\begin{definition}[Definition:Quotient (Algebra)/Real]
Let $x, y \in \R$ be real numbers such that $y \ne 0$.
The '''quotient of $x$ on division by $y$''' is defined as the value of $q$ in the expression:
:$\forall x, y \in \R, y \ne 0: \exists! q \in \Z, r \in \R: x = q y + r, 0 \le r < \size y$
From the definition of the Modulo Operation:
:$x \bmod y := x - y \floor {\dfrac x y} = r$
it can be seen that the '''quotient of $x$ on division by $y$''' is defined as:
:$q = \floor {\dfrac x y}$
\end{definition}