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[parent] division (Definition)

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

The quotient (or ratio) $ \frac{a}{b}$ of $ a$ and $ b$ may be defined as such a number (or element of the field) $ x$ 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 $ a$, because
$\displaystyle b(b^{-1}a) = (bb^{-1})a = 1a = a.$



"division" is owned by pahio.
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See Also: inverse of a product, division in group, conjugation (mnemonic), difference, uniqueness of division algorithm in Euclidean domain

Also defines:  quotient, ratio, fundamental property of quotient, reduction

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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 MSC00A05 (General :: General and miscellaneous specific topics :: General mathematics)
 12E99 (Field theory and polynomials :: General field theory :: Miscellaneous)

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