Why does a negative times a negative always equal a positive?

Answer 1

There are many intuitive explanations that show it works. For example, you can look at debt as a negative number or you can use the number line to see why negative times negative is a positive. None of these is really a proof. In fact, since any negative number is really the same positive number multiplied by -1, i.e. -5=5x-1 when we really need to ask only why is -1x-1=1. One good, but often not satisfying, answer is that if we use any other convention, then it won't work. The fact is, it is a convention.

For the mathematical purist, here is a nice proof that two negatives multiplied together equal a positive

Let a and b be any two real numbers. Consider the number n defined by

n = ab + (-a)(b) + (-a)(-b).

We can write

n= ab + (-a)[ (b) + (-b) ] if we just actor out -a

= ab + (-a)(0)

= ab + 0

= ab.

Also,

n = [ a + (-a) ]b + (-a)(-b) if we factor out b

= 0 * b + (-a)(-b)

= 0 + (-a)(-b)

= (-a)(-b).

So we have

n = ab

and

n = (-a)(-b)

Hence, by the transitivity of equality, we have

ab = (-a)(-b) and the proof is complete.