Eli5 Why do we need complex numbers in real life?
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There are a few different applications that come to mind. Admittedly, most of them require a really good understanding of physics and calculus. So I will try to write this as if none of that is prior knowledge. I hope that others can give more tangible examples.
Most of it boils down to Eulers identity. Basically, we can replace sin(x) and cos(x) with something that uses complex numbers, which is actually easier sometimes. For example, I studied sound waves in graduate school and we used complex numbers to model how a wave moves. As a very general statement, anything that rotates or oscillates can be modeled using complex numbers. I actually still use complex numbers daily!
Another example, complex numbers are used for lots of calculations in signal processing. Every time your phone receives a signal. It uses something called the Fourier transform, which uses complex numbers, to “translate” the signal to something it understands.
So if you are learning it in school now, or teaching it to someone, it may be difficult to motivate learn complex numbers without saying “trust me, in a decade this might matter”, but if you or they become an engineer or scientist, complex numbers have a lot of uses!
TLDR: engineers and scientists use them for a lot of complex (no pun intended) calculation because they actually make things easier sometimes. 🙂
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