Definition:Addition/Real Modulo Addition

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

• Modulo Addition is Well-Defined

• 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 $24$