Why is a negative number times negative number always a positive?

Answer 1

That's the way it is defined. And it's defined that way to maintain consistency in different properties of numbers. However, the following argument makes it look reasonable, at least. Let's assume it is already given that a positive number times a negative number is negative. (If this doesn't sound reasonable to you, please ask a separate question.)

Now consider the following sequence:

-10 x 3 = -30

-10 x 2 = -20

-10 x 1 = -10

-10 x 0 = 0

-10 x -1 = ...

Now consider the question: What number should you fill in to complete the sequence? The left number in each multiplication is constant, the middle number decreases by one at a time. The rightmost column obviously obviously increases by 10 at a time; if you continue the sequence, you inevitably get 10 - a positive number. Similarly, you get positive numbers if you replace the sequence with any other combination of numbers, such that the first number is negative, and the second number decreases at a constant rate.