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I have been trying to read Penrose’s

Road to Reality

. He spends a lot of time talking about complex numbers.

Nature uses numbers that contain the square root of -1.

$$
i=\sqrt{-1}
$$

Any number can be expressed as:

In a limited way, complex numbers are 2 dimensional. Some guy defined the “Complex Plane” where the y-axis is the imaginary part and the x-axis is the real number line. Numbers can be plotted on this plane. The numbers can be expressed in polar coordinates:

where:

I learned about these numbers in circuit and electromagnetics courses at A&M. A signal that varies with the sine of the frequency time the time can be expressed as the real part of a point in the Complex Plane with a fixed value (distance from the origin) rotating about the origin at the signal’s frequency.