inverse numberInverseNumber2004-12-14 05:53:512009-01-21 08:49:55Definitiondivisionreciprocal number% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe {\em inverse number} or {\em reciprocal number} of a non-zero real or complex number $a$ may be denoted by $a^{-1}$, and it \PMlinkescapetext{means} the quotient $\frac{1}{a}$ (so, it is really the $-1^\mathrm{th}$ power of $a$).
\begin{itemize}
\item Two numbers are inverse numbers of each other if and only if their product is equal to 1 (cf. opposite inverses).
\item If $a$ ($\neq 0$) is given in a quotient form $\frac{b}{c}$, then its inverse number is simply
$$\left(\frac{b}{c}\right)^{-1} = \frac{c}{b}.$$
\item Forming the inverse number is also a multiplicative function, i.e.
$$(bc)^{-1} = b^{-1}c^{-1}$$
(to be more precise, it is an automorphism of the multiplicative group of $\mathbb{R}$ resp. $\mathbb{C}$).
\end{itemize}