Definition:Quotient (Algebra)

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