Definition:Set of Residue Classes/Least Absolute

Definition
Let $\left[\!\left[{a}\right]\!\right]_m$ be the residue class of $a$ (modulo $m$).

Except when $r = \dfrac m 2$, we can choose $r$ to be the integer in $\left[\!\left[{a}\right]\!\right]_m$ which has the smallest absolute value.

In that exceptional case we have:
 * $-\dfrac m 2 + m = \dfrac m 2$

and so:
 * $-\dfrac m 2 \equiv \dfrac m 2 \pmod m$

Thus $r$ is defined as the least absolute residue of $a$ (modulo $m$) :
 * $-\dfrac m 2 < r \le \dfrac m 2$

Also see

 * Definition:Least Positive Residue