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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

## Argument 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 $$z$$:

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

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 $$2\pi$$

The argument of $$0$$ is $$0$$.

### 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

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

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