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 $\dfrac a b$.

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 := \begin{cases}

x - y \left \lfloor {\dfrac 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.