Definition:Remainder

Let $$a, b \in \Z$$.

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 $$\frac {a}{b}$$.

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


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

See the definition of the Modulo Operation.

Note
Some sources call this the principal remainder.