Proof of $+0 = -0$ How do you prove  $+ 0 = - 0$  ? 
I have no clue where to start from. (I am a 11th Grader). Can it be done only using concepts I have learned till now or will I need some more concepts?
 A: $0$ is the unique(!) number with the property $$\tag1x+0=x$$ for all $x$.
For any $y$, $-y$ is the unique(!) number with the property $$\tag2y+(-y)=0.$$
From $(1)$ with $x=0$, we get the following:
$$0+0=0$$
Also, from $(2)$ with $y=0$, we get the following:
$$0+(-0)=0$$
Thus, by the Transitive Property of Equality, we can set the left side of both of the previous equations equal to each other:
$$0+0=0+(-0)$$
Hence, by the Subtractive Property of Equality, we subtract $0$ from both sides of this equation to see that $-0=0$.
(NOTE: The $+$ in $+0$ being redundant and/or misleading)
A: From the definition of $0$, $0$ is the additive identity of $\Bbb{R}$, meaning for all $a \in \Bbb{R}$: $$0+a=a$$
Also, from the definition of negative numbers, for all $b \in \Bbb{R}$:
$$b+(-b)=0$$
Thus, if we let $b=0$ in the second equation, we get:
$$0+(-0)=0$$
Also, if we let $a=-0$ in the first equation, we get:
$$0+(-0)=-0$$
Thus, by the Transitive Property of Equality with the last two equations, $-0=0$.
