Definition:Complex Modulus

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This page is about complex modulus. For other uses, see modulus.

Definition

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.


It is to be noted that the modulus of a complex number $z$ is the distance between the origin and the point representing $z$ on an Argand diagram.


Examples

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 $7 + 24 i$

$\cmod {7 + 24 i} = 25$


Complex Modulus of $3iz - z^2$

Let:

$w = 3 i z - z^2$

where $z = x + i y$.

Then:

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


Complex Modulus of $\tan \theta + i$

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

where:

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


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

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

where:

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


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.


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.


Sources

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