Algebra: Sets, Symbols, and the Language of Thought

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Overview

The History of Mathematics is a fascinating survey of the development of math through discovery, innovation, collaboration, and experimentation. The set presents a compelling overview of myriad aspects of mathematics using understandable language and appealing line illustrations and photographs. Algebra is a thorough examination not only of how and why this branch of mathematics arose and flourished in different cultures at different times but also of its useful applications in science and in society. Author John Tabak pinpoints the foundations of algebra more than 4,000 years ago in Mesopotamia and traces its progress in other ancient civilizations around the world, including the Hindu, Greek, and Arabic cultures, and at different points in history. Mostly rhetorical in its early forms, algebra in the symbolic form that we use today was formalized in the 17th century. Over the past three centuries, algebra has taken two diverging paths: one toward increasingly higher levels of abstraction, the other toward more concrete computational methods. Tabak explores algebra as an indispensable part of every branch of mathematics and a widely utilized tool in scientific and engineering research. Algebra includes an index, a chronology of notable events, a glossary of terms, a helpful list of Internet resources, and an array of historical and current print sources. Keyed to current principles and standards in teaching math, The History of Mathematics set is essential for young readers who require information on relevant topics in mathematics.
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Editorial Reviews

School Library Journal
Gr 10-Up-In uniform but independent volumes, Tabak offers sustained, meaty historical and methodological studies of the major branches of mathematics. Beginning, in some cases, with prehistoric evidence, he traces: developments in our very idea of what algebra is; the nature of ancient, projective (including non-Euclidian), and coordinate geometry; the uses of mathematics in finding precise ways to describe natural laws; the "invention" of rational, irrational and imaginary numbers, and the idea of infinity; and advances in probability theory and statistics. In each volume he analyzes the insights and accomplishments of many thinkers, ancient and modern, providing a generous array of illuminating demonstrations and examples while keeping extraneous biographical details-and, for that matter, illustrations-to a minimum, then closes with a time line, a specialized glossary, and annotated, multimedia reading lists. General readers may prefer to start with such single-volume histories as William Berlinghoff and Fernando Gouvea's Math through the Ages (Oxton House, 2002) or Ivor Grattan-Guinness's Rainbow of Mathematics (Norton, 2000), but for serious students of the subject, and collections supporting strong science programs, these make appropriate additions to both reference and circulating shelves.-John Peters, New York Public Library Copyright 2004 Reed Business Information.
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Product Details

  • ISBN-13: 9780816049547
  • Publisher: Facts on File, Incorporated
  • Publication date: 6/28/2004
  • Series: Facts on File History of Mathematics Series
  • Pages: 240
  • Age range: 14 - 17 Years
  • Product dimensions: 6.28 (w) x 9.34 (h) x 0.81 (d)

Table of Contents

Introduction: Algebra as Language xi
1 The First Algebras 1
Mesopotamia: The Beginnings of Algebra 2
Mesopotamians and Second-Degree Equations 5
The Mesopotamians and Indeterminate Equations 7
Clay Tablets and Electronic Calculators 8
Egyptian Algebra 10
Chinese Algebra 12
Rhetorical Algebra 16
2 Greek Algebra 18
The Discovery of the Pythagoreans 19
The Incommensurability of [radical]2 24
Geometric Algebra 25
Algebra Made Visible 27
Diophantus of Alexandria 31
3 Algebra from India to Northern Africa 35
Brahmagupta and the New Algebra 38
Mahavira 42
Bhaskara and the End of an Era 44
Islamic Mathematics 46
Poetry and Algebra 47
Al-Khwarizmi and a New Concept of Algebra 50
A Problem and a Solution 53
Omar Khayyam, Islamic Algebra at Its Best 54
Leonardo of Pisa 59
4 Algebra as a Theory of Equations 60
The New Algorithms 63
Algebra as a Tool in Science 69
Francois Viete, Algebra as a Symbolic Language 71
Thomas Harriot 75
Albert Girard and the Fundamental Theorem of Algebra 79
Further Attempts at a Proof 83
Using Polynomials 88
5 Algebra in Geometry and Analysis 91
Rene Descartes 95
Descartes on Multiplication 98
Pierre de Fermat 102
Fermat's Last Theorem 105
The New Approach 106
6 The Search for New Structures 110
Niels Henrik Abel 112
Evariste Galois 114
Galois Theory and the Doubling of the Cube 117
Doubling the Cube with a Straightedge and Compass Is Impossible 120
The Solution of Algebraic Equations 122
Group Theory in Chemistry 127
7 The Laws of Thought 130
Aristotle 130
Gottfried Leibniz 133
George Boole and the Laws of Thought 137
Boolean Algebra 141
Aristotle and Boole 144
Refining and Extending Boolean Algebra 146
Boolean Algebra and Computers 149
8 The Theory of Matrices and Determinants 153
Early Ideas 155
Spectral Theory 159
The Theory of Matrices 166
Matrix Multiplication 172
A Computational Application of Matrix Algebra 175
Matrices in Ring Theory 177
Chronology 179
Glossary 197
Further Reading 203
Index 213
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