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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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 \( z = \sqrt(2) e^{i\pi/4} \)

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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