Negative Zero in the set of real numbers So according to the laws of algebra for a given $x$, one denotes by $-x$ the number such that $x+y = 0$ and is called negative of x. The subtraction operation is given by $z - x = z + (-x)$.
It is easy to prove that $-0 = 0$, thus they have the same value. But this mean that -0 and 0 are equivalent or are they different elements, $-0 \in \mathbb{R}$ and $0 \in \mathbb{R}$ s.t. $-0 = 0$?

Since two objects are equal if and only if they are the same objects, $-0$ and $0$ are in fact the same object.
And just because you can write something in several ways doesn't mean each way gives off a different object. $2=1+1$ and $4=2+2=2\cdot 2=2^2=\sqrt{15+1}$.
