# 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:

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 known as

Some sources call this the principal remainder.