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,

which is the “fundamental property of quotient”.

The quotient of the numbers $a$ and $b$ ($\neq 0$) is a uniquely determined number, since if one had

then we could write

from which the supposition $b\neq 0$ would imply $x-y=0$, i.e. $x=y$.

The explicit general expression for $\frac{a}{b}$ is

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

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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 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.

  $$\frac{1}{5+2\sqrt{2}}=\frac{5-2\sqrt{2}}{(5-2\sqrt{2})(5+2\sqrt{2})}=\frac{5-2\sqrt{2}}{25-8}=\frac{5-2\sqrt{2}}{17};$$

  in the first case one aspires after a real and in the second case after a rational denominator.

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  The division is neither associative nor commutative, but it is right distributive over addition:

  $$\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$$