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Complex Number Exponential Form

Tool for converting complex numbers into exponential notation form re^i 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

## Complex Number Converter

### What is the exponential form of a complex number? (Definition)

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 both a complex modulus calculator tool and a complex argument calculator tool.

### 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 $\theta \in \mathbb{R}$

### How to convert Cartesian coordinates into polar coordinates?

The conversion of complex cartesian coordinates into complex 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).

### 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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