The Britannica Guide to Algebra and Trigonometry PDF
Preview The Britannica Guide to Algebra and Trigonometry
Published in 2011 by Britannica Educational Publishing (a trademark of Encyclopædia Britannica, Inc.) in association with Rosen Educational Services, LLC 29 East 21st Street, New York, NY 10010. Copyright © 2011 Encyclopædia Britannica, Inc. Britannica, Encyclopædia Britannica, and the Thistle logo are registered trademarks of Encyclopædia Britannica, Inc. All rights reserved. Rosen Educational Services materials copyright © 2011 Rosen Educational Services, LLC. All rights reserved. Distributed exclusively by Rosen Educational Services. For a listing of additional Britannica Educational Publishing titles, call toll free (800) 237-9932. First Edition Britannica Educational Publishing Michael I. Levy: Executive Editor J.E. Luebering: Senior Manager Marilyn L. Barton: Senior Coordinator, Production Control Steven Bosco: Director, Editorial Technologies Lisa S. Braucher: Senior Producer and Data Editor Yvette Charboneau: Senior Copy Editor Kathy Nakamura: Manager, Media Acquisition William L. Hosch: Associate Editor, Mathematics and Computer Sciences Rosen Educational Services Alexandra Hanson-Harding: Editor Bethany Bryan: Editor Nelson Sá: Art Director Cindy Reiman: Photography Manager Matthew Cauli: Designer, Cover Design Introduction by John Strazzabosco Library of Congress Cataloging-in-Publication Data The Britannica guide to algebra and trigonometry / edited by William L. Hosch. p. cm.—(Math explained) “In association with Britannica Educational Publishing, Rosen Educational Services.” Includes bibliographical references and index. ISBN 978-1-61530-219-2 (eBook) 1. Algebra. 2. Trigonometry. I. Hosch, William L. II. Title: Algebra and trigonometry. QA155.B75 2010 512—dc22 2009047905 Cover © www.istockphoto.com/Stefan Klein; p. 12 © www.istockphoto.com/Fernando Batista; p. 22 © www.istockphoto.com/Alejandro Raymond; pp. 23, 91, 163, 202, 227, 259, 268, 270, 273 © www.istockphoto.com/James Thew. C ONTENTS Introduction 12 Chapter 1: Algebra 23 History of Algebra 23 The Emergence of Formal Equations 23 Problem Solving in Egypt and 25 Babylon 25 Greece and the Limits of Geometric Expression 27 The Equation in India and China 31 Islamic Contributions 32 Commerce and Abacists in the European Renaissance 35 Cardano and the Solving of Cubic and Quartic Equations 38 Viète and the Formal Equation 40 The Concept of Numbers 41 36 Classical Algebra 43 Analytic Geometry 43 The Fundamental Theorem of Algebra 45 Impasse with Radical Methods 47 Galois Theory 48 Applications of Group Theory 51 Fundamental Concepts of Modern Algebra 54 Systems of Equations 57 Quaternions and Vectors 61 The Close of the Classical Age 63 Structural Algebra 63 47 Precursors to the Structural Approach 64 The Structural Approach Dominates 66 Algebraic Superstructures 67 New Challenges and Perspectives 69 Branches of Algebra 70 Elementary Algebra 71 Algebraic Quantities 71 Algebraic Expressions 73 Solving Algebraic Equations 75 79 Solving Systems of Algebraic Equations 76 Linear Algebra 78 88 Vectors and Vector Spaces 78 Linear Transformations and Matrices 81 Eigenvectors 82 Modern Algebra 83 Basic Algebraic Structures 83 Field Axioms 85 Rings 86 Group Theory 88 Chapter 2: Great Algebraists 91 Early Algebraists (Through the 16th Century) 91 Bhaskara II 91 Brahmagupta 93 Girolamo Cardano 94 Diophantus of Alexandria 96 Lodovico Ferrari 99 Scipione Ferro 100 al-Karaji 101 - - al-Khwarizmı 102 Liu Hui 103 95 Mahavira 105 Qin Jiushao 106 Classical Algebraists (17th–19th Centuries) 107 Niels Henrik Abel 107 Bernhard Bolzano 110 George Boole 111 Arthur Cayley 113 Évariste Galois 116 Carl Friedrich Gauss 119 Sir William Rowan Hamilton 124 Charles Hermite 129 108 Felix Klein 130 Leopold Kronecker 131 Ernst Eduard Kummer 132 Sophus Lie 133 Joseph Liouville 135 Paolo Ruffini 137 Seki Takakazu 138 James Joseph Sylvester 140 François Viète 142 Algebraists of the Structural Period (20th Century– ) 143 Emil Artin 143 Richard Ewen Borcherds 144 116 Nicolas Bourbaki 144 Richard Dagobert Brauer 145 Élie-Joseph Cartan 146 George Dantzig 147 Leonard Eugene Dickson 148 Jean Dieudonné 149 Georg Frobenius 149 Aleksandr Osipovich Gelfond 150 David Hilbert 151 Saunders Mac Lane 154 Gregori Aleksandrovich Margulis 155 Emmy Noether 156 157 Daniel Gray Quillen 158 Alfred Tarski 159 Hermann Weyl 160 Efim Isaakovich Zelmanov 161 Chapter 3: Algebraic Terms and Concepts 163 Algebraic Equation 163 Algebraic Number 163 Associative Law 164 Automorphism 164 Binomial Theorem 165 167 Boolean Algebra 166 Complex Number 168 174 Commutative Law 168 Cramer’s Rule 168 Degree of Freedom 169 Determinant 170 Discriminant 171 Distributive Law 171 Eigenvalue 172 Equation 172 Factor 173 Fundamental Theorem of Algebra 173 Gauss Elimination 174 Group 175 Group Theory 175 182 Hodge Conjecture 176 Homomorphism 176 Ideal 178 Imaginary Number 179 Injection 179 Irrational Number 180 Linear Equation 180 Liouville Number 181 Matrix 182 Multinomial Theorem 186 Parameter 187 Pascal’s Triangle 187 Polynomial 190 Quadratic Equation 191 Quaternion 192 Rational Number 192 Ring 193 Root 193 Square Root 195 Surjection 195 Synthetic Division 196 System of Equations 197 199 Variable 197 Vector 197 Vector Operations 200 Vector Space 200 Chapter 4: Trigonometry 202 History of Trigonometry 202 Classical Trigonometry 202 Ancient Egypt and the Mediterranean World 203 India and the Islamic World 206 Passage to Europe 207 217 Modern Trigonometry 209 From Geometric to Analytic Trigonometry 209 Application to Science 212 Principles of Trigonometry 214 Trigonometric Functions 214 Trigonometric Functions of an Angle 216 Tables of Natural Functions 219 Plane Trigonometry 219 Spherical Trigonometry 221 Analytic Trigonometry 222 223 Coordinates and Transformation of Coordinates 223
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