Eli5: why do we incorporate imaginary numbers in various physics equations.
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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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