# equation

Equation

An 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

 $\displaystyle E_{1}=E_{2},$ (1)

where one has 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 left hand side and the right hand side of the equation (1).

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 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 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 case is the equation

 $\dim{V}=2$

where $V$ is a certain or a vector space; both elements of the extended real number system.

Root of equation

If an equation (1) in $M$ one indeterminate, say $x$, then a value of $x$ which satisfies (1), i.e. makes it true, is called a root or a solution of the equation. Especially, if we have a polynomial equation$f(x)=0$,  we may speak of the or the $x_{0}$; it is the multiplicity of the zero $x_{0}$ of the polynomial $f(x)$. A multiple root has multiplicity greater than 1.

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.

 Title equation Canonical name Equation Date of creation 2013-03-22 15:28:33 Last modified on 2013-03-22 15:28:33 Owner pahio (2872) Last modified by pahio (2872) Numerical id 30 Author pahio (2872) Entry type Definition Classification msc 20N02 Related topic Equality2 Related topic AlgebraicEquation Related topic DiophantineEquation Related topic TrigonometricEquation Related topic DifferenceEquation Related topic DifferentialEquation Related topic IntegralEquation Related topic FunctionalEquation Related topic HomogeneousEquation Related topic ProportionEquation Related topic FiniteDifference Related topic RecurrenceRelation Related topic CharacteristicEquation Defines equate Defines side Defines root Defines solution Defines root of an equation Defines left hand side Defines right hand side Defines multiplicity of the root Defines order of the root Defines multiple root