Definition:Modulo Addition/Definition 3

Definition

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

Let $\Z_m$ be the set of integers modulo $m$:

$\Z_m = \set {0, 1, \ldots, m - 1}$

The operation of addition modulo $m$ is defined on $\Z_m$ as:

$a +_m b := a + b - k m$

where $j$ is the largest integer such that $k m \le a + b$.

Also denoted as

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

The notation for addition of two residue classes modulo $m$ is not usually $\eqclass a m +_m \eqclass b m$.

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

Also see

• Equivalence of Definitions of Integer Modulo Addition

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Example $2.3$