opposite number OppositeNumber 2005-02-18 07:52:10 2006-12-08 07:23:06 Definition complex addition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The {\em opposite number} of a number $a$ is such a number\, $-a$\, that $$-a\!+\!a = 0.$$ Some properties: \begin{itemize} \item $-0 = 0$ \item $-(-a) = a$ \item $-(a\!+\!b) = (-a)\!+\!(-b)$ \item $-(a\!\cdot\!b) = a\!\cdot\!(-b) = (-a)\!\cdot\!b$ \item $-(a\!-\!b) = b\!-\!a$ \item $-\sum_{j = 1}^n a_j = \sum_{j = 1}^n (-a_j)$ \item $-\int_a^b f(x)\,dx = \int_b^a f(x)\,dx$ \end{itemize} Exactly similar properties (except the last) are valid in every ring.\, The fourth property implies the \textbf{Corollary.}\, If one changes the sign of one factor of a ring product, then the sign of the whole product changes.