I volunteer as a math tutor on the side. One of the questions I am frequently asked is this: Why do you get a positive number when multiplying two negative numbers? Well, allow me to settle it once and for all! 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.
Example: You owe 3 people $5 each. So you are "Negative 15" (3 × -$5 = -$15). They then say "we like you so much we forgive the debt" ... you have just had 3 subtractions of -5, so it is like you have added $15 (-3 × -$5 = +$15). You now have no debt: -$15 + $15 = $0
That's addition. The best explanation I found was on perspectives.com in the Science section, page 2, posted by "thoughtless" April 11, 2012. Think of a train where west of the station is positive; its position two hours ago is minus 2. I can't repeat it here, but it is clearly multiplication of two negatives.
Actually it can be proven very easily in any ring. First, we can prove that (-1)*x = -x. If we add x to (-1)*x, we should get 0. Indeed using the distributive rule, x+(-1)*x = 1*x+(-1)*x = (1+(-1))*x = 0*x = 0. Then you need to show -(-1) = 1. Well, -1+1=0, so 1 is the additive inverse of (-1), i.e. -(-1) = 1. (This of course holds for any x). Thus, using our two derived rules, (-1)*(-1) = -(-1) = 1.
I tend to go to English... as chances are if you are learning this, your understanding of the language is better than that of algebra... "You can't NOT look" = "You HAVE TO look" It is the opposite, of opposite. Two negatives result in a positive in the same way when multiplying... for sake of logic. That is at least an easier way perhaps to get it across to someone young.
The problem with algebra education is that a rigorous axiomatic approach isn't taught. Instead of fostering logic, kids are told to accept something is true "just because." Take the Pythagorean Theorem, for example. Every kid who's ever taken a geometry class has learned to use it. But how many of them can actually show why it's true?
