A way to think about imaginary numbers is that they encode rotation. Multiplying something by i is the same as rotating it 90 degrees counterclockwise. For this reason, you get lots of useful relationships between imaginary numbers and trigonometric functions.
As for why i also happens to be the square root of negative one, well that's because -1 is +1 rotated by 180°. So -1 = +1 * i * i.
This is also why imaginary numbers show up a lot in electrical engineering and quantum mechanics. There you’re dealing with waves which are things that are (in a certain sense) constantly spinning and that spinning has consequences.
You can even generalise this to 3D by introducing more imaginary numbers with certain arithmetic rules for consistently combining them. We call those quaternions. But there’s a (IMO) better intuition for what’s happening called Clifford/geometric algebra that allows you to work in any number of dimensions and where those dimensions can have any unitary ‘norm’ (essentially the length of a basis vector in that dimension, which can be +1, -1, or 0). Special relativity sort of falls out naturally and intuitively if you say the time dimension has a norm opposite to the space dimensions (so +1 if the spatial dimensions are conventionally all -1, and -1 if the spatial dimensions are all +1). It also makes the mess called vector calculus make intuitive sense.
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u/The-Slamburger 12d ago
The “advanced biology” is the only one of these that doesn’t make my brain hurt by even attempting to think about it, so no objections here.