Why does Negative times a negative equal a positive?

Answer 1

There are a lot of long examples that help to visualize why a negative times a negative is a positive, but this is just going to be an algebraic proof.

Let x = a*b + (-a)*b + (-a)*(-b)

If we factor out the (-a) for the second part of the equation, we are left with:

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

b+(-b) = 0, so the resulting equation is:

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

Any number times zero is zero, so:

x = a*b

Next, we go back to the original equation, and factor our the "b" from the first part, leaving:

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

a+(-a) = 0, so:

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

0*b = 0, so:

x = (-a)*(-b)

Now we see that x equals both a*b and (-a)*(-b), meaning:

a*b = (-a)*(-b)

So the product of 2 negative numbers must be equal the the product of their positive counterparts, i.e., a positive result.