Why negative of a negative number is positive? I am intrigued in seeking the philosophy underlying it. When I was trying to prove it mathematically, I was failed but later I started to analyze vectors that what they are? As an outgrowth of vector's study I found my answer i.e. If there is a vector $A$, then its reverse is the vector -$A$ but what is the reverse of $-A$ again? I used the convention of prefixing a minus sign before $-A$ and figured out that it is the vector we were familiar with i.e. $A$ and no vector can just be opposite to $-A$ i.e. no vector can be at $180$ degree to the vector $-A$ rather than $A$ or '$-(-A)$'. And so i satisfied myself but i am puzzled thus far that whether I am right to explain Negative number as the additive inverse or not. Can you tell me if other philosophical fact satisfies it.

A _negative_ (integer) number is the "inverse" with respect to the addition :

> $x + (-x) = 0$.

Thus, consider the inverse of a negative number :

> $(-x) + [- (-x)] = 0$;

by property of $=$ and commutativy of addition we have that :

> $x + (-x) = [- (-x)] + (-x)$

and thus :

> > $x = [- (-x)]$.