Definition:Division over Euclidean Domain

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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

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