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单词 ModulusOfComplexNumber
释义

modulus of complex number


DefinitionLet z be a complex numberMathworldPlanetmathPlanetmath, and letz¯ be the complex conjugateMathworldPlanetmath of z.Then the modulus, or absolute valueMathworldPlanetmathPlanetmath, of z is defined as

|z|:=zz¯.

There is also the notation

modz

for the modulus of z.

If we write z in polar form as  z=reiϕ  with  r0,ϕ[0, 2π), then  |z|=r. It follows that the modulus is a positive real number or zero.Alternatively, if a is the real part of z, and b the imaginary part, then

|z|=a2+b2,(1)

which is simply the Euclidean norm of the point  (a,b)2.It follows that the modulus satisfies the triangle inequalityMathworldMathworldPlanetmath

|z1+z2||z1|+|z2|,

also

|z||z|,|z||z|,|z||z|+|z|.

Modulus is :

|z1z2|=|z1||z2|,|z1z2|=|z1||z2|

Since , the definition of modulus includes the real numbers. Explicitly, if we write  x  in polar form,  x=reiϕ,  r>0,  ϕ[0,2π),  then  ϕ=0  or  ϕ=π, so  eiϕ=±1. Thus,

|x|=x2={xx>00x=0-xx<0.
Titlemodulus of complex number
Canonical nameModulusOfComplexNumber
Date of creation2013-03-22 13:36:39
Last modified on2013-03-22 13:36:39
Ownermatte (1858)
Last modified bymatte (1858)
Numerical id17
Authormatte (1858)
Entry typeDefinition
Classificationmsc 32-00
Classificationmsc 30-00
Classificationmsc 12D99
Synonymcomplex modulus
Synonymmodulus
Synonymabsolute value of complex number
Synonymabsolute value
Synonymmodulus of a complex number
Related topicAbsoluteValue
Related topicSubadditive
Related topicSignumFunction
Related topicComplexConjugate
Related topicPotentialOfHollowBall
Related topicConvergenceOfRiemannZetaSeries
Related topicRealPartSeriesAndImaginaryPartSeries
Related topicArgumentOfProductAndSum
Related topicArgumentOfProductAndQuotient
Related topicEqualityOfComplexNumbers
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更新时间:2025/5/5 4:36:52