Definition:Complex Modulus

Definition
Let $$z = a + \imath b$$ be a complex number.

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


 * $$\left|{z}\right| \ \stackrel {\mathbf {def}} {=\!=} \ \sqrt {a^2 + b^2}$$.

Real Number
Note that when $$y = 0$$, i.e. when $$z$$ is wholly real, this becomes $$\left|{z}\right| = \sqrt{x^2} = \left|{x}\right|$$, which is consistent with the definition of the absolute value of $x$.

Also see

 * Modulus of Complex-Valued Function