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 conjugat*e 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 .