# Definition:Addition/Real Modulo Addition

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## Definition

Let $z \in \R$.

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

The addition operation is defined on $\R_z$ as follows:

- $\eqclass a z +_z \eqclass b z = \eqclass {a + b} z$

This operation is called **addition modulo $z$**.

## Also denoted as

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.

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

The usual notation is $a + b \pmod z$.

## Also see

- Definition:Modulo Addition, defined on an integer modulus.

## Sources

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $24$