Definition:Complex Modulus

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This page is about Complex Modulus. For other uses, see Modulus.


Let $z = a + i b$ be a complex number, where $a, b \in \R$.

Then the (complex) modulus of $z$ is written $\cmod z$ and is defined as the square root of the sum of the squares of the real and imaginary parts:

$\cmod z := \sqrt {a^2 + b^2}$

The complex modulus is a real-valued function, and, as and when appropriate, can be referred to as the complex modulus function.

Also known as

The complex modulus is also known as the complex absolute value, or just absolute value.

Others use that term only for the absolute value of a real number.

The notation $\bmod z$ is sometimes seen for $\cmod z$, but on $\mathsf{Pr} \infty \mathsf{fWiki}$ $\cmod z$ is preferred.

Some linguistic purists object to the term complex modulus on the grounds that it is not the modulus itself which is complex.

Such schools of thought prefer the term modulus of a complex number.


Complex Modulus of $i$

$\cmod i = \cmod {-i} = 1$

Complex Modulus of $-5$

$\left\vert{-5}\right\vert = 5$

Complex Modulus of $1 + i$

$\left\vert{1 + i}\right\vert = \sqrt 2$

Complex Modulus of $4 + 3 i$

$\cmod {4 + 3 i} = 5$

Complex Modulus of $-4 + 2 i$

$\cmod {-4 + 2 i} = 2 \sqrt 5$

Complex Modulus of $3iz - z^2$


$w = 3 i z - z^2$

where $z = x + i y$.


$\cmod w^2 = x^4 + y^4 + 2 x^2 y^2 - 6 x^2 y - 6 y^3 + 9 x^2 + 9 y^2$

Complex Modulus of $\tan \theta + i$

$\left\vert{\tan \theta + i}\right\vert = \left\vert{\sec \theta}\right\vert$


$\theta \in \R$ is a real number
$\tan \theta$ denotes the tangent function
$\sec \theta$ denotes the secant function.

Complex Modulus of $\dfrac {1 + 2 i t - t^2} {1 + t^2}$

$\cmod {\dfrac {1 + 2 i t - t^2} {1 + t^2} } = 1$


$t \in \R$ is a real number.

Also see

  • Results about complex modulus can be found here.

Special cases

Technical Note

$\mathsf{Pr} \infty \mathsf{fWiki}$ has a $\LaTeX$ shortcut for the symbol used to denote complex modulus:

The $\LaTeX$ code for \(\cmod z\) is \cmod z .