Negative Math: How Mathematical Rules Can Be Positively Bent [Hardcover]

Alberto A. Martínez (Author)

Book Description

ISBN-10: 0691123098 | ISBN-13: 978-0691123097 | Publication Date: November 7, 2005

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.

Key Features:

• 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

Editorial Reviews

From Publishers Weekly

It's a rare person who describes negative numbers (or any numbers) as "unassuming but fun," and he is likely the same person who would notice that negative numbers "stand as just about the only kind of numbers about which a book has not been written." That man is Martinez, and in this book, he touches on mathematics history and great mathematical squabbles about the "evident meaning" of negative numbers, all with the goal of sexing up negative numbers and proposing a "meaningful math" that could rekindle the "connection between mathematical truth and physical experience." No small feat, and the outcome is a qualified success: he writes with clarity and provides context (French novelist Henri Beyle resented the notion that two negatives make a positive) that helps layreaders to deal with abstruse subject matter, but many of his canny re-interpretations of mathematical laws depend on questionable means, such as rejiggering "the definition that we choose to give to the = sign." English majors who never understood why they were required to take math classes may enjoy Martinez's blend of humanism and philosophy, and number-people will certainly want to give this a look.
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Review

It is fair to say that Negative Math completely blew my mind. . . . Martínez's superb writing makes even the most subtle arguments and paradoxes seem obvious, but don't expect this short book to be easy sailing. It will set your mind racing, although every page is absolutely worth the effort. -- Plus Magazine, University of Cambridge

this is a serious-minded and interesting book. . . . The first part of the book, which I enjoyed immensely, is a history of the struggles of mathematicians to cope with the idea of negative numbers. It is enormously encouraging . . . intriguing and provocative. . . -- The Mathematical Intelligencer

The author has committed himself to having this writing and this subject matter accessible to the general reader, and he has succeeded to a remarkable degree . . . For the teacher currently involved with these concepts, this innovative work should provide useful background and prove to be an outstanding read. -- The Mathematics Teacher

a book that is at once scholarly and readable . . . anyone with an interest in intellectual history would benefit . . . Martínez's book has the potential to cause the generation of many golden fibers that can be used in weaving the fabric of mathematics. -- Books & Culture

It is interesting and to a certain extent inspiring to look at this fundamental transformation of mathematics with the eyes of algebra and not as usual from the point of view of non-Euclidean geometry . . .  whoever follows author will be inspired and forced to think about problems which he never put himself before. -- Zentralblatt MATH


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

See all Editorial Reviews

Product Details

• Hardcover: 280 pages
• Publisher: Princeton University Press (November 7, 2005)
• Language: English
• ISBN-10: 0691123098
• ISBN-13: 978-0691123097
• Product Dimensions: 9.4 x 6.4 x 1.1 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 3.9 out of 5 stars  See all reviews (8 customer reviews)
• Amazon Best Sellers Rank: #1,197,878 in Books (See Top 100 in Books)

More About the Author

I'm originally from San Juan, Puerto Rico. I write about science and history, and I'm especially interested in neglected controversies and disagreements in the elements of physics and math. I've been a Research Fellow at The Smithsonian, M.I.T., Harvard, Boston University, and Caltech, and I'm a tenured professor at the University of Texas at Austin. My new book is about myths in science: Science Secrets: The Truth About Darwin's Finches, Einstein's Wife, and Other Myths.

Here's my website at UT:
https://webspace.utexas.edu/aam829/1/m/About.html
It includes articles, book overviews, interviews, and book reviews.

Customer Reviews

Most Helpful Customer Reviews

27 of 28 people found the following review helpful:

4.0 out of 5 stars Very uneven, but interesting, January 4, 2006

By 

Bruce R. Gilson (Wheaton, MD United States) - See all my reviews
(VINE VOICE)    (REAL NAME)   

This review is from: Negative Math: How Mathematical Rules Can Be Positively Bent (Hardcover)

This book was quite hard to review. There are parts I found extremely interesting, and other parts I thought were full of sloppy thinking and misleading analogies. But overall, I think it is a worthwhile book to read. I think it is appropriate to divide the book into four separate sections, each of which deserves to be reviewed separately.

The first part is an attempt to show that some of the rules of algebra (particularly the rules for manipulating signs) are really counter-intuitive, and also an attempt to gain the perspective of an elementary algebra student who cannot understand why the rules are what they are. It is this part that I think is the worst part of the book, and in his attempts to show that the rules are counter-intuitive, all he manages to do is show that _his_ intuition works quite differently from _my_ intuition. This part is the section in which I found the sloppy thinking and resort to false analogy which I mentioned earlier. It seemed to me that there were things the author just didn't understand, but as I read further in the book, I found that he actually understood them even though he didn't seem to at first. This section would get three stars if I felt generous, or even two, if I were to review it alone.

The second part (actually intermixed with the first in its location in the book) describes the difficulties that mathematicians (even great ones) had in comprehending the concept of negative and imaginary numbers, and as such it provides some historical background for the rest of the book, which justifies its inclusion. If I were to review this part by itself, it would get three stars, meaning "it's OK," but it hardly justifies the book.

The best part of the book is the third. This is a very interesting attempt to come up with an algebra that differs from the usual, where he has to maintain consistency, and so he looks deeply into questions as to what further modifications to traditional algebra have to be made to go along with a postulated change. Much like the introduction of non-Euclidean geometry, the process leads to an odd-looking algebra, but one which fits together, and it is this part that I liked well enough to rate as five-star, bringing the overall rating for the book to four. This part made the book worthwhile for me.

