Tool for converting complex numbers into exponential notation form and vice versa by calculating the values of the module and the main argument of the complex number.

Complex Number Exponential Form - dCode

Tag(s) : Arithmetics, Geometry

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Tool for converting complex numbers into exponential notation form and vice versa by calculating the values of the module and the main argument of the complex number.

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

Example: $ 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 $

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Source : https://www.dcode.fr/complex-number-exponential-form

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