Definition:Division over Euclidean Domain/Quotient
< Definition:Division over Euclidean Domain(Redirected from Definition:Partial Quotient in Euclidean Domain)
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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
- 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.