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:
 * $r = x \bmod y$

Also see

 * Definition:Modulo Operation
 * Definition:Quotient (Algebra)