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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 $(0;z)$.

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Complex Number Argument -

Tag(s) : Arithmetics, Geometry

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# Complex Number Argument

## Complex from Argument and Modulus Calculator

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

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

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

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 $0$ (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 sometimes some consider it to be 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.

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