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
Think of the way we use negative numbers in systems in which there’s an importance to direction, or in other words an importance to if two different objects are “in sync”, i.e, possessing the same sign or “out of sync”, i.e, possessing differing signs.
You can think of a examples of this like banking, in which money you have is positive compared to money you owe which is treated as negative.
Negative numbers help us in this scenario because they allow us to encode the directionality of the money in the numbers themselves, and then treat negative money the same exact way as regular money.
In the same way, the complex numbers allow us to talk about systems in which objects can be in sync in a non binary way – as in not just, completely in sync, or completely out of sync.
An example of this would be to take two identical sine waves, move one of them so that it’s shifted compared to the other, and then observing how their sum behaves.
The waves can be in varying degrees of synchronisation or phase as it’s usually called, and this greatly affects how their sum looks like, they can be totally in phase, and then their sum is just twice of them, or they can be totally out of phase and then their sum is zero, or they can be neither this nor that and then their sum is somewhere in between.
Here’s a quick Desmos visualisation of this to help demonstrate what I mean :
https://www.desmos.com/calculator/sjzhuhiumh
You can interact with the slider to control the size of the parameter a and thus decide how much phase will there be between the sine wave.
Now it turns out that many physical phenomena can be described using this concept of being in varying degrees of phase, and in which case complex numbers can help us in our description because they are defined in a way which allows us to talk about angles between them – essentially the phase between them.
As for actual fields in which they are relevant, many commenters have given in length explanations about their usage in electrical power systems, so I thought I’d mention that complex numbers are also used extensively in quantum physics due to the deep importance of interference in quantum systems, I won’t go into detail about this because I don’t think it can be expressed well in this format but in essence many systems are alike in certain ways to the two sine waves I mentioned earlier.
Sorry if I deviated a bit from your question but I think talking about complex numbers is essentially talking about phase and why it matters.
Hope this helps you in any way with your paper!
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