What complex numbers are and what they are used for.
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> what complex numbers are
Define a new number “i”, with the property that i^2 = -1. This allows us to talk about all the numbers of the form ai + b where a and b are normal real numbers; those are the “complex numbers”.
>what they are used for
One place they show up is rotation. Here’s a neat property of complex numbers: if you plot them on a 2-d grid with the +i direction straight up and the +1 direction to the right, then complex numbers are points on this grid. You can then express them in polar coordinates (i.e. as a distance r from 0,0 and an angle theta above the +1 direction). For example, i+1 would be r=sqrt(2), theta = 45 degrees. Then multiplying two complex numbers (r_1, theta_1) * (r_2, theta_2) gives (r_1 * r_2, theta_1 + theta_2). So multiplying complex numbers that are 1 distance from 0,0 is like adding angles.
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