divisionDivision2004-09-06 17:16:542007-01-20 16:21:07Definitionfieldquotientratiofundamental property of quotientreduction% this is the default PlanetMath preamble. as your knowledge
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% define commands here{\em 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 {\em quotient} (or {\em 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
$$b(b^{-1}a) = (bb^{-1})a = 1a = a.$$
\begin{itemize}
\item 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 $a$.
\item The quotient of $a$ and $b$ does not change if both numbers (elements) are multiplied (or divided, which \PMlinkescapetext{action} is called {\em 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 \PMlinkname{square root polynomials}{SquareRootOfSquareRootBinomial}, 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.
\item The division is neither associative nor commutative, but it is right distributive over addition:
$$\frac{a+b}{c} = \frac{a}{c}+\frac{b}{c}$$
\end{itemize}