Definition:Division over Euclidean Domain/Remainder

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:

$r$ is the remainder of the division of $a$ by $b$.

Also see

• Definition:Quotient of Division over Euclidean Domain

• Definition:Remainder of Integer Division

• Definition:Euclidean Valuation

• Results about remainders 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.