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

In that exceptional case we have $\displaystyle -\frac m 2 + m = \frac m 2$ and so $\displaystyle -\frac m 2 \equiv \frac m 2 \pmod m$.

So we define $r$ as the least absolute residue of $a$ (modulo $m$) if:
 * $\displaystyle -\frac m 2 < r \le \frac m 2$

Compare with

 * Definition:Least Positive Residue