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.

Complex Number Exponential Form - dCode

Tag(s) : Arithmetics, Geometry

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*!

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.

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} $

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).

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 $

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, breaker, 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, mobile, tablet, iPhone or Android app!

Reminder : dCode is free to use.

The copy-paste of the page "Complex Number Exponential Form" or any of its results, is allowed (even for commercial purposes) as long as you cite dCode!

Exporting results as a .csv or .txt file is free by clicking on the *export* icon

Cite as source (bibliography):

*Complex Number Exponential Form* on dCode.fr [online website], retrieved on 2023-09-27,

exponential,notation,argument,modulus,complex,number

https://www.dcode.fr/complex-number-exponential-form

© 2023 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.

Feedback