What complex numbers are and what they are used for.
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They are used to pack an extra dimension into algebra and trigonometry so you don’t have to use integral calculus to solve engineering problems. The real part of the number represents the x axis, the imaginary part of the number represents the y axis, and Euler’s formula of e^ix = cos x+ i sin x relates them to pi and the natural log because e^(i*pi)=-1.
Natural logarithms model natural processes perfectly with simple equations. For example, charging a capacitor with a fixed current results in a voltage that has an e^x shape. When we want to calculate the current that results from changing the voltage on a capacitor, we have to use the formula I=C*dV/dt, then integrate.
In electrical engineering we use complex numbers to describe the behavior of AC circuits by mapping the periodic cycle of a sine wave onto a unit circle. This allows us to use trigonometry to transform amplitude and phase angles into vectors. This allows us to calculate the impedance of a capacitor or inductor at a specific frequency using simple linear algebra instead of solving an integral, because Euler’s formula maps e^x to complex space. Thus, an electrical circuit that would take hours to solve with calculus takes minutes to solve with linear algebra, all because Euler’s formula relates imaginary numbers to logarithms to x-y coordinates.
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