# Definition:Quotient (Algebra)

## Definition

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

From the Division Theorem:

$\forall a, b \in \Z, b \ne 0: \exists_1 q, r \in \Z: a = q b + r, 0 \le r < \size b$

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

### Real Arguments

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

The quotient of $x$ on division by $y$ is defined as the value of $q$ 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:

$x \bmod y := x - y \left \lfloor {\dfrac x y}\right \rfloor = r$

it can be seen that the quotient of $x$ on division by $y$ is defined as:

$q = \left \lfloor {\dfrac x y}\right \rfloor$