Search for a tool
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.

Results

Complex Number Exponential Form -

Tag(s) : Arithmetics, Geometry

Share dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Thanks to your feedback and relevant comments, dCode has developed the best 'Complex Number Exponential Form' tool, so feel free to write! Thank you!

# Complex Number Exponential Form

## Complex Number Converter

### 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: 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.

### 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 complex Cartesian coordinates into complex polar coordinates?

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).

### 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 $e^{i(-\pi/2)} = \cos{-\pi/2} + i\sin{-\pi/2} = -i$

## Source code

dCode retains ownership of the "Complex Number Exponential Form" source code. Except explicit open source licence (indicated Creative Commons / free), the "Complex Number Exponential Form" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "Complex Number Exponential Form" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Complex Number Exponential Form" are not public, same for offline use on PC, tablet, iPhone or Android !
The copy-paste of the page "Complex Number Exponential Form" or any of its results, is allowed as long as you cite the online source https://www.dcode.fr/complex-number-exponential-form
Reminder : dCode is free to use.

## Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!