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…
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In three-phase AC current you have three sine waves, but they each reach their peak at a different time. This is called phase-shift.
Complex numbers let you do maths on all of them at once whilst preserving the phase.
Complex numbers lets you do math on a 2D plane. This is useful when you have things that can be represented using vectors. The real part represents the left- or right-ness of the vector, while the imaginary part represents the up- or down-ness.
A pendulum is an oscillation between kinetic and potential energy, it’s continuously converting one to another. An electromagnetic oscillation is similar, it converts energy between electrical and magnetic field. Because such things have two coupled but inherently different components it’s really convenient to represent them on a 2D number plane. Oscillation reduces to a vector rotation because the total energy in the system is constant represented by length of the vector, but shifting from one form to another represented by sine-cosine relation.
i suggest you read up on fourier transformations. any signal that contains periodicity can be mapped to the frequency domain, which is equivalent to decomposing it into a bunch of sine/cosine waves. remember euler’s formula for polar coordinates – that’s where the complex numbers come in
AC circuit analysis. There are components, inductances and capacitors, that need knowledge of calculus to understand how they work. But if the current changes like sine wave, all calculations are exactly the same as they are for resistors and direct current, only in complex numbers. And – wonder – no calculus needed. Just simple addition and multiplication.
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).
The pendulum is a great example of a physical system governed by second-order differential equations of motion. If you assume small-angle motion, you can approximate the EOM with a linear differential equation. Then you use an awesome tool called the Laplace transform (a more general form of the Fourier transform that someone else mentions) to transform the linear differential equation into a polynomial algebraic equation. This is where complex numbers earn their keep bigtime.
The pendulum leads to a quadratic equation. You can tell all you need to know about the pendulum’s behavior by finding the roots of that quadratic equation. There will always be two roots (because all quadratic equations have two roots). If the pendulum is undamped, the roots are pure imaginary. If there is energy dissipation, the roots are complex conjugates, with negative real parts. If something is putting energy in (“negative damping”), then the roots are complex conjugates with positive real parts.
You can transform back into differential equation land, thus finding the time-domain solution to the differential equation, but you can pick off a lot of useful information just by where the roots are in the complex plane. Does any root have a positive real part? Then the system is unstable. How far are the roots from the origin? That gives you the natural frequency of the system. How far are the roots from the imaginary axis? That tells you how well damped the motion is.
Basically all control systems use complex numbers to develop their control laws.
When you write the differential equations for almost all real-world physical systems with controllers, complex numbers are going to show up in the solution if you’re using a Laplace transform, which is the most common way to do the analysis.
These are *everywhere*…your thermostat, your oven, your car, your phone, anything with an engine or motor, anything with a servo…they’re ubiquitous.
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