The modulus of a complex number
, also called the complex norm, is denoted
and defined by
![|x+iy|=sqrt(x^2+y^2).](/images/equations/ComplexModulus/NumberedEquation1.svg) |
(1)
|
If
is expressed as a complex exponential (i.e., a phasor),
then
![|re^(iphi)|=|r|.](/images/equations/ComplexModulus/NumberedEquation2.svg) |
(2)
|
The complex modulus is implemented in the Wolfram Language as Abs[z],
or as Norm[z].
The square
of
is sometimes called the absolute square.
Let
and
be two complex numbers. Then
so
![|(c_1)/(c_2)|=(|c_1|)/(|c_2|).](/images/equations/ComplexModulus/NumberedEquation3.svg) |
(5)
|
Also,
so
![|c_1c_2|=|c_1||c_2|](/images/equations/ComplexModulus/NumberedEquation4.svg) |
(8)
|
and, by extension,
![|z^n|=|z|^n.](/images/equations/ComplexModulus/NumberedEquation5.svg) |
(9)
|
The only functions satisfying identities of the form
![|f(x+iy)|=|f(x)+f(iy)|](/images/equations/ComplexModulus/NumberedEquation6.svg) |
(10)
|
are
,
, and
(Robinson 1957).
See also
Absolute Square,
Absolute Value,
Complex Argument,
Complex
Number,
Imaginary Part,
Maximum
Modulus Principle,
Minimum Modulus Principle,
Real Part
Related Wolfram sites
http://functions.wolfram.com/ComplexComponents/Abs/
Explore with Wolfram|Alpha
References
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, p. 16, 1972.Krantz, S. G. "Modulus of
a Complex Number." §1.1.4 n Handbook
of Complex Variables. Boston, MA: Birkhäuser, pp. 2-3, 1999.Robinson,
R. M. "A Curious Mathematical Identity." Amer. Math. Monthly 64,
83-85, 1957.Referenced on Wolfram|Alpha
Complex Modulus
Cite this as:
Weisstein, Eric W. "Complex Modulus."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ComplexModulus.html
Subject classifications