Complex Number Argument

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 $.

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

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

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 $

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 $

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

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

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)