Why is a negative times a negative a positive?

Answer 1

Here is a proof.

Let a and b be any two real numbers. Consider the number x defined as

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

We can write this out differently as

x = ab + (-a)[ (b) + (-b) ]

Then, by factoring out -a , we find that x= ab + (-a)(0) = ab + 0 = ab.

Also,

x = [ a + (-a) ]b + (-a)(-b)

And by factoring out b, we find that x=0 * b + (-a)(-b) = 0 + (-a)(-b) = (-a)(-b).

Therefore x = ab and x = (-a)(-b)

Then, by the transitivity of equality, we have

ab = (-a)(-b).