Definition:Congruence (Number Theory)/Modulo Operation

Definition
Let $z \in \R$. Let $\bmod$ be defined as the modulo operation:


 * $x \bmod y := \begin{cases}

x - y \left \lfloor {\dfrac x y}\right \rfloor & : y \ne 0 \\ x & : y = 0 \end{cases}$

Then congruence modulo $z$ is the relation on $\R$ defined as:
 * $\forall x, y \in \R: x \equiv y \pmod z \iff x \bmod z = y \bmod z$

The real number $z$ is called the modulus.

Also see

 * Congruence Modulo $m$ for integral $m$


 * Equivalence of Congruence Definitions


 * Congruence Modulo $m$ is an Equivalence Relation