Definition:Remainder

Definition
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:
 * $$x \, \bmod \, y \ \stackrel {\mathbf {def}} {=\!=} \ \begin{cases}

x - y \left \lfloor {\frac x y}\right \rfloor & : y \ne 0 \\ x & : y = 0 \end{cases}$$ from whence it can be seen that $$x \, \bmod \, y$$ and the remainder of $$x$$ on division by $$y$$ are the same thing.

Note
Some sources call this the principal remainder.