A complex number can be written in Cartesian coordinates as , where is the real part and is the imaginary part. Or it can be written in polar coordinates as , where is the absolute value and is the point on the unit circle corresponding to the angle measured in the usual anti-clockwise direction from the positive -semiaxis. The angle is called the argument, but unfortunately it is only defined up to multiples of . The unimodular number could be referred to as the direction of .
The complex conjugate of is the complex number .
Then the Cartesian decomposition correspond to the following additive trick:
While the polar decomposition corresponds to a multiplicative trick:
By this I mean that is twice the real part of , while is twice the imaginary part of times . And on the other hand, is the square of the absolute value of , and is the square of the direction of .