Negative Math: How Mathematical Rules Can Be Positively Bent

280 Pages·2005·2.13 MB·English

by  Alberto A. Martinez

negative math

negative math
HOW MATHEMATICAL RULES
CAN BE POSITIVELY BENT
An easy introduction to the study of developing
algebraic rules to describe relations among things
ALBERTO A. MARTÍNEZ
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright © 2006 by Princeton University Press
Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock,
Oxfordshire OX20 1SY
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Martínez, Alberto A., date.
Negative math : how mathematical rules can be positively bent / Alberto A. Martínez.
p. cm.
“An easy introduction to the study of developing algebraic rules to describe relations
among things.”
Includes bibliographical references and index.
ISBN-13: 978-0-691-12309-7 (cloth : alk. paper)
ISBN-10: 0-691-12309-8 (cloth : alk. paper)
1. Mathematics. 2. Numbers, Negative. I. Title.
QA155.M28 2005
510–dc22 2005043377
British Library Cataloging-in-Publication Data is available
This book has been composed in Sabon with Hel Neue Family Display
Printed on acid-free paper. ∞
pup.princeton.edu
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
You can use a spoon to drive a screw into a wall. With
practice, you can become skillful at it. You can also
learn many juggling tricks with the spoon, and thus im-
press and bewilder people who don’t juggle spoons.
And you can make all of this more puzzling by calling
the spoon a “fork.” And you can write books about it
and form societies with other people who also juggle
spoons called forks. And even then, sure, you can use a
spoon to drive screws into a wall.
But a screwdriver is better. And even if you’ve never
seen a screwdriver, you can just as well invent one. It
might resemble the spoon in some ways though not
in others. So you can keep your spoon as well; for eat-
ing soup, for juggling, or even, occasionally, for driving
screws into walls. At least until you have more skill
with a better tool.

contents
figures ix
chapter 1 Introduction 1
chapter 2 The Problem 10
chapter 3 History: Much Ado About
Less than Nothing 18
The Search for Evident Meaning 36
chapter 4 History: Meaningful and Meaningless
Expressions 43
Impossible Numbers? 66
chapter 5 History: Making Radically New
Mathematics 80
From Hindsight to Creativity 104
chapter 6 Math Is Rather Flexible 110
Sometimes –1 Is Greater than Zero 112
Traditional Complications 115
Can Minus Times Minus Be Minus? 131
Unity in Mathematics 166
viii — CONTENTS
chapter 7 Making a Meaningful Math 174
Finding Meaning 175
Designing Numbers and Operations 186
Physical Mathematics? 220
notes 235
further reading 249
acknowledgments 259
index 261
figures
figure 1. A number line. 11
figure 2. The sum of a positive number and an imaginary
number, represented as a line. 45
figure 3. The geometric “sum” of two straight lines. 63
figure 4. The complex number 4 + 5i, represented as a
diagonal line. 75
figure 5. A rectangular triangle. 76
figure 6. A number line. 114
figure 7. A diagram of the Pythagorean theorem. 144
figure 8. A triangle constructed on a plane of positive
and imaginary numbers. 146
figure 9. A representation of numbers as units on
perpendicular lines. 148
figure 10. Lines that are “rotated” 90° from one another. 148
figure 11. Lines determined by the same two numbers,
but in different order. 150
figure 12. Lines “rotated” 90° from one another, without
imaginary numbers. 151