Definition:Division over Euclidean Domain/Quotient

This page is about Partial Quotient in Euclidean Domain. For other uses, see Partial Quotient.

Definition

Let $\struct {D, +, \circ}$ be a Euclidean domain:

whose zero is $0_D$

whose Euclidean valuation is denoted $\nu$.

Let $a, b \in D$ such that $b \ne 0_D$.

Let $q$ and $r$ be the result of division of $a$ by $b$:

$a = q \circ b + r$ where either $\map \nu r < \map \nu b$ or $r = 0_D$.

Then:

$q$ is the quotient of the division of $a$ by $b$.

Also known as

A quotient in the context of division over a Euclidean domain is also known as a partial quotient when the remainder is not zero.

Also see

• Definition:Remainder of Division over Euclidean Domain

• Definition:Quotient of Integer Division

• Definition:Euclidean Valuation

• Results about quotients of division over a Euclidean domain can be found here.

Sources

• This article incorporates material from Euclidean valuation on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.