Definition:Division over Euclidean Domain
This page is about Division over Euclidean Domain. For other uses, see division.
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 $b \ne 0_D$.
By the definition of Euclidean valuation:
- $\exists q, r \in D: a = q \circ b + r$
such that either:
- $\map \nu r < \map \nu b$
or:
- $r = 0_D$
The process of finding $q$ and $r$ is known as division of $a$ by $b$, and we write:
- $a \div b = q \rem r$
Quotient
- $q$ is the quotient of the division of $a$ by $b$.
Remainder
- $r$ is the remainder of the division of $a$ by $b$.
Notation
The operation of division can be denoted as:
- $a / b$, which is probably the most common in the general informal context
- $\dfrac a b$, which is the preferred style on $\mathsf{Pr} \infty \mathsf{fWiki}$
- $a : b$, which is usually used when discussing ratios
- $a \div b$, which is rarely seen outside grade school, but can be useful in contexts where it is important to be specific.
Examples
Integer Division
Let $a, b \in \Z$ be integers such that $b \ne 0$.
From the Division Theorem:
- $\exists_1 q, r \in \Z: a = q b + r, 0 \le r < \size b$
where $q$ is the quotient and $r$ is the remainder.
The process of finding $q$ and $r$ is known as (integer) division of $a$ by $b$, and we write:
- $a \div b = q \rem r$
Polynomial Division
Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$.
Let $X$ be transcendental over $F$.
Let $F \sqbrk X$ be the ring of polynomials in $X$ over $F$.
Let $\map A x$ and $\map B x$ be polynomials in $F \sqbrk X$ such that the degree of $B$ is non-zero.
From the Division Theorem for Polynomial Forms over Field:
- $\exists \map Q x, \map R x \in F \sqbrk X: \map A x = \map Q x \map B x + \map R x$
such that:
- $0 \le \map \deg R < \map \deg B$
where $\deg$ denotes the degree of a polynomial.
The process of finding $\map Q x$ and $\map R x$ is known as polynomial division of $\map A x$ by $\map B x$, and we write:
- $\map A x \div \map B x = \map Q x \rem \map R x$
Also see
- Results about division over a Euclidean domain can be found here.
Linguistic Note
The verb form of the word division is divide.
Thus to divide is to perform an act of division.
Sources
- This article incorporates material from Euclidean valuation on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.