Search for a tool
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.

Results

Complex Number Exponential Form -

Tag(s) : Arithmetics, Geometry

Share
dCode and you

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!

Team dCode likes feedback and relevant comments; to get an answer give an email (not published). It is thanks to you that dCode has the best Complex Number Exponential Form tool. Thank you.

# Complex Number Exponential Form

## Complex Number Converter

### From polar coordinates (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.

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

### 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 source code of the script Complex Number Exponential Form online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Complex Number Exponential Form script for offline use on PC, iPhone or Android, ask for price quote on contact page !