Definition:Remainder

Definition
Let $a, b \in \Z$ be integers such that $b \ne 0$.

From the Division Theorem, we have that:


 * $\forall a, b \in \Z, b \ne 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < \left|{b}\right|$

The value $r$ is defined as the remainder of $a$ on division by $b$, or the remainder of $\dfrac a b$.

Real Arguments
When $x, y \in \R$ the remainder is still defined:

Also known as
Some sources call this the principal remainder.

Also see

 * Definition:Integer Division
 * Definition:Quotient (Algebra)