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Negative Math: How Mathematical Rules Can Be Positively Bent

4.1 out of 5 stars 8 customer reviews
ISBN-13: 978-0691123097
ISBN-10: 0691123098
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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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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

Product Details

  • Hardcover: 288 pages
  • Publisher: Princeton University Press (November 27, 2005)
  • Language: English
  • ISBN-10: 0691123098
  • ISBN-13: 978-0691123097
  • Product Dimensions: 6.4 x 1 x 9.4 inches
  • Shipping Weight: 1.2 pounds (View shipping rates and policies)
  • Average Customer Review: 4.1 out of 5 stars  See all reviews (8 customer reviews)
  • Amazon Best Sellers Rank: #884,587 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

By Bruce R. Gilson on January 4, 2006
Format: 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.
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Format: 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

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.
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Format: 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.
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