The Britannica Guide to Algebra and Trigonometry

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ISBN: 9781615301133 | 1615301135
Cover: Hardcover
Copyright: 8/15/2010

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Calculating and manipulating the unknown has been the enterprise of the field of algebra since its earliest inception in Babylon and ancient Egypt. Trigonometry draws on principles presented in algebra and uses angle measurements to elaborate on geometric calculations. Essential to further mathematical and scientific study, both algebra and trigonometry provide crucial tools in managing variables and understanding the relationships between them. This volume presents the fundamentals of these fascinating areas of mathematics while chronicling their respective histories.

Introduction	p. 12
Algebra	p. 23
History of Algebra	p. 23
The Emergence of Formal Equations	p. 23
Problem Solving in Egypt and Babylon	p. 25
Greece and the Limits of Geometric Expression	p. 27
The Equation in India and China	p. 31
Islamic Contributions	p. 32
Commerce and Abacists in the European Renaissance	p. 35
Cardano and the Solving of Cubic and Quartic Equations	p. 38
Viète and the Formal Equation	p. 40
The Concept of Numbers	p. 41
Classical Algebra	p. 43
Analytic Geometry	p. 43
The Fundamental Theorem of Algebra	p. 45
Impasse with Radical Methods	p. 47
Galois Theory	p. 48
Applications of Group Theory	p. 51
Fundamental Concepts of Modern Algebra	p. 54
Systems of Equations	p. 57
Quaternions and Vectors	p. 61
The Close of the Classical Age	p. 63
Structural Algebra	p. 63
Precursors to the Structural Approach	p. 64
The Structural Approach Dominates	p. 66
Algebraic Superstructures	p. 67
New Challenges and Perspectives	p. 69
Branches of Algebra	p. 70
Elementary Algebra	p. 71
Algebraic Quantities	p. 71
Algebraic Expressions	p. 73
Solving Algebraic Equations	p. 75
Solving Systems of Algebraic Equations	p. 76
Linear Algebra	p. 78
Vectors and Vector Spaces	p. 78
Linear Transformations and Matrices	p. 81
Eigenvectors	p. 82
Modern Algebra	p. 83
Basic Algebraic Structures	p. 83
Field Axioms	p. 85
Rings	p. 86
Group Theory	p. 88
Great Algebraists	p. 91
Early Algebraists (Through the 16th Century)	p. 91
Bhaskara II	p. 91
Brahmagupta	p. 93
Girolamo Cardano	p. 94
Diophantus of Alexandria	p. 96
Lodovico Ferrari	p. 99
Scipione Ferro	p. 100
al-Karaji	p. 101
al-Khwarizmi	p. 102
Liu Hui	p. 103
Mahavira	p. 105
Qin Jiushao	p. 106
Classical Algebraists (17th-19th Centuries)	p. 107
Niels Henrik Abel	p. 107
Bernhard Bolzano	p. 110
George Boole	p. 111
Arthur Cayley	p. 113
Évariste Galois	p. 116
Carl Friedrich Gauss	p. 119
Sir William Rowan Hamilton	p. 124
Charles Hermite	p. 129
Felix Klein	p. 130
Leopold Kronecker	p. 131
Ernst Eduard Kummer	p. 132
Sophus Lie	p. 133
Joseph Liouville	p. 135
Paolo Ruffini	p. 137
Seki Takakazu	p. 138
James Joseph Sylvester	p. 140
François Viète	p. 142
Algebraists of the Structural Period (20th Century-)	p. 143
Emil Artin	p. 143
Richard Ewen Borcherds	p. 144
Nicolas Bourbaki	p. 144
Richard Dagobert Brauer	p. 145
Élie-Joseph Cartan	p. 146
George Dantzig	p. 147
Leonard Eugene Dickson	p. 148
Jean Dieudonné	p. 149
Georg Frobenius	p. 149
Aleksandr Osipovich Gelfond	p. 150
David Hilbert	p. 151
Saunders Mac Lane	p. 154
Gregori Aleksandrovich Margulis	p. 155
Emmy Noether	p. 156
Daniel Gray Quillen	p. 158
Alfred Tarski	p. 159
Hermann Weyl	p. 160
Efim Isaakovich Zelmanov	p. 161
Algebraic Terms and Concepts	p. 163
Algebraic Equation	p. 163
Algebraic Number	p. 163
Associative Law	p. 164
Automorphism	p. 164
Binomial Theorem	p. 165
Boolean Algebra	p. 166
Complex Number	p. 168
Commutative Law	p. 168
Cramer's Rule	p. 168
Degree of Freedom	p. 169
Determinant	p. 170
Discriminant	p. 171
Distributive Law	p. 171
Eigenvalue	p. 172
Equation	p. 172
Factor	p. 173
Fundamental Theorem of Algebra	p. 173
Gauss Elimination	p. 174
Group	p. 175
Group Theory	p. 175
Hodge Conjecture	p. 176
Homomorphism	p. 176
Ideal	p. 178
Imaginary Number	p. 179
Injection	p. 179
Irrational Number	p. 180
Linear Equation	p. 180
Liouville Number	p. 181
Matrix	p. 182
Multinomial Theorem	p. 186
Parameter	p. 187
Pascal's Triangle	p. 187
Polynomial	p. 190
Quadratic Equation	p. 191
Quaternion	p. 192
Rational Number	p. 192
Ring	p. 193
Root	p. 193
Square Root	p. 195
Surjection	p. 195
Synthetic Division	p. 196
System of Equations	p. 197
Variable	p. 197
Vector	p. 197
Vector Operations	p. 200
Vector Space	p. 200
Trigonometry	p. 202
History of Trigonometry	p. 202
Classical Trigonometry	p. 202
Ancient Egypt and the Mediterranean World	p. 203
India and the Islamic World	p. 206
Passage to Europe	p. 207
Modern Trigonometry	p. 209
From Geometric to Analytic Trigonometry	p. 209
Application to Science	p. 212
Principles of Trigonometry	p. 214
Trigonometric Functions	p. 214
Trigonometric Functions of an Angle	p. 216
Tables of Natural Functions	p. 219
Plane Trigonometry	p. 219
Spherical Trigonometry	p. 221
Analytic Trigonometry	p. 222
Coordinates and Transformation of Coordinates	p. 223
Polar Coordinates	p. 223
Transformation of Coordinates	p. 225
Great Trigonometricians	p. 227
Aryabhata I	p. 227
al-Battani	p. 229
Abraham de Moivre	p. 230
Leonhard Euler	p. 231
James Gregory	p. 234
Hipparchus	p. 238
Lover of Truth	p. 238
Solar and Lunar Theory	p. 240
Other Scientific Work	p. 243
Menelaus of Alexandria	p. 245
Ptolemy	p. 246
Astronomer	p. 248
Mathematician	p. 250
Geographer	p. 251
Regiomontanus	p. 252
Nasir al-Din al-Tusi	p. 256
Trigonometric Terms and Concepts	p. 259
Alfonsine Tables	p. 259
Almagest	p. 259
Law of Cosines	p. 262
Fourier Series	p. 262
Hyperbolic Function	p. 262
Law of Sines	p. 263
Trigonometric Function	p. 263
Trigonometry Table	p. 264
Triangulation	p. 266
Glossary	p. 268
Bibliography	p. 270
Index	p. 273
Table of Contents provided by Ingram. All Rights Reserved.

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