Eli5: why do we incorporate imaginary numbers in various physics equations.
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To add to the other answers, a big reason we use imaginary numbers in physics is to make the maths easier.
If you want an analogy, imagine walking along a path, and finding something blocking it. You could try to climb over it, or break it apart, or move it, but that is a lot of work. Or you could just hop off the path briefly, walk around the thing blocking the path, and then hop back onto it.
So there are a bunch of areas of physics where we *could* work entirely with real numbers, but the maths would be a pain. It is much easier to turn our real thing into a complex thing, work with that, and then drop the imaginary part at the end.
This is particularly true with waves. Waves involve trig functions (particularly cosine and sine). Trig functions are a pain to work with (as you might know if you’ve had to deal with them in maths). But there is an identity that lets us link trig functions with complex exponentials:
> e^(ix) = cos(x) + i.sin(x)
exponentials are lovely to work with. Much easier and simpler than trig things (most of the time). So if we’re dealing with some nasty trig waves, we can do a bit of mathematical trickery to replace them with complex exponentials, do all our calculations and work with them (much easier than with the trig thing), and then just drop the imaginary part at the end.
It’s also worth emphasising that imaginary numbers aren’t any more “imaginary” than any other number (other than maybe 1 and 0). It is a little unfortunate that the labels “imaginary” and “real” stuck. “Complex” is a better term; complex numbers are like other numbers, just a bit more complicated.
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