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.