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 $a, b \in \R$.

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 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.

It is to be noted that the term complex modulus is technically inaccurate, on the grounds that it is not the modulus itself which is complex.

However, it has been chosen for $\mathsf{Pr} \infty \mathsf{fWiki}$ for its relative ambiguity when distinguishing it from other uses of the term modulus.

Most sources prefer instead the term modulus of a complex number, which, while accurate and unambiguous, is more unwieldy and cumbersome than complex modulus.


Complex Modulus of $i$

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

Complex Modulus of $-5$

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

Complex Modulus of $1 + i$

$\cmod {1 + i} = \sqrt 2$

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 .

Linguistic Note

The plural of modulus is moduli.