Shadow

Tried to explain 3-(-5)=8 to my daughter today to no avail. Where I went wrong was to try to explain the whole thing on a number line, but I confused myself before confusing Jessie — I guess she was not bothered by the math challenge at all; she just watched me fumbling for an explanantion and laughed to her tears. She finally agreed  3-(-5)=3-(0-5)=3-0+5=8, but I know she was just tired of me.

I had to turn to wikipedia for answer, and as always, it never fails me.  

Subtracting a negative is equivalent to adding the corresponding positive:

5 − (−2) = 5 + 2 = 7(if you have a net worth of $5 and you get rid of a debt of $2, then your new net worth is $7).

Also:

−8 − (−3) = −5(if you have a debt of $8 and get rid of a debt of $3, then you still have a debt of $5).

I admit Wikipedia’s examples make more sense than what I tried to do, but I still think the number line will work. It gotta work. It’s the mother of all numerical representations! (Or is it?)

While on Wikipedia, I found that I am not the first — or last — person to puzzle over the concept of negativity. 

Negative numbers were not well-understood until modern times. As recently as the 18th century, the Swiss mathematician Leonhard Euler believed that negative numbers were greater than infinity, and it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless.⁴

Wikipedia lead me to another really interesting math site for middleschool students, where they have several Q&A sections on negative numbers. I find this discussion fascinatingly unsettling:

Math Forum: Ask Dr. Math FAQ: Negative Times a Negative

A Mathematical Explanation

If we can agree that a negative number is just a positive number multiplied by -1, then we can always write the product of two negative numbers this way:

   (-a)(-b) = (-1)(a)(-1)(b) = (-1)(-1)ab 

For example,

    -2 * -3 = (-1)(2)(-1)(3)
	
            = (-1)(-1)(2)(3)
	
            = (-1)(-1) * 6

So the real question is,

   (-1)(-1) = ?

and the answer is that the following convention has been adopted:

   (-1)(-1) = +1

This convention has been adopted for the simple reason that any other convention would cause something to break.

For example, if we adopted the convention that (-1)(-1) = -1, the distributive property of multiplication wouldn’t work for negative numbers:

   (-1)(1 + -1) = (-1)(1) + (-1)(-1)
	
        (-1)(0) = -1 + -1
	
              0 = -2

As Sherlock Holmes observed, "When you have excluded the impossible, whatever remains, however improbable, must be the truth."

Since everything except +1 can be excluded as impossible, it follows that, however improbable it seems, (-1)(-1) = +1.

It’s inadequate in my mind, not because of the lack of mathematical rigor, but because it basically says "it has to be the case" without giving me any intuition. Fascinating, because often times this is the level at which we learn and operate symbol systems — "this is how it is; take it or leave it". The reasons, the design principles are beyond most of us, and we rarely bother to ask. Eventually, Jessie will grow up like me, knowing (-1)*(-1)=1 like second nature but never asking why. Only the learners and (some) teachers, and maybe a few mathematicians, think deeper questions.

Such as this boy, who asked for a proof that 1 + 1 = 1. And he got it here on the Dr. Math forum.

This entry was posted on Tuesday, August 14th, 2007 at 11:13 pm and is filed under Odds'n'Ends, Education. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site. Edit this entry.