modulus of complex number

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


There is also the notation


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




Modulus is :


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,

Title modulus of complex number
Canonical name ModulusOfComplexNumber
Date of creation 2013-03-22 13:36:39
Last modified on 2013-03-22 13:36:39
Owner matte (1858)
Last modified by matte (1858)
Numerical id 17
Author matte (1858)
Entry type Definition
Classification msc 32-00
Classification msc 30-00
Classification msc 12D99
Synonym complex modulus
Synonym modulus
Synonym absolute value of complex number
Synonym absolute value
Synonym modulus of a complex number
Related topic AbsoluteValue
Related topic Subadditive
Related topic SignumFunction
Related topic ComplexConjugate
Related topic PotentialOfHollowBall
Related topic ConvergenceOfRiemannZetaSeries
Related topic RealPartSeriesAndImaginaryPartSeries
Related topic ArgumentOfProductAndSum
Related topic ArgumentOfProductAndQuotient
Related topic EqualityOfComplexNumbers