Search for a tool
Complex Number Argument

Tool for calculating the value of the argument of a complex number. The argument of a nonzero complex number $z$ is the value (in radians) of the angle $\theta$ between the abscissa of the complex plane and the line formed by $(0;z)$.

Results

Complex Number Argument -

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 community Discord for help requests!

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

# Complex Number Argument

## Complex from Argument and Modulus Calculator

Tool for calculating the value of the argument of a complex number. The argument of a nonzero complex number $z$ is the value (in radians) of the angle $\theta$ between the abscissa of the complex plane and the line formed by $(0;z)$.

### How to calculate the argument of a complex number?

The argument is an angle $\theta$ qualifying the complex number $z$ in the complex plane:

$\arg(z) = 2\arctan \left( \frac{\Im(z)}{\Re(z) + |z|} \right) = \theta \mod 2\pi$

with $\Re(z)$ the real part and $\Im(z)$ the imaginary part of $z$.

Example: Take $z = 1+i$, the real part is $1$, the imaginary part is $1$ and the modulus of the complex number $|z|$ equals $\sqrt(2)$, so $\arg(z) = 2 \arctan \left( \frac{1}{1 + \sqrt(2) } \right) = \frac{\pi}{4}$

The result of the $\arg(z)$ calculation is a value between $-\pi$ and $+\pi$ and the theta value is modulo $2\pi$

In electricity, the argument is equivalent to the phase (and the module is the effective value).

### What are the properties of arguments?

Take $z$, $z_1$ and $z_2$ be non-zero complex numbers and $n$ is a natural integer. The remarkable properties of the argument function are:

$\arg( z_1 \times z_2 ) \equiv \arg(z_1) + \arg(z_2) \mod 2\pi$

$\arg( z^n ) \equiv n \times \arg(z) \mod 2\pi$

$\arg( \frac{1}{z} ) \equiv -\arg(z) \mod 2\pi$

$\arg( \frac{z_1}{z_2} ) \equiv \arg(z_1) - \arg(z_2) \mod 2\pi$

If $a$ is a strictly positive real and $b$ a strictly negative real, then

$\arg(a \cdot z) \equiv \arg(z) \mod 2\pi$

$\arg(b \cdot z) \equiv \arg(z) +\pi \mod 2\pi$

### What is the argument of the number 0?

The argument of $0$ is $0$

## Source code

dCode retains ownership of the online 'Complex Number Argument' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Complex Number Argument download for offline use on PC, tablet, iPhone or Android !

## Need Help ?

Please, check our community Discord for help requests!