unknown quantity a real and imaginary history of algebra - john derbyshire

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If you have comments, questions or just want more information about the books published by the National Academies Press and the Joseph Henry Press, you may contact our customer service department toll-free at 888-624-8373, visit us online, or send an email to comments@nap.edu. This book plus many more are available at http://www.jhpress.org. Copyright © All rights reserved. Distribution, posting, or copying is strictly prohibited without written permission of the National Academies Press. Request reprint permission for this book. ISBN: 0-309-65688-5, 400 pages, 6 x 9, (2006) This PDF is available from the Joseph Henry Press at: http://www.nap.edu/catalog/11540.html http://www.nap.edu/catalog/11540.html We ship printed books within 1 business day; personal PDFs are available immediately. Unknown Quantity: A Real and Imaginary History of Algebra John Derbyshire UNKNOW N QUANTITY John Derbyshire Joseph Henry Press Washington, D.C. A REAL AND IMAGINARY HISTORY OF ALGEBRA UNKNOW N QUANTITY Joseph Henry Press • 500 Fifth Street, NW • Washington, DC 20001 The Joseph Henry Press, an imprint of the National Academies Press, was created with the goal of making books on science, technology, and health more widely available to professionals and the public. Joseph Henry was one of the early founders of the National Academy of Sciences and a leader in early American science. Any opinions, findings, conclusions, or recommendations expressed in this volume are those of the author and do not necessarily reflect the views of the National Academy of Sciences or its affiliated institutions. Library of Congress Cataloging-in-Publication Data Derbyshire, John. The unknown quantity : a real and imaginary history of algebra / by John Derbyshire. p. cm. Includes bibliographical references and index. ISBN 0-309-09657-X (hardback : alk. paper) 1. Algebra—History. 2. Equations—History. 3. Algebra, Universal—History. 4. Algebra, Abstract— History. 5. Geometry, Algebraic—History. I. Title. QA151.D47 2006 512.009—dc22 2005037018 Cover image: J-L Charmet / Photo Researchers, Inc. Copyright © 2006 by John Derbyshire. All rights reserved. Printed in the United States of America. For Rosie VII CONTENTS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Math Primer: Numbers and Polynomials . . . . . . . . . . . . . . . . . 7 Part 1 The Unknown Quantity 1Four Thousand Years Ago . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2The Father of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3Completion and Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Math Primer: Cubic and Quartic Equations . . . . . . . . . . . 57 4Commerce and Competition . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5Relief for the Imagination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Part 2 Universal Arithmetic 6The Lion’s Claw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Math Primer: Roots of Unity . . . . . . . . . . . . . . . . . . . . . . . 109 7The Assault on the Quintic . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Math Primer: Ve ctor Spaces and Algebras . . . . . . . . . . . . 134 8The Leap into the Fourth Dimension . . . . . . . . . . . . . . . . . . 145 viii UNKNOWN QUANTITY 9An Oblong Arrangement of Terms . . . . . . . . . . . . . . . . . . . . 161 10 Victoria’s Brumous Isles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Part 3 Levels of Abstraction Math Primer: Field Theory . . . . . . . . . . . . . . . . . . . . . . . . 195 11 Pistols at Dawn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 12 Lady of the Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Math Primer: Algebraic Geometry . . . . . . . . . . . . . . . . . . 241 13 Geometry Makes a Comeback . . . . . . . . . . . . . . . . . . . . . . . . 253 14 Algebraic This, Algebraic That . . . . . . . . . . . . . . . . . . . . . . . 279 15 From Universal Arithmetic to Universal Algebra . . . . . . . . . 298 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Picture Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 INTRODUCTION §I.1 THIS BOOK IS A HISTORY OF ALGEBRA, written for the curious nonmathematician. It seems to me that the author of such a book should begin by telling his reader what algebra is. So what is it? Passing by an airport bookstore recently, I spotted a display of those handy crib sheets used by high school and college students, the ones that have all the basics of a subject printed on a folding triptych laminated in clear plastic. There were two of these cribs for algebra, titled “Algebra—Part 1” and “Algebra—Part 2.” Parts 1 and 2 com- bined (said the subheading) “cover principles for basic, intermediate, and college courses.” 