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: 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 :


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


A: It's useful to train oneself to use the phrase "additive inverse". Using that terminology, we first learn about positive numbers, and then we learn about negative numbers which are the additive inverses of positive numbers. Finally, your question can then be translated into the statement that the additive inverse of the additive inverse of $x$ equals $x$, which is true because
$$x + (-x) = 0
$$
and from this we that $x$ is the additive inverse of $-x$ which is the additive inverse of $x$.
A: The negative of a number is defined to be the unique solution of the equation $a+x=0$ and the solution of this equation is denoted by $-a$. Similarly the solution of the equation $(-a)+y=0$ is $-(-a)$.
Therefore we have $a+(-a)=(-a)+-(-a)$ and since $+$ is commutative in $\mathbb{R}$ and also both cancellation laws hold therefore we get,

$a+(-a)=(-a)+-(-a)\implies a+(-a)=-(-a)+(-a)\implies a=-(-a)$

