The Britannica Guide to Algebra and Trigonometry

281 Pages·2010·13.69 MB·English

Published in 2011 by Britannica Educational Publishing
(a trademark of Encyclopædia Britannica, Inc.)
in association with Rosen Educational Services, LLC
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First Edition
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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