Definition:Complex Modulus

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

Then the (complex) modulus of $z$ is written $\left\vert{z}\right\vert$ and is defined as the square root:


 * $\left\vert{z}\right\vert := \sqrt {a^2 + b^2}$

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

Also known as
The complex modulus is also known as the complex absolute value. Others use that term only for the absolute value of a real number.

The notation $\bmod z$ is sometimes seen for $\left\vert{z}\right\vert$, but on it is not recommended.

Also see

 * Modulus is Norm, showing that the modulus defines a norm on the field of complex numbers.


 * Modulus of Complex-Valued Function


 * Modulus in Terms of Conjugate, in which $\left\vert{z}\right\vert$ is defined without having to explicitly reference the components $a$ and $b$ within $z$ as follows:
 * $\left\vert{z}\right\vert := \sqrt {z \overline z}$
 * where $\overline z$ denotes the complex conjugate of $z$.


 * Absolute Value of a real number: when $b = 0$, i.e. when $z$ is wholly real, this definition becomes $\left\vert{z}\right\vert = \sqrt{x^2} = \left\vert{x}\right\vert$, which is consistent with the definition of the absolute value of $x$.