Why does multiplying two negative numbers equal a positive number?

Question

Why does multiplying two negative numbers equal a positive number?

In: 1210

Answer 1

Basically because a negative number is supposed to behave in the opposite way to a positive number when it comes to addition.

So we know that for any number, say -1, we have

-1 × (-1 + 1) = -1 × 0 = 0

Now, we know that for multiplication and addition we can say

0 = -1 × (-1 + 1) = (-1 × -1) + (-1 × 1)

We know that -1 × 1 = -1 therefore the equation is only fulfilled if -1 × -1 = 1.

So we decide the rules that addition and multiplication have to fulfill and then define -1 as the thing that fulfills the rules and is the opposite of 1 for addition.