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

Feedback and suggestions are welcome so that dCode offers the best 'Complex Number Argument' tool for free! Thank you!

# Complex Number Argument

## Argument Calculator

 Result Format Automatic Selection Exact Value (when possible) Approximate Numerical Value Scientific Notation

## Complex from Argument and Modulus Calculator

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

The argument is an angle $\theta$ qualifying the complex number $z$ in the complex plane is noted arg or Arg is calculated with the formula:

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

with $\Re(z)$ the real part, $\Im(z)$ the imaginary part and $|z|$ the complex modulus of $z$.

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

To determine the argument of a complex number $z$, apply the above formula to find $\arg(z)$.

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 are the complex argument values to know?

Some arguments are trivial (argument of 1, argument of -1, argument of i, argument of -i, etc.) and can be memorized:

— $\arg( 1 ) = 0$

— $\arg( 2 ) = 0$

— $\arg( n ) = 0$ (with $n$ a positive real number)

— $\arg( -1 ) = \pi$

— $\arg( -2 ) = \pi$

— $\arg( -n ) = \pi$ (with $n$ a non zero positive real number)

— $\arg( i ) = \pi / 2$

— $\arg( - i ) = - \pi / 2$

— $\arg( 1+i ) = \pi / 4$

— $\arg( 1-i ) = - \pi / 4$

— $\arg( -1+i ) = 3 \pi / 4$

— $\arg( -1-i ) = - 3 \pi / 4$

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

The argument of $0$ is $0$ (the number 0 has a real and complex part of zero and therefore a null argument).

### What does an argument equal to 0 mean?

If the argument of a complex number is $\arg(z) = 0$ then the number has no imaginary part (it is a real number).

### What is the principal argument?

The argument is an angle, usually in radians. The angles repeat every $2 \pi$ so there is an infinite number of them.

The principal/main argument is the one between $-\pi$ and $\pi$ (but some people take the one between $0$ and $2 \pi$)

To calculate the main argument from a non-principal argument add or subtract $2 \pi$ as many times as necessary (modulo $2 \pi$ calculation)

dCode always calculates the principal argument.

## Source code

dCode retains ownership of the "Complex Number Argument" source code. Except explicit open source licence (indicated Creative Commons / free), the "Complex Number Argument" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Complex Number Argument" 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 Argument" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.

## Cite dCode

The copy-paste of the page "Complex Number Argument" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Complex Number Argument on dCode.fr [online website], retrieved on 2024-09-14, https://www.dcode.fr/complex-number-argument

## Need Help ?

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