algebra & geometry: an introduction to university mathematics PDF
Preview Algebra & geometry: an introduction to university mathematics
Algebra & Geometry An Introduction to University Mathematics Algebra & Geometry An Introduction to University Mathematics Mark V. Lawson Heriot-Watt University Edinburgh, UK CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2016 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20160603 International Standard Book Number-13: 978-1-48224649-0 (eBook - EPUB3) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher 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 storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com In Memoriam WILLIAM HENRY LAWSON 24th April 1915, Ypres CHARLES PEIRCE ECCLESHALL 6th June 1915, Gallipoli Contents Preface Prolegomena S I IDEAS ECTION C 1 ▪ The nature of mathematics HAPTER 1.1 MATHEMATICS IN HISTORY 1.2 MATHEMATICS TODAY 1.3 THE SCOPE OF MATHEMATICS 1.4 WHAT THEY (PROBABLY) DIDN’T TELL YOU IN SCHOOL 1.5 FURTHER READING C 2 ▪ Proofs HAPTER 2.1 MATHEMATICAL TRUTH 2.2 FUNDAMENTAL ASSUMPTIONS OF LOGIC 2.3 FIVE EASY PROOFS 2.4 AXIOMS 2.5 UN PETIT PEU DE PHILOSOPHIE 2.6 MATHEMATICAL CREATIVITY 2.7 PROVING SOMETHING FALSE 2.8 TERMINOLOGY 2.9 ADVICE ON PROOFS C 3 ▪ Foundations HAPTER 3.1 SETS 3.2 BOOLEAN OPERATIONS 3.3 RELATIONS 3.4 FUNCTIONS 3.5 EQUIVALENCE RELATIONS 3.6 ORDER RELATIONS 3.7 QUANTIFIERS 3.8 PROOF BY INDUCTION 3.9 COUNTING 3.10 INFINITE NUMBERS C 4 ▪ Algebra redux HAPTER 4.1 RULES OF THE GAME 4.2 ALGEBRAIC AXIOMS FOR REAL NUMBERS 4.3 SOLVING QUADRATIC EQUATIONS 4.4 BINOMIAL THEOREM 4.5 BOOLEAN ALGEBRAS 4.6 CHARACTERIZING REAL NUMBERS S II THEORIES ECTION C 5 ▪ Number theory HAPTER 5.1 REMAINDER THEOREM 5.2 GREATEST COMMON DIVISORS 5.3 FUNDAMENTAL THEOREM OF ARITHMETIC 5.4 MODULAR ARITHMETIC 5.5 CONTINUED FRACTIONS C 6 ▪ Complex numbers HAPTER 6.1 COMPLEX NUMBER ARITHMETIC 6.2 COMPLEX NUMBER GEOMETRY 6.3 EULER’S FORMULA 6.4 MAKING SENSE OF COMPLEX NUMBERS C 7 ▪ Polynomials HAPTER 7.1 TERMINOLOGY 7.2 REMAINDER THEOREM 7.3 ROOTS OF POLYNOMIALS 7.4 FUNDAMENTAL THEOREM OF ALGEBRA 7.5 ARBITRARY ROOTS OF COMPLEX NUMBERS 7.6 GREATEST COMMON DIVISORS OF POLYNOMIALS 7.7 IRREDUCIBLE POLYNOMIALS 7.8 PARTIAL FRACTIONS 7.9 RADICAL SOLUTIONS 7.10 ALGEBRAIC AND TRANSCENDENTAL NUMBERS 7.11 MODULAR ARITHMETIC WITH POLYNOMIALS C 8 ▪ Matrices HAPTER 8.1 MATRIX ARITHMETIC 8.2 MATRIX ALGEBRA 8.3 SOLVING SYSTEMS OF LINEAR EQUATIONS 8.4 DETERMINANTS 8.5 INVERTIBLE MATRICES 8.6 DIAGONALIZATION 8.7 BLANKINSHIP’S ALGORITHM C 9 ▪ Vectors HAPTER 9.1 VECTORS GEOMETRICALLY 9.2 VECTORS ALGEBRAICALLY 9.3 GEOMETRIC MEANING OF DETERMINANTS 9.4 GEOMETRY WITH VECTORS 9.5 LINEAR FUNCTIONS 9.6 ALGEBRAIC MEANING OF DETERMINANTS 9.7 QUATERNIONS C 10 ▪ The principal axes theorem HAPTER 10.1 ORTHOGONAL MATRICES 10.2 ORTHOGONAL DIAGONALIZATION 10.3 CONICS AND QUADRICS Epilegomena Bibliography Index
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