Definition:Quotient (Algebra)

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

From the Division Theorem:


 * $\forall a, b \in \Z, b \ne 0: \exists_1 q, r \in \Z: a = q b + r, 0 \le r < \left|{b}\right|$

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:


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

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 quotient of $x$ on division by $y$ are the same thing.

Also see

 * Definition:Integer Division
 * Definition:Remainder