algebra sets, symbols, and the language of thought THE HISTORY OF M AT H E M A T I C S www.com THE HISTORY OF algebra sets, symbols, and the language of thought John Tabak, Ph.com ALGEBRA: Sets, Symbols, and the Language of Thought Copyright © 2004 by John Tabak, Ph. Permissions appear after relevant quoted material. All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval systems, without permission in writing from the pub- lisher.
For information contact: Facts On File, Inc. 132 West 31st Street New York NY 10001 Library of Congress Cataloging-in-Publication Data Tabak, John. Algebra : sets, symbols, and the language of thought / John Tabak. — (History of mathematics) Includes bibliographical references and index.T33 2004 512—dc222003017338 Facts On File books are available at special discounts when purchased in bulk quanti- ties for businesses, associations, institutions or sales promotions.
Please call our Special Sales Department in New York at (212) 967-8800 or (800) 322-8755. You can find Facts On File on the World Wide Web at http://www.com Text design by David Strelecky Cover design by Kelly Parr Illustrations by Sholto Ainslie Printed in the United States of America MP FOF 10 9 8 7 6 5 4 3 2 1 This book is printed on acid-free paper.com To Diane Haber, teacher, mathematician, and inspirator.com 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 √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-Khwārizmı̄ and a New Concept of Algebra 50 A Problem and a Solution 53 Omar Khayyám, Islamic Algebra at Its Best 54 Leonardo of Pisa 59 www.com 4 Algebra as a Theory of Equations 60 The New Algorithms 63 Algebra as a Tool in Science 69 François Viète, 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 René 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 Évariste 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 www.com 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 www.com INTRODUCTION ALGEBRA AS LANGUAGE algebra n. a generalization of arithmetic in which letters representing numbers are combined according to the rules of arithmetic 2. any of various systems or branches of mathematics or logic concerned with the properties and relationships of abstract enti- ties (as complex numbers, matrices, sets, vectors, groups, rings, or fields) manipulated in symbolic form under operations often analogous to those of arithmetic (By permission.
From Merriam-Webster’s Collegiate Dictionary, 10th ed.: Merriam-Webster, 2002) Algebra is one of the oldest of all branches of mathematics. Its history is as long as the history of civilization, perhaps longer. The well-known historian of mathematics B. van der Waerden believed that there was a civilization that preceded the ancient civilizations of Mesopotamia, Egypt, China, and India and that it was this civilization that was the root source of most early mathe- matics.
This hypothesis is based on two observations: First, there were several common sets of problems that were correctly solved in each of these widely separated civilizations. Second, there was an important incorrectly solved problem that was common to all of these lands. Currently there is not enough evidence to prove or disprove his idea. We can be sure, however, that algebra was used about 4,000 years ago in Mesopotamia.
We know that some remarkably similar problems, along with their algebraic solutions, can be found on Egyptian papyri, Chinese paper, and xi www.com xii ALGEBRA Mesopotamian clay tablets. We can be sure that algebra was one of the first organized intellectual activities carried out by these early civilizations. Algebra, it seems, is as essential and as “natural” a human activity as art, music, or religion. No branch of mathematics has changed more than algebra.
Geometry, for example, has a history that is at least as old as that of algebra, and although geometry has changed a lot over the millennia, it still feels geometric. A great deal of geometry is still concerned with curves, surfaces, and forms. Many contemporary books and articles on geometry, as their ancient counterparts did, include pictures, because modern geometry, as the geometry of these ancient civilizations did, still appeals to our intuition and to our experience with shapes. It is very doubtful that Greek geometers, who were the best mathematicians of antiquity, would have understood the ideas and techniques used by contemporary geometers.
Geometry has changed a great deal during the intervening millennia. Still, it is at least probable that those ancient Greeks would have recognized modern geometry as a kind of geometry. The same cannot be said of algebra, in which the subject matter has changed entirely. Four thousand years ago, for example, Mesopotamian mathematicians were solving problems like this: Given the area and perimeter of a plot of rectangular land, find the dimensions of the plot.
This type of problem seems practical; it is not. Despite the refer- ence to a plot of land, this is a fairly abstract problem. It has little practical value. How often, after all, could anyone know the area and perimeter of a plot of land without first knowing its dimen- sions? So we know that very early in the history of algebra there was a trend toward abstraction, but it was a different kind of abstraction than what pervades contemporary algebra.
