I’m writing a paper on real world applications of complex numbers and can’t understand it at all. If you have any other real world applications, that would be super helpful too. Thanks!
In: Mathematics
Much of it has to do with the function e^(ix) . e is of course “euler’s number”, 2.71818. Now it may be bizarre to have an exponent with an imaginary number in it, this idea doesn’t work within our standard understanding of exponentiation, but we do a little mathematical wizardry and distance ourselves from the idea of repeated multiplication while looking into the properties e^x has and you get this weird function e^(ix) = cos x + i sin x
Why? How? This can be explained better with calculus but even without there is a general sense in which this is true but that lies more with the fundamentals of what this mysterious number e is. If you are more interested, there is an excellent YouTube series by 3blue1brown in YouTube about this topic (Lockdown math) that he recently released during this quarantine. But take it as a given, e^(ix) = cos x + i sin x
Stepping back, this function can be considered a rotation about the complex plane. What does this mean? Well if you increase x, you always get a complex number with distance 1 from the origin if you Pythagorean theorem the real and imaginary parts, and it seems to “walk around in a circle” in this sense. This turns out to be a useful pattern to keep in mind often.
This function is convenient because it gives us a shorthand way to keep track of both sine and cosine in a single function. There are a lot of things in electronics that can be modeled by sine and cosine, you will commonly see this in “RLC circuits”, resistor inductor capacitor circuits because these tend to be modeled quite nicely by sine and cosine, some parts sine and some parts cosine. These circuits are periodic and have properties that are “circle like,” for instance the current flowing through can be modeled with e^(ix).
This function can be viewed as an exchange in some way, the real “values” being exchanged for imaginary “values,” then back to real “values” and continuing the cycle. For instance in a circuit with a capacitor and inductor, real current one moment flows into a capacitor to in this sense become “imaginary current”, then at a point this capacitor begins to discharge into the inductor to become real current once again.
This concept of rotation just happens to be very common and [even things that don’t seem to be rotations often will happen to be sums of different rotations](https://youtu.be/-qgreAUpPwM). And rotations just have a number of useful calculus properties that just appear in nature.
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