Definition:Modulus of Complex Number

Definition

Let $z = a + i b$ be a complex number.

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

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

Real Number

Note that when $b = 0$, i.e. when $z$ is wholly real, this 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 $x$.

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:

$\left\vert{z}\right\vert := \sqrt {z \times \overline z}$

where $\overline z$ denotes the complex conjugate of $z$.