Definition:Addition/Real Modulo Addition

Definition
The addition operation on $\R_z$, the set of set of all residue classes modulo $z$, is defined by the rule:


 * $\left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{b}\right]\!\right]_z = \left[\!\left[{a + b}\right]\!\right]_z$

Although the operation of addition modulo $z$ is denoted by the symbol $+_z$, if there is no danger of confusion, the symbol $+$ is often used instead.

More usually, though, the notation $a + b \left({\bmod\, z}\right)$ is used instead of $\left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{b}\right]\!\right]_z$.

It means the same thing and, although obscuring the true meaning behind modulo arithmetic, is more streamlined and less unwieldy.

See modulo addition.