Definition:Modulo Addition

Definition
Let $z \in \R$.

Let $\R_z$ be the set of all residue classes modulo $z$ of $\R$.

We define the addition operation on $\R_z$, defined as follows:


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

This can be shown to be a well-defined operation.

This operation is called addition modulo $z$.

Comment
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.

In fact, the notation for addition of two residue classes modulo $z$ is not usually $\left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{b}\right]\!\right]_z$.

What is more normally seen is $a + b \left({\bmod\, z}\right)$.

Using this notation, what this result says is:

and it can be proved in the same way.

Similarly: $a - c \equiv b - d \left({\bmod\, z}\right)$.

Warning
Compare this with Modulo Multiplication, which is defined only on an integer modulus.