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) $.
Complex Number Argument - dCode
Tag(s) : Arithmetics, Geometry
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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) $.
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).
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)
dCode always calculates the principal argument.
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Cite as source (bibliography):
Complex Number Argument on dCode.fr [online website], retrieved on 2024-12-02,