1 I read through the material they contained. Some of the topics might not be considered properly algebraic by a professional math- ematician. “Functions,” for example, and “Sequences and Series” be- long to what professional mathematicians call “analysis.” On the whole, though, this is a pretty good summary of basic algebra and reveals the working definition of the word “algebra” in the modern American high school and college-basics curriculum: Algebra is the part of advanced mathematics that is not calculus. In the higher levels of math, however, algebra has a distinctive quality that sets it apart as a discipline by itself. One of the most fa- 1 [...]... books I have relied on most heavily are histories of algebra written by mathematicians for mathematicians: A History of Algebra by B L van der Waerden (1985) and The Beginnings and Evolution of Algebra by Isabella Bashmakova and Galina Smirnova (translated by Abe Shenitzer, 2000) I shall refer to these books in what follows just by the names of their authors (“van der Waerden says ”) One other major... Mathematical Cuneiform Texts in 1945, and this has remained a standard English-language work on Babylonian mathematics Investigations have of course continued, and the brilliance of the Babylonians is now clear to everyone In particular, we now know that they were masters of some techniques that can reasonably be called algebraic §1.4 Neugebauer discovered that the Hammurabi-era mathematical texts were of. .. to all well-instructed adherents of the great Western monotheistic religions This was the world of Abraham and Isaac, Jacob and Joseph, Ur and Haran, Sodom and Gomorrah The Western civilization of that time encompassed all of the Fertile Crescent, that nearly continuous zone of cultivable land that stretches northwest from the Persian Gulf up the plains of the Tigris and Euphrates, across the Syrian... especially in algebra The adjective “algebraic,” when used by mathematicians, can usually be translated as “concerned with polynomials.” Examine a theorem in algebra, even one at the very highest level By peeling off a couple of layers of meaning, you will very likely uncover a polynomial Polynomial has a fair claim to being the single most important concept in algebra, both ancient and modern Part... levels of abstraction Because the development of algebra was irregular and haphazard, in the way of all human things, I found it difficult to keep to a strictly chronological approach, especially through the 19th century I hope that my narrative makes sense nonetheless and that the reader will get a clear view of all the main lines of development §I.3 My aim is not to teach higher algebra to the reader... Euphrates, around the earlier of those dates Hammurabi10 was an Amorite, speaking a dialect of Akkadian He brought all of Mesopotamia under his rule and made Babylon the great city of the age This was the first Babylonian empire.11 This first Babylonian empire was a great record-keeping civilization Their writing was in the style called cuneiform, or wedgeshaped That is to say, written words were patterns... rarefied and demanding of all mental disciplines, whose objects are abstractions of abstractions of abstractions, yet whose results have a power and beauty that are all too little known outside the world of professional mathematicians Most amazing, most mysterious of all, these ethereal mental objects seem to contain, within their nested abstractions, the deepest, most fundamental secrets of the physical... centuries ago,9 and were written by people living in Mesopotamia and Egypt To a person of our time, that world seems inconceivably remote The year 1800 BCE was almost as far back in Julius Caesar’s past as Caesar is in ours Outside a small circle of specialists, the only widespread knowledge of that time and those places is the fragmentary and debatable account given in the Book of Genesis and thereby... Noether—some of the biggest names in early 20th-century math In the mid-1920s, Neugebauer’s interest turned to the mathematics of the ancient world He made a study of ancient Egyptian and published a paper about the Rhind Papyrus, of which I shall say more in a moment Then he switched to the Babylonians, learned Akkadian, and embarked on a study of tablets from the Hammurabi era The fruit of this work was the...2 UNKNOWN QUANTITY mous quotations in 20th-century math is this one, by the great German mathematician Hermann Weyl It appeared in an article he published in 1939 In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.2 Perhaps the reader knows that topology is a branch of geometry, sometimes called “rubber-sheet geometry,” dealing . unknown quantity : a real and imaginary history of algebra / by John Derbyshire. p. cm. Includes bibliographical references and index. ISBN 0-3 0 9-0 9657-X (hardback : alk. paper) 1. Algebra History. . at: http://www.nap.edu/catalog/11540.html http://www.nap.edu/catalog/11540.html We ship printed books within 1 business day; personal PDFs are available immediately. Unknown Quantity: A Real and Imaginary History of Algebra John Derbyshire. time, algebra has become the most rarefied and de- manding of all mental disciplines, whose objects are abstractions of abstractions of abstractions, yet whose results have a power and beauty that