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单词 Equation
释义

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 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

(AB)B=AB

(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

f(x)=y.

A somewhat case is the equation

dimV=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 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

x2+1=x

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

x:=1±i32,

which, by the way, are the primitive sixth roots of unityMathworldPlanetmath.  Their multiplicities are 1.

Titleequation
Canonical nameEquation
Date of creation2013-03-22 15:28:33
Last modified on2013-03-22 15:28:33
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id30
Authorpahio (2872)
Entry typeDefinition
Classificationmsc 20N02
Related topicEquality2
Related topicAlgebraicEquation
Related topicDiophantineEquation
Related topicTrigonometricEquation
Related topicDifferenceEquation
Related topicDifferentialEquation
Related topicIntegralEquation
Related topicFunctionalEquation
Related topicHomogeneousEquation
Related topicProportionEquation
Related topicFiniteDifference
Related topicRecurrenceRelation
Related topicCharacteristicEquation
Definesequate
Definesside
Definesroot
Definessolution
Definesroot of an equation
Definesleft hand side
Definesright hand side
Definesmultiplicity of the root
Definesorder of the root
Definesmultiple root
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更新时间:2025/5/4 10:02:21