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Complex Number Exponential Form

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.

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Complex Number Exponential Form -

Tag(s) : Arithmetics, Geometry

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Complex Number Exponential Form

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Complex Number Converter

From a Complex Number a+ib


From values a and b (in a+ib)



From modulus and argument



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.

Answers to Questions

What is the exponential form of a 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.

What is Euler's formula?

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 \( \theta \in \mathbb{R} \)

What are the properties of complex exponentiation?

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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