equation Equation 2005-08-17 09:21:45 2008-06-05 03:56:36 Definition groupoid equate side root solution root of an equation left hand side right hand side multiplicity of the root order of the root multiple root equation root solution % 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 \theoremstyle{definition} \newtheorem*{thmplain}{Theorem} \textbf{Equation} An {\em equation} concerns usually elements of a certain set $M$, where one can say if two elements are equal.\, In the simplest case, $M$ has one binary operation ``$*$'' producing as result some elements of $M$, and these can be compared.\, Then, an equation in\, $(M,\,*)$\, is a proposition of the form \begin{align} E_1 = E_2, \end{align} where one has {\em equated} two expressions $E_1$ and $E_2$ formed with ``$*$'' of the elements or indeterminates of $M$.\, We call the expressions $E_1$ and $E_2$ respectively the {\em left hand side} and the {\em right hand side} of the equation (1).\\ \textbf{Example.}\, Let $S$ be a set and $2^S$ the set of its subsets.\, In the groupoid\, $(2^S,\,\smallsetminus)$,\, where ``$\smallsetminus$'' is the set difference, we can write the equation $$(A\!\smallsetminus\!B)\!\smallsetminus\!B = A\!\smallsetminus\!B$$ (which is always true).\\ Of course, $M$ may be equipped with more operations or be a module with some ring of multipliers --- then an equation (1) may \PMlinkescapetext{contain} them. But one need not assume any algebraic structure for the set $M$ where the expressions $E_1$ and $E_2$ are values or where they \PMlinkescapetext{represent generic} elements.\, Such a situation would occur e.g. if one has a continuous mapping $f$ from a topological space $L$ to another $M$; then one can consider an equation $$f(x) = y.$$ A somewhat \PMlinkescapetext{comparable} case is the equation $$\dim{V} = 2$$ where $V$ is a certain or a \PMlinkescapetext{generic} vector space; both \PMlinkescapetext{sides represent} elements of the extended real number system.\\ \textbf{Root of equation} If an equation (1) in $M$ \PMlinkescapetext{contains} one indeterminate, say $x$, then a value of $x$ which satisfies (1), i.e. makes it true, is called a {\em root} or a {\em solution} of the equation. Especially, if we have a polynomial equation\, $f(x) = 0$,\, we may speak of the \PMlinkescapetext{{\em multiplicity}} or the {\em \PMlinkescapetext{order of a root}} $x_0$; it is the multiplicity of the zero $x_0$ of the polynomial $f(x)$. A {\em multiple root} has multiplicity greater than 1.\\ \textbf{Example.}\, The equation $$x^2\!+\!1 = x$$ in the system $\mathbb{C}$ of the complex numbers has as its roots the numbers $$x := \frac{1\!\pm\!i\sqrt{3}}{2},$$ which, by the way, are the primitive sixth roots of unity.\, Their multiplicities are 1.