Symmetric interval on an integer ring $\mathbb{Z}_p$?

Question

$\mathbb{Z}_p=\{0,1,..,p-1\}$ for some positive integer $p$.

How can we define $\mathbb{Z}_p$ is the interval of $(-\frac{p}{2}, \frac{p}{2}] \cap \mathbb{Z}$?

For example. if $p=7$, I can understand $\mathbb{Z}_8=\{0,1,2,3,4,5,6, 7\}$. But, I couldn't understand the mapping between $\{0,1,2,3,4,5,6, 7\} $ and $\{-3, -2, -1, 0, 1, 2, 3, 4\}.$

What is the mapping of each element?

Answer 1

$$\mathbb{Z}_8=\{0,1,2,3,4,5,6, 7\}\equiv \{-3, -2, -1, 0, 1, 2, 3, 4\}$$

Note that in $ \mathbb{Z}_8 $ we have $5=-3, 6=-2,$ and $7=-1$

The use of negative makes some computations easier specially when dealing with absolute values or $\pm $ notation.

For example $$ \mathbb{Z}_5 = 0,\pm 1, \pm2 $$

$$ \mathbb{Z}_9 = 0,\pm 1, \pm 2, \pm 3, \pm 4$$