Finally, the author ends with one very long chapter that probably summarizes what _he_ wants the book to be, though the previous section is what _I_ want of the book. He advocates a concept of a mathematics that would be suited to explaining problems of physics in a more natural manner, even if it might look different from traditional mathematics. I would have been happier if this part were shorter, though I think the author himself probably considered this part to be the major thesis of the book and this is why he devoted so much of the book to this part. This part is actually interesting enough that I'd rate it 4 stars, though as I said I'd prefer it more streamlined.

That is an overview of the book: very uneven, with both very good parts and bad, but if everything is all combined, the total package is a pretty good one.

20 of 20 people found the following review helpful:

5.0 out of 5 stars Very unusual, January 16, 2006

By 

Marco DiCola - See all my reviews

This review is from: Negative Math: How Mathematical Rules Can Be Positively Bent (Hardcover)

The writing is strangely impersonal, though I'm not sure what's weird about it. At least it's easy to understand, no big equations, no calculus. The book has several historical parts, especially on negative numbers, which used to be considered fictitious and "evil" and I had not imagined that there used to be so many controversies and disagreements on them for centuries. The author essentially traces the birth of revolutionary algebras to controversies on the negative sign, just as non-Euclidean geometries came from controversies on the parallel postulate. Euler, D'Alembert, Carnot, De Morgan, Playfair, Hamilton, Frege and a bunch of others show up. There are good passages on Bishop Berkeley defending free-thinking in
mathematics.

Before reading this book, I always read and believed that minus times minus must be positive because of the distributive rule. But this book argues that the distributive rule is no more special than the commutative rule, and that accordingly it too can be restricted or rejected just like the universality of the commutative rule was rejected when the theories of quaternions and vectors were invented. Then the book presents an algebra in which minus times minus is minus, something which I haven't seen before. It's essentially a non-commutative algebra, and I expected that therefore the distributive rule would not hold, but surprisingly it does, albeit in a non-commutative form. A good part of this, I think, is where the author shows that this algebra can trace weird curves that are not given by analytic geometry. The last part of the book seems to criticize physics for restraining itself to mathematical methods that historically were not designed for its purposes. Apparently the author wants us to rush out to develop new "artificial" mathematics; but it's not clear to me what's so bad about the methods already in use. Still he wants to develop physical algebras just like mathematicians have developed physical geometries.

I give it a high rating because again and again I found myself agreeing with ideas that at first seemed ridiculous, such as that -1 is not necessarily less than zero. It has a lot of unusual reasonable ways of looking at the elements of math. Oh, but one more thing, I don't exactly understand what's the point of the spoon.

11 of 12 people found the following review helpful:

5.0 out of 5 stars Can you bend the rules of math?, March 9, 2007

By 

Hagios (Rhode Island) - See all my reviews

This review is from: Negative Math: How Mathematical Rules Can Be Positively Bent (Hardcover)

Consider the number line. It is very symmetrical. Zero is in the middle, with the positive numbers to the right, and the negative numbers to the left. But that symmetry rapidly breaks down under multiplication; if multiplication were symmetrical then multiplying two negative numbers should produce another negative number, but it doesn't. The symmetry breaks down even more when you take square roots. The square root of 4 is plus or minus 2. The square root of -4 isn't even a real number; it is plus or minus 2i. Here is another violation of the symmetry of the number line : 2^2 is 4, but -2^-2 is 1/4. That is weird.

Martinez develops an algebra that restores the symmetry of the number line under multiplication, while simultaneously dispensing with imaginary numbers. All you have to do is change the rule of multiplication so that a negative number times a negative number is still negative. Now the square root of -4 is -2. This also gets rid of the double roots for square numbers. It also makes -2^-2 = -4.

This algebra runs into some problems. For one thing, multiplication is not commutative. This seems odd, but we're already familiar with non commutative operations. Some examples include subtraction, division, and matrix multiplication. Martinez smoothes over this issue and a couple other potential pitfalls. He also shows that you can actually create simpler solutions to some problems in mathematics. Martinez's algebra also does a better job of corresponding to the real world. Thinking of negative numbers as "moving in the other direction" results in an arithmetic that does a better job of applying to the real world.

I would highly recommend this book because I'll never think of mathematics, or numbers, in the same way again. Martinez's experiment really forces you to think about what numbers mean, and what possible real world or geometric interpretation they may have. But having said that, I think his experiment is destined to be a failure.

Here is an example of how Martinez's algebra breaks down. 5 = (10 - 5), so 5 x 5 should be the same as (10 - 5) x (10 - 5). But according to Martinez's algebra, it is 75 (you can work this out - use the sign of the first number only for the inner terms). This experiment also provides insight into how negative numbers might work. We can think of -5 x -5 as ( 0 - 5) x (0 - 5) = 25. But according to Martinez's algebra, it is defined to be -25. [UPDATE: Martinez kindly explains in the comment that the artificial algebra uses a different distribution rule than traditional algebra, a nuance that I missed in this review)

Martinez's algebra also breaks down because you cannot use logarithms as a shorthand for division. Lets pick an easy example to demonstrate the point, dividing 4 by 8. The way you do this is to express them both to a common base and subtract the exponents. So you get log( 2^2 ) - log( 2^3 ) = log( 2^-1 ) = -1. Then you use re-exponentiate to get your answer. In traditional algebra, 2^-1 = 1/2 = 4/8. That is the correct answer. But with Martinez's algebra, 2^-1 = -2, which is the wrong answer.

I should point out at this point that Martinez successfully works out a few kinks that originally appear as though they would doom his system. So it is quite possible that someone with more mathematical maturity could figure out a way around these obstacles. But my instinct is that Martinez's system has run into a dead end. The real lesson I have taken from the experiment is that math isn't easy to bend, but I thoroughly enjoyed the attempt.