what are some applications of complex numbers?

Question

So I’m looking for some applications that are relatively easy to understand, ideally in like CS or physics, but anything works. I’ve seen it be talked about in discussions of oscillations (I have a pretty good basic understanding of pendulum motion, for example, but I don’t understand how complex numbers come in), or in AC currents (I have like 0 background in electricity and magnetism apart from basic high school physics with super basic DC systems), but I can never understand how we use complex numbers for that…

In: 0

Answer 1

Waves. Including in electricity, probability, and various other places.

You might know of the result:

> e^(iθ) = cos θ + i.sin θ

Trig functions like sine and cosine can be a real pain to work with, particularly if you are multiplying them together (or differentiating them). There are identities that help, but they are messy.

But if we have a trig function (and there are some tricks we can use to rearrange them into a single sine or cosine) we can can pretend that it is the real part of some complex exponential.

Exponentials a really easy to work with, particularly when you multiply them (you just add the arguments) and differentiate them (they stay the same).

A whole bunch of problems involving waves can become so much simpler if you turn them into complex exponentials, do the maths there, and then take the real part when we get our final answer.

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There are also some neat tricks we can do with analysing waves (particularly with signals) involving complex numbers.

If we have a bunch of waves (i.e. signals) stacked on top of each other, each with their own frequencies and magnitudes, there is a neat maths trick called a Fourier Transform (well, a bunch of related tricks called Fourier and Laplace and other transforms) that we can use that takes our wavy function and pulls out the specific frequencies. We can also use it to construct any shaped wave out of sine and cosine waves of different frequencies. [This diagram](https://commons.wikimedia.org/wiki/File:Fouriertransform_av_Trikromatisk_str%C3%A5ling.gif) is in Norwegian, but you can see the idea: We have a weird signal (middle graph) that is made up of three different waves (left graph). If we do some sneaky maths involving complex numbers we can extract those three signals into their specific frequencies (the right graph).