What is the difference between $0$ and $-0$? Why is there a $0$ and a $-0$? I thought zero meant nothing, so how can we have negative nothing?
 A: Normally, given a number $a$, the number $-a$ is the additive inverse of $a$: i.e., $-a$ is the number that satisfies $a + (-a) = e$, where $e$ is the additive identity. Since $e + e = e$ by definition of an additive identity, we see that $e = -e$. In the case of $\Bbb{Z}$, $\Bbb{Q}$, $\Bbb{R}$, $\Bbb{C}$, and related objects, $0$ is the additive identity, so $0 = -0$.
A: People often have an unspoken assumption, that, basically
$$\text{If it looks different, it is different}$$
until they learn otherwise, and this is just one of many cases where that isn't true. $0=-0$. And $1=1.0$. And $1/2=2/4=3/6$. And $0.999...=1$. There are plenty of cases where there are multiple ways of writing the same thing down. Writing the negative in $-0$ is perfectly possible, but redundant, because it's just $0$ anyway. So in a sense you can have "negative nothing", but it's just the same as good old regular "nothing".
N.B. In some computer systems this isn't technically the case. If you represent negative numbers with one's complement then you do in fact have two zeroes, and it is sensible to call one of them $-0$. Mathematically we still have $0=-0$ (i.e. they still behave the same way under various operations), but they're stored as two distinct numbers. This is kind of inconvenient, and so we normally use two's complement instead, where there's only one zero.
A: For a number $x$, $-x$ denotes the unique $y$ such that $x+y=0$. In the case for $x=0$ we have that $-x=0$ as well.
A: Aha, computer science question!
This shows that a sign-magnitude (or one's complement) representation is being used, leading to two zeros with a different sign bit.
(I'm ducking under the desk now to avoid being struck by projectile office supplies).
A: In floating point arithmetic $0$ and $-0$ are different. 
If you take a positive number $x$ you will get $x/0=\infty$ and $x/(-0)=-\infty$.
A: To take the question (possibly a little too) literally, the difference between two numbers, $x$ and $y$ is given by $|x-y|$. Letting $x=0, y=-0$:
$$\begin{align*}
|x-y| &= |0-(-0)| \\
      &= |0+0| \\
      &= |0| \\
      &= 0
\end{align*}$$
So there's no difference between them at all :)
