Why is the additive inverse of a positive number always negative?

Answer 1

The short answer is because of the definition. A longer answer is:

Start of with a negative number X whose additive inverse is Y. SUPPOSE Y is not positive.

Then X + Y is

one negative, X,

less another number that is at most 0, Y.

So their sum is less than or equal to X. That is, X + Y is at least as negative as X (and possibly even more negative).

But if Y is the additive inverse of X then, by definition (of additive inverses), X + Y MUST be 0.

That is a contradiction. So somewhere down the line there is an incorrect supposition. The only supposition above is that Y is not positive. So that supposition must be incorrect. Therefore Y is positive.

Answer 2

When you put them opposite each other on the number line the negatives are opposite to the positives.