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

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 $a$.

The quotient of $a$ and $b$ does not change if both numbers (elements) are multiplied (or divided, which action is called reduction) by any $k\neq 0$:
$\frac{ka}{kb}=(kb)^{{1}}(ka)=b^{{1}}k^{{1}}ka=b^{{1}}a=\frac{a}{b}$ So we have the method for getting the quotient of complex numbers,
$\frac{a}{b}=\frac{\bar{b}a}{\bar{b}b},$ where $\bar{b}$ is the complex conjugate of $b$, and the quotient of square 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.

The division is neither associative nor commutative, but it is right distributive over addition:
$\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$
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