Unknown Quantity: A Real and Imaginary History of Algebra

391 Pages·2006·3.91 MB·English

by  John Derbyshire

UNKNOWN
QUANTITY

UNKNOWN
QUANTITY
A R E A L A N D
I M A G I N A R Y H I S T O R Y
O F A L G E B R A
John Derbyshire
Joseph Henry Press
Washington, D.C.
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

C
ONTENTS
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Math Primer: Numbers and Polynomials . . . . . . . . . . . . . . . . . 7
Part 1
The Unknown Quantity
1 Four Thousand Years Ago . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 The Father of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Completion and Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Math Primer: Cubic and Quartic Equations . . . . . . . . . . . 57
4 Commerce and Competition . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5 Relief for the Imagination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Part 2
Universal Arithmetic
6 The Lion’s Claw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Math Primer: Roots of Unity . . . . . . . . . . . . . . . . . . . . . . . 109
7 The Assault on the Quintic . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Math Primer: Vector Spaces and Algebras . . . . . . . . . . . . 134
8 The Leap into the Fourth Dimension . . . . . . . . . . . . . . . . . . 145
VII
viii UNKNOWN QUANTITY
9 An 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
I
NTRODUCTION
§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