division
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,
$$b\cdot \frac{a}{b}=a,$$ 
which is the “fundamental property of quotient”.
The quotient of the numbers $a$ and $b$ ($\ne 0$) is a uniquely determined number, since if one had
$$\frac{a}{b}=x\ne y=\frac{a}{b},$$ 
then we could write
$$b(xy)=bxby=aa=0$$ 
from which the supposition $b\ne 0$ would imply $xy=0$, i.e. $x=y$.
The explicit general expression for $\frac{a}{b}$ is
$$\frac{a}{b}={b}^{1}\cdot a$$ 
where ${b}^{1}$ is the inverse number (the multiplicative inverse^{}) of $a$, because
$$b({b}^{1}a)=(b{b}^{1})a=1a=a.$$ 

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For positive numbers the quotient may be obtained by performing the division algorithm^{} with $a$ and $b$. If $a>b>0$, then $\frac{a}{b}$ indicates how many times $b$ fits in $a$.

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The quotient of $a$ and $b$ does not change if both numbers (elements) are multiplied (or divided, which is called reduction^{}) by any $k\ne 0$:
$$\frac{ka}{kb}={(kb)}^{1}(ka)={b}^{1}{k}^{1}ka={b}^{1}a=\frac{a}{b}$$ So we have the method for getting the quotient of complex numbers^{},
$$\frac{a}{b}=\frac{\overline{b}a}{\overline{b}b},$$ where $\overline{b}$ is the complex conjugate of $b$, and the quotient of http://planetmath.org/SquareRootOfSquareRootBinomialsquare root polynomials, e.g.
$$\frac{1}{5+2\sqrt{2}}=\frac{52\sqrt{2}}{(52\sqrt{2})(5+2\sqrt{2})}=\frac{52\sqrt{2}}{258}=\frac{52\sqrt{2}}{17};$$ in the first case one aspires after a real and in the second case after a rational denominator.

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The division is neither associative nor commutative^{}, but it is right distributive over addition^{}:
$$\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$$
Title  division 
Canonical name  Division 
Date of creation  20140808 17:51:29 
Last modified on  20140808 17:51:29 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  29 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 00A05 
Classification  msc 12E99 
Related topic  InverseFormingInProportionToGroupOperation 
Related topic  DivisionInGroup 
Related topic  ConjugationMnemonic 
Related topic  Difference2 
Related topic  UniquenessOfDivisionAlgorithmInEuclideanDomain 
Defines  quotient 
Defines  ratio 
Defines  fundamental property of quotient 
Defines  reduction 