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 operationMathworldPlanetmath*” producing as result some elements of M, and these can be compared.  Then, an equation in  (M,*)  is a propositionPlanetmathPlanetmath of the form

E1=E2, (1)

where one has equated two expressions E1 and E2 formed with “*” of the elements or indeterminates of M.  We call the expressions E1 and E2 respectively the left hand side and the right hand side of the equation (1).

Example.  Let S be a set and 2S the set of its subsets.  In the groupoid(2S,),  where “” is the set differenceMathworldPlanetmath, we can write the equation


(which is always true).

Of course, M may be equipped with more operationsMathworldPlanetmath or be a module with some ring of multipliers — then an equation (1) may them.

But one need not assume any algebraic structurePlanetmathPlanetmath for the set M where the expressions E1 and E2 are values or where they elements.  Such a situation would occur e.g. if one has a continuous mapping f from a topological spaceMathworldPlanetmath L to another M; then one can consider an equation


A somewhat case is the equation


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 equationf(x)=0,  we may speak of the or the x0; it is the multiplicityMathworldPlanetmath of the zero x0 of the polynomialMathworldPlanetmathPlanetmathPlanetmath f(x). A multiple root has multiplicity greater than 1.

Example.  The equation


in the system of the complex numbersMathworldPlanetmathPlanetmath has as its roots the numbers


which, by the way, are the primitive sixth roots of unityMathworldPlanetmath.  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