Eli5: why do we incorporate imaginary numbers in various physics equations.
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Imaginary/complex numbers are used a lot to represent time. For instance with waves (say electrical/magnetic waves for less theoretical) you alter the phase to adjust position, and phase equations use imaginary/complex numbers.
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When you listen to recorded music for instance, the speaker you play it thru needs amplification and besides just the impedance of the the speaker it also has phase, so 2 speakers with the same frequency response and impedance could still sound different based on their phase (though newer tech like class-D amps seem to mostly be phase-independent).
Similar reason why we incorporate negative numbers.
Negative numbers are useful because they let us represent a thing having “two states.” Credit/debit, above/below sea level, towards/away, etc.
Complex numbers are used to represent cyclical things. If you multiply with -1, you end up flipping between the two directions (-1, +1, -1, +1…) but when you multiply with the imaginary unit i instead, you rotate between four directions (+1, i, -1, -i, +1…) which is very useful if we want to represent, say, the way a wave works. Quantum physics, in particular, deals with wave functions all the time.
The introduction of imaginary and complex numbers provides richer mathematical tools. Functions that are hard to analyze or integrate in the real domain might be easier in the complex domain. Think of it like cooking. Sometimes you need many pots and pans as opposed to a single wok or casserole in order to obtain a dish that will end up in a single plate (real numbers).
Often the problem with be entirely represented using real numbers, and the solution will be entirely represented in real numbers. You probably could solve the problem using only real numbers (at least most of the time), but if we use complex numbers it makes the problem easier to solve.
Complex numbers work particularly well to describe things relating to circular motion, or oscillation (which can be circular motion in one less dimension), because they give tools to handle the mathematics of this more simply than natural numbers do.
Its a way to describe multidimensional coupled problems.
For example, in electromagnetics you have two components evolving in time, electric field and magnetic field, voltage and current, charge and magnetic flux. And those two components are coupled together so you need to sometimes treat things as a single complex number that has two components, real and imaginary.
Another example is rotations in 3D space, you have three degrees of freedom, but there is coupling, there are multiple ways to get to one and the same rotation. So to describe rotations there a little bit more complicated type of complex numbers is used called quaternions.
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.
We incorporate complex numbers because they are the most convenient way to describe something oscillating back and forth. And, for convenience, we want our equations to be as simple as possible.
In particular, for classical physics, oscillation in one dimension is equivalent to a rotation in suitably chosen two dimensions. And, complex numbers “just happen” to describe just exactly that two-dimensional set of numbers. With quantum mechanics, the “just happen” is even more profound and it is very hard to come up with a simple explanation for what the second dimension really stands for.
That is, we could just as well use a specific two-dimensional number system with a two-dimensional notation like “(a,b)”. But, this would be messy as most of the time our variables are either of the form (a,0) or (0,b). To avoid the unnecessary zeros, we just add letter “i” or sometimes with electricity letter “j” to the second dimension.
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