Definition:Quotient (Integer Division)

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:

$\forall x, y \in \R, y \ne 0: \exists! q \in \Z, r \in \R: x = q y + r, 0 \le r < \size y$

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:

$q = \floor {\dfrac x y}$

Also see

• Definition:Integer Division
• Definition:Remainder

Linguistic Note

The word quotient derives from the Latin word meaning how often.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): quotient: 2.
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): division
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): quotient
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): division
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): quotient
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): quotient