Today mathematicians want to know how algebra “works.” Their goal is to understand the logical structure of algebraic systems. The search for these logical structures has occupied much of the last hundred years of algebraic research. Today mathematicians who do research in the www.com Introduction xiii field of algebra often focus their attention on the mathematical structure of sets on which one or more abstract operations have been defined—operations that are somewhat analogous to addi- tion and multiplication. We can illustrate the difference between modern algebra and ancient algebra by briefly examining a very important subfield of contemporary algebra.
It is called group theory, and its subject is the mathematical group. Roughly speaking, a group is a set of objects on which a single operation, somewhat similar to ordinary multiplication, is defined. Investigating the mathematical proper- ties of a particular group or class of groups is a very different kind of undertaking from solving the rectangular-plots-of-land prob- lem described earlier. The most obvious difference is that group theorists study their groups without reference to any nonmathe- matical object—such as a plot of land or even a set of numbers— that the group might represent.
Group theory is solely about (mathematical) groups. It can be a very inward looking discipline. By way of contrast with the land problem, we include here a famous statement about finite groups. (A finite group is a group with only finitely many elements.) The following statement was first proved by the French mathematician Augustin-Louis Cauchy (1789–1857): Let the letter G denote a finite group.
Let N represent the number of elements in G. Let p represent a prime number. If p (evenly) divides N then G has an element of order p. You can see that the level of abstraction is much higher in this statement than in the rectangular-plot-of-land problem.
To many well-educated laypersons it is not even clear what the statement means or even whether it means anything at all. Ancient mathematicians, as would most people today, would have had a difficult time seeing what group theory, one of the most important branches of contemporary mathematical research, and the algebraic problems of antiquity have in com- mon. In many ways, algebra, unlike geometry, has evolved into something completely new.com xiv ALGEBRA As algebra has become more abstract, it has also become more important in the solution of practical problems. Today it is an indispensable part of every branch of mathematics.
The sort of algebraic notation that we begin to learn in middle school—“let x represent the variable”—can be found at a much higher level and in a much more expressive form throughout all contemporary mathematics. Furthermore it is now an important and widely uti- lized tool in scientific and engineering research. It is doubtful that the abstract algebraic ideas and techniques so familiar to mathe- maticians, scientists, and engineers can even be separated from the algebraic language in which those ideas are expressed. Algebra is everywhere.
This book begins its story with the first stirrings of algebra in ancient civilizations and traces the subject’s development up to modern times. Along the way, it examines how algebra has been used to solve problems of interest to the wider public. The book’s objective is to give the reader a fuller appreciation of the intellec- tual richness of algebra and of its ever-increasing usefulness in all of our lives.com 1 the first algebras Mesopotamian ziggurat at Ur. For more than two millennia Mesopotamia was the most mathematically advanced culture on Earth.
(The Image Works) How far back in time does the history of algebra begin? Some scholars begin the history of algebra with the work of the Greek mathematician Diophantus of Alexandria (ca. It is easy to see why Diophantus is always included. His works contain problems that most modern readers have no difficulty rec- ognizing as algebraic. Other scholars begin much earlier than the time of Diophantus.
They believe that the history of algebra begins with the mathe- matical texts of the Mesopotamians. The Mesopotamians were a people who inhabited an area that is now inside the country of Iraq. Their written records begin about 5,000 years ago in the city-state of Sumer. The Sumerian method of writing, called 1 www.com 2 ALGEBRA cuneiform, spread throughout the region and made an impact that outlasted the nation of Sumer.
The last cuneiform texts, which were written about astronomy, were made in the first century A., about 3,000 years after the Sumerians began to represent their language with indentations in clay tablets. The Mesopotamians were one of the first, perhaps the first, of all literate civilizations, and they remained at the forefront of the world’s mathematical cultures for well over 2,000 years. Since the 19th century, when archaeologists began to unearth the remains of Mesopotamian cities in search of clues to this long-forgotten culture, hundreds of thousands of their clay tablets have been recovered. These include a number of mathematics tablets.
Some tablets use mathematics to solve scientific and legal problems—for example, the timing of an eclipse or the division of an estate. Other tablets, called problem texts, are clearly designed to serve as “textbooks.” Mesopotamia: The Beginnings of Algebra We begin our history of algebra with the Mesopotamians. Not everyone believes that the Mesopotamians knew algebra.