Negative Math: How Mathematical Rules Can Be Positively Bent

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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.




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Negative Math: How Mathematical Rules Can Be Positively Bent Overviews

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

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Customer Review

not sure the book has the right conclusions - Deep Roy - Minneapolis, MN United States
the author claims that there's no device which measures an imaginary number; well, for EE's, an imaginary number (phasor) can be measured with an oscilloscope since oscilloscopes can measure amplitude and phase; this representation can then be converted to an imaginary number. the author also claims that you can't have -5 apples in a box and this seems sinister since -5, for example, would mean that there's an IOU in the box after the vender had 5 apples and bartered 10 apples for something he wanted. in this case, when the vender got a dozen more apples, 5 of them would be owed. perhaps I should read more than amazon's sample pages but the ones I read didn't seem that complete.

A wonderful and vital perspective - Eric Belcastro - Bridgeville, PA United States
I will leave out information that I feel other reviewers have already mentioned and just concentrate on a few points.

This writing really is wonderful, in my opinion. I think that, as a popular writing it is certainly accessible to the general public, but simplicity can be deceptive. The topics may not involve dense opaque notation and detailed treatment from abstract algebra, but if such approaches were resulted to, there would not have been any room to really deal with the historical, philosophical, and physical implications of new algebras. Often texts of that nature present the subject in its most compact form, which is rather out of step with any true learning process which gains an intuitive understanding first, and then builds up in layers upon the first insights, questioning and reworking as the process continues. I feel that the comment by one reviewer that the author is in error by not discussing the topic in abstract algebra form (that would, coincidentally make the book utterly incomprehensible to the common reader) is symptomatic of what happens when someone becomes very proficient at a subject and completely loses all touch with the reality of the people who don't have her/his expertise. It is not the job of this book to take that approach. That is your job. Take the ideas and fly with them, or reject them; either which way, I think you will come away from it with a deeper perspective. Even mathematicians dealing with hypercomplex algebras could deal with a short retreat to consider these basic ideas anew.

I am thankful to Mr. Martinez for writing this book. I hope it is well received.

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