# Definition:Remainder/Real

## Definition

Let $x, y \in \R$ be real numbers such that $y \ne 0$.

The remainder of $x$ on division by $y$ is defined as the value of $r$ in the expression:

$\forall x, y \in \R, y \ne 0: \exists! q \in \Z, r \in \R: x = q y + r, 0 \le r < \left|{y}\right|$

From the definition of the Modulo Operation:

$x \bmod y := x - y \left \lfloor {\dfrac x y}\right \rfloor$

it can be seen that the remainder of $x$ on division by $y$ is defined as:

$r = x \bmod y$