A student in class asks the math teacher: "Shouldn't minus times minus make minus?" Teachers soon convince most students that it does not. Yet the innocent question brings with it a germ of mathematical creativity. What happens if we encourage that thought, odd and ungrounded though it may seem?
Few books in the field of mathematics encourage such creative thinking. Fewer still are engagingly written and fun to read. This book succeeds on both counts. Alberto Martinez shows us how many of the mathematical concepts that we take for granted were once considered contrived, imaginary, absurd, or just plain wrong. Even today, he writes, not all parts of math correspond to things, relations, or operations that we can actually observe or carry out in everyday life.
Negative Math ponders such issues by exploring controversies in the history of numbers, especially the so-called negative and "impossible" numbers. It uses history, puzzles, and lively debates to demonstrate how it is still possible to devise new artificial systems of mathematical rules. In fact, the book contends, departures from traditional rules can even be the basis for new applications. For example, by using an algebra in which minus times minus makes minus, mathematicians can describe curves or trajectories that are not represented by traditional coordinate geometry.
Clear and accessible, Negative Math expects from its readers only a passing acquaintance with basic high school algebra. It will prove pleasurable reading not only for those who enjoy popular math, but also for historians, philosophers, and educators.
- Uses history, puzzles, and lively debates to devise new mathematical systems
- Shows how departures from rules can underlie new practical applications
- Clear and accessible
- Requires a background only in basic high school algebra
Alberto A. Martinez teaches history of science and mathematics at the University of Texas, Austin. He studies history to better understand scientific creativity and to clarify ambiguities in the elements of physics and algebra.
"Alberto A. Martínez . . . shows that the concept of negative numbers has perplexed not just young students but also quite a few notable mathematicians. . . . The rule that minus times minus makes plus is not in fact grounded in some deep and immutable law of nature. Martínez shows that it's possible to construct a fully consistent system of arithmetic in which minus times minus makes minus. It's a wonderful vindication for the obstinate smart-aleck kid in the back of the class."--Greg Ross, American Scientist
"Alberto Martinez . . . has written an entire book about the fact that the product of two negative numbers is considered positive. He begins by reminding his readers that it need not be so. . . . The book is written in a relaxed, conversational manner. . . . It can be recommended to anyone with an interest in the way algebra was developed behind the scenes, at a time when calculus and analytic geometry were the main focus of mathematical interest."--James Case, SIAM News
"[Negative Math] is very readable and the style is entertaining. Much is done through examples rather than formal proofs. The writer avoids formal mathematical logic and the more esoteric abstract algebras such as group theory."--Mathematics Magazine
"An excellent book, truly readable and accurate. I repeatedly found myself intrigued and informed by Martínez's examples and approaches, which succeed in transforming competent historical analysis into an informative and thought-provoking meditation on mathematical meaning."--Joan L. Richards, Brown University
"Beautifully written. Accurate and reliable. The author's point, that mathematics is constructed according to our judgment of what will serve us, is very important and little understood."--Reuben Hersh, University of New Mexico
Table of Contents:
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
Chapter 7: Making a Meaningful Math 174
Finding Meaning 175
Designing Numbers and Operations 186
Physical Mathematics? 220
Further Reading 249
This book has been translated into: