Definition:Modulo Addition/Definition 1

Definition
Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$:
 * $\Z_m = \left\{{\left[\!\left[{0}\right]\!\right]_m, \left[\!\left[{1}\right]\!\right]_m, \ldots, \left[\!\left[{m-1}\right]\!\right]_m}\right\}$

where $\left[\!\left[{x}\right]\!\right]_m$ is the residue class of $x$ modulo $m$.

The operation of addition modulo $m$ is defined on $\Z_m$ as:
 * $\left[\!\left[{a}\right]\!\right]_m +_m \left[\!\left[{b}\right]\!\right]_m = \left[\!\left[{a + b}\right]\!\right]_m$

Also denoted as
Although the operation of addition modulo $m$ is denoted by the symbol $+_m$, if there is no danger of confusion, the symbol $+$ is often used instead.

The notation for addition of two integers modulo $m$ is not usually $\left[\!\left[{a}\right]\!\right]_m +_m \left[\!\left[{b}\right]\!\right]_m$.

What is more normally seen is $a + b \pmod m$.

Also see

 * Equivalence of Definitions of Integer Modulo Addition