Why does a negative times a negative equal a positive?

Answer 1

Number Line

Imagine a number line on which you walk. Multiplying x * y is taking x steps, each of size y. Negative steps require you to face the negative end of the line before you start walking, and negative step sizes indicate they should go backwards (i.e., heel first). So, (-x) * (-y) means to stand on zero, face in the negative direction, and then take x backward steps, each of size y.

A ProofLet a and b be any two real numbers. Consider the number x defined by x = ab + (-a)(b) + (-a)(-b).

We can write

x = ab + (-a)[ (b) + (-b) ] (factor out -a) = ab + (-a)(0) = ab + 0 = ab.

Also,

x = [ a + (-a) ]b + (-a)(-b) (factor out b) = 0 * b + (-a)(-b) = 0 + (-a)(-b) = (-a)(-b).

So we have

x = ab

and

x = (-a)(-b)

Hence, by the transitivity of equality, we have

ab = (-a)(-b).

Because the opposite of a negative is a positive.