Why -- equals plus?

Answer 1

I suppose you mean, if you multiply them. There is really no way to "prove" this; you should consider it a definition. However, it is a definition that makes sense. For example, with this definition, the distributive law continues to be valid for all integers.

Consider this sequence:

3 x 3 = 9
2 x 3 = 6
1 x 3 = 3
0 x 3 = 0
-1 x 3 = -3
-2 x 3 = -6

Each time I reduce the first factor by one, and the final product reduces by three. It therefore seems reasonable, for example, that -1 x 3 = -3.

Next step, another sequence:

3 x -3 = -9
2 x -3 = -6
1 x -3 = -3
0 x -3 = 0
-1 x -3 = 3
-2 x -3 = 6

This time I reduced the second factor by one each time; the product increases by 3 each time. So, it seems at least reasonable that a negative number times a negative number gives a positive product. As hinted before, several math laws, like the distributive property, maintain their validity if multiplication is defined this way.