Complex Number Exponential Form

The exponential notation of a complex number $ z $ of argument $ \ theta $ and of modulus $ r $ is: $$ z = r \operatorname{e}^{i \theta} $$

Example: The complex number $ z $ written in Cartesian form $ z = 1+i $ has for modulus $ \sqrt(2) $ and argument $ \pi/4 $ so its complex exponential form is $ z = \sqrt(2) e^{i\pi/4} $

dCode offers a complex modulus calculator and a complex argument calculator tools.

Euler's formula applied to a complex number connects the cosine and the sine with complex exponential notation: $$ e^{i\theta } = \cos {\theta} + i \sin {\theta} $$ with $ \theta \in \mathbb{R} $

The conversion of cartesian coordinates into polar coordinates for the complex numbers $ z = ai + b $ (with $ (a, b) $ the cartesian coordinates) is precisely to write this number in complex exponential form in order to retrieve the module $ r $ and the argument $ \theta $ (with $ (r, \theta) $ the polar coordinates).

If the complex number has no imaginary part: $ e^{i0} = e^{0} = 1 $ or $ e^{i\pi} = \cos(\pi) + i\sin(\pi) = -1 $

If the complex number has no real part: $ e^{i(\pi/2)} = \cos{\pi/2} + i\sin{\pi/2} = i $ or $ e^{i(-\pi/2)} = \cos{-\pi/2} + i\sin{-\pi/2} = -i $