Numbers: Computers, Philosophers, and the Search for Meaning

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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. Numbers is an insightful look at the properties and uses of numerical quantities, from fractions to algebraic numbers, transcendental numbers, and complex numbers. We rely on numbers to carry out countless daily activities -- from mapping the universe to running word-processing programs to buying lunch. Author John Tabak points out that numbers are a human invention, as seen through the compelling histories of Babylonian, Roman, and Arab thinkers and their influential systems for representing numbers. The book examines in detail the number pi, the evolution of the idea of infinity, the representation of numbers in computers, the metric and American systems of measurement, and the application of some historical concepts of numbers in such modern forms as cryptography and hand calculators. Numbers 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 for further research. 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: 9780816049554
  • Publisher: Facts on File, Incorporated
  • Publication date: 6/28/2004
  • Series: Facts on File Math Library
  • Pages: 240
  • Age range: 11 - 17 Years
  • Product dimensions: 6.10 (w) x 9.20 (h) x 0.80 (d)

Table of Contents

Acknowledgments x
Introduction: Number and Imagination xi
Part 1 Numbers for Computation 1
1 The First Problems 3
2 Early Counting Systems 11
A Mesopotamian Education 13
The Mesopotamian Number System 14
Mesopotamian Mathematics Homework 18
The Egyptian Number System 20
A Problem from the Ahmes Papyrus 24
The Mayan Number System 26
The Chinese Number System 30
A Problem from the Nine Chapters 32
3 Our Place Value Number System 34
Explaining the New System 38
4 Analytical Engines 46
Calculators, Computers, and the Human Imagination 48
Charles Babbage and the Analytical Engine 50
An Early Electronic Representation of Our Number System 52
Floating-Point Representation 56
Floating-Point Arithmetic and Your Calculator 58
Why Computers? 60
Part 2 Extending the Idea of a Number 65
5 An Evolving Concept of a Number 67
Irrational Numbers 70
Pythagoras of Samos 73
The Irrationality of [radical]2 75
6 Negative Numbers 77
Ancient Mathematical Texts from the Indian Subcontinent 82
Out of India 83
7 Algebraic Numbers 85
Tartaglia, Ferrari, and Cardano 88
Girard and Wallis 93
Euler and d'Alembert 97
The Debate over "Fictitious" Numbers 98
The Complex Numbers: A Modern View 103
Using Complex Numbers 104
8 Transcendental Numbers and the Search for Meaning 107
Dedekind and the Real Number Line 111
Part 3 The Problem of Infinity 117
9 Early Insights 119
10 Galileo and Bolzano 130
Infinity as a Number 136
The Life and Opinions of Tristram Shandy, Gentleman 140
11 Georg Cantor and the Logic of the Infinite 143
There Are No More Rational Numbers than Natural Numbers 146
There Are More Real Numbers than Natural Numbers 148
The Russell Paradox 154
Resolving the Russell Paradox 159
12 Cantor's Legacy 165
Kurt Godel 170
Formal Languages Today 173
Alan Turing 174
Chronology 181
Glossary 199
Further Reading 203
Index 213
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