how do imaginary numbers work, and what are they used for
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Mathematicians don’t like being told they can’t do something. They like taking existing patterns and finding ways to generalise them, or come up with new patterns.
Tell a mathematician that there is no number that, when you add it to 5, gives you 4, and they’ll create -1, and negative numbers.
No number that multiplies 2 to give 1? Now we need fractions.
No number that squares to 2? We get irrational numbers. And so on.
Mathematicians take a “gap”, and try to come up with an extension to the existing rules that covers this gap in a way that is consistent and (ideally) useful.
Imaginary numbers are another step in this process. We find there is no existing number that squares to give -1, so we define one (or arguably two), called *i*. Personally I’m not a huge fan of the term “imaginary” to define it (and “real” for other numbers), as that implies *i* is more of a mathematical construction than all the other numbers. But anyway.
Once we have this new number (or two numbers), given by *i^2 = -1*, we apply all the same rules of numbers we had before and see what happens. And we end up with some pretty neat results.
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As for what they are used for, often they are useful as a way of getting around nastier maths. You have a nasty maths problem, you use some trick to hop into the complex plane (where imaginary numbers work), where the problem becomes much easier to solve, and then you can hop back into the real line to finish off the problem.
A common example of this is with waves. Waves are generally defined by trig functions (sines and cosines). Those can be quite messy to work with. But there is a neat result from complex numbers that says:
> e^(ix) = cos(x) + i sin(x)
So if we have a messy trig wave, we can do some trig algebra to turn it into just a cosine. We can then say that the thing we are looking at is just the real part of some complex exponential. And exponentials are far easier to work with than trig functions (in some ways). We do all our maths with the complex exponential, and then just have to remember to drop the imaginary part at the end (although sometimes we don’t bother, and leave it as implied).
A lot of wave-based physics uses imaginary numbers in this way (including things like Fourier transforms for signals), and as quantum mechanics is all about wave-like behaviour, imaginary numbers make quantum mechanics a lot easier.
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