Eulers Formula

Eulers formula is the following:

however, it can be rewritten (and in which form is more important to us) as for following:

This can be plotted on the complex plane as per the following:

We can see theta represents the angle from the x "real" axis in radians.

This equation states, that along the unit circle (a circle centered at 0,0 with a radius of 1), you can define a point either by using the left side or the right side of this equation, and convert between the two.

If we want to have a different length vector other than 1, we need to multiply by said length. On the left side, we would be left with the following:

Where A would be the magnitude (or Amplitude, or length), and theta would be the phase. This simpler way of writing complex numbers into a form that represents Amplitude and Phase is extremely useful for Fourier Analysis. In fact, these 2 variables of A and theta may be referred to as "Fourier coefficients."

Before continuing, be sure to understand the previous equation applying Eulers Formula, and how changing the amplitude and phase values will change the vector/point that would be drawn on the complex plane.