PlanetMath: division

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division

(Definition)

Division is the operation which assigns to every two numbers (or more generally, elements of a field) and their quotient or ratio, provided that the latter, , is distinct from zero.

The quotient (or ratio) $\frac{a}{b}$ of and may be defined as such a number (or element of the field) that $b \cdot x = a$ . Thus,

$\displaystyle b \cdot \frac{a}{b} = a,$

which is the “fundamental property of quotient”. The explicit general expression for $\frac{a}{b}$ is

$\displaystyle \frac{a}{b} = b^{-1}\cdot a$

where $b^{-1}$ is the inverse number (the multiplicative inverse) of

, because

$\displaystyle b(b^{-1}a) = (bb^{-1})a = 1a = a.$

For positive numbers the quotient may be obtained by performing the division algorithm with and . If , then $\frac{a}{b}$ indicates how many times fits in .
The quotient of and does not change if both numbers (elements) are multiplied (or divided, which action is called reduction) by any $k \neq 0$ :
$\displaystyle \frac{ka}{kb} = (kb)^{-1}(ka) = b^{-1}k^{-1}ka = b^{-1}a = \frac{a}{b}$
So we have the method for getting the quotient of complex numbers,
$\displaystyle \frac{a}{b} = \frac{\bar{b}a}{\bar{b}b},$
where $\bar{b}$ is the complex conjugate of , and the quotient of square root polynomials, e.g.
$\displaystyle \frac{1}{5+2\sqrt{2}} = \frac{5-2\sqrt{2}}{(5-2\sqrt{2})(5+2\sqrt{2})} = \frac{5-2\sqrt{2}}{25-8} = \frac{5-2\sqrt{2}}{17};$
in the first case one aspires after a real and in the second case after a rational denominator.
The division is neither associative nor commutative, but it is right distributive over addition:
$\displaystyle \frac{a+b}{c} = \frac{a}{c}+\frac{b}{c}$

"division" is owned by pahio.

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Also defines:

quotient, ratio, fundamental property of quotient, reduction

This object's parent.

Attachments:: inverse number (Definition) by pahio
long division (Theorem) by alozano
division by zero (Example) by Algeboy
table of division up to 12 (Data Structure) by PrimeFan

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Cross-references: addition, right distributive, commutative, associative, denominator, rational, real, complex conjugate, complex numbers, division algorithm, positive, multiplicative inverse, inverse number, expression, field, numbers, operation
There are 212 references to this entry.

This is version 22 of division, born on 2004-09-06, modified 2007-01-20.
Object id is 6148, canonical name is Division.
Accessed 20005 times total.

Classification:

AMS MSC:	00A05 (General :: General and miscellaneous specific topics :: General mathematics)
	12E99 (Field theory and polynomials :: General field theory :: Miscellaneous)

Pending Errata and Addenda

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