Complex Numbers

A complex number can be written in Cartesian coordinates as z=x+iy, where x is the real part and y is the imaginary part. Or it can be written in polar coordinates as z=re^{i\theta}, where r is the absolute value and e^{i\theta} is the point on the unit circle corresponding to the angle \theta measured in the usual anti-clockwise direction from the positive x-semiaxis. The angle \theta is called the argument, but unfortunately it is only defined up to multiples of 2\pi i. The unimodular number e^{i\theta} could be referred to as the direction of z.

The complex conjugate of z is the complex number \bar{z}=x-iy=r/e^{i\theta}.

Then the Cartesian decomposition correspond to the following additive trick:

\displaystyle 2z=(z+\bar{z})+(z-\bar{z})

While the polar decomposition corresponds to a multiplicative trick:

\displaystyle z^2=(z\cdot\bar{z})(z/\bar{z})

By this I mean that z+\bar{z} is twice the real part of z, while z-\bar{z} is twice the imaginary part of z times i. And on the other hand, z\cdot\bar{z} is the square of the absolute value of z, and z/\bar{z} is the square of the direction of z.

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