differenceDifference22007-09-27 09:15:562009-11-16 22:15:11Definitionsubtractionminuendsubtrahend% this is the default PlanetMath preamble. as your knowledge
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The {\em difference of} two numbers $a$ and $b$ is a number $d$ such that
$$b\!+\!d = a.$$
The difference of $a$ (the {\em minuend}) and $b$ (the {\em subtrahend}) is denoted by $a\!-\!b$.
The definition is \PMlinkescapetext{similar} for the elements $a,\,b$ of any \PMlinkescapetext{additive} Abelian group (e.g. of a vector space).\,The difference of them is always unique.\\
\textbf{Note 1.}\, Forming the difference of numbers (resp. elements), i.e. subtraction, is in a certain sense converse to the addition operation:
$$(x\!+\!y)\!-\!y \;=\; x$$
\textbf{Note 2.}\, As for real numbers, one may say that the difference \emph{between} $a$ and $b$ is $|a\!-\!b|$ (which is the same as $|b\!-\!a|$); then it is always nonnegative.\, For all complex numbers, such a phrase would be nonsense.\\
\textbf{Some \PMlinkescapetext{identities}}
\begin{itemize}
\item $b\!+\!(a\!-\!b) = a$
\item $a\!-\!b = a\!+\!(-b)$
\item $-(a\!-\!b) = b\!-\!a$
\item $n(a\!-\!b) = na\!-\!nb \quad (n\in\mathbb{Z})$
\item $a\!-\!a = 0$
\end{itemize}