# What is the modulus of a complex number?

If a complex number is expressed in polar coordinates (i.e. as $r \left(\cos \theta + i \sin \theta\right)$), then it's just the radius ($r$).
If a complex number is expressed in rectangular coordinates - i.e. in the form $a + i b$ - then it's the length of the hypotenuse of a right angled triangle whose other sides are $a$ and $b$.
From Pythagoras theorem we get: $| a + i b | = \sqrt{{a}^{2} + {b}^{2}}$.