After watching a lot of Numberphile and 3B1B videos I said to myself, you know what, I'm going back to college to get a maths degree. I switched at last moment to actuarial sciences when applying, because it's looked like a good professional move and was the best decision on my life.

Can we all at least agree that counting numbers are a joke? Sometimes they start at zero … sometimes they start at one …

I really don't think it is.

Considering we're trying to find lengths, shouldn't we be doing absolute value squared?

The short version is: we use some weird abstractions (i.e., ways of representing complex things) to do math and make sense of things.

The longer version:

Electromagnetic signals are how we transmit data wirelessly. Everything from radio, to wifi, to xrays, to visible light are all made up of electromagnetic signals.

Electromagnetic waves are made up of two components: the electrical part, and the magnetic part. We model them mathematically by multiplying one part (the magnetic part, I think) by the constant i, which is defined as sqrt(-1). These are called "complex numbers", which means there is a "real" part and a "complex" (or "imaginary") part. They are often modeled as the diagram OP posted, in that they operate at "right angles" to each other, and this makes a lot of the math make sense. In reality, the way the waves propegate through the air doesn't look like that exactly, but it's how we do the math.

It's a bit like reading a description of a place, rather than seeing a photograph. Both can give you a mental image that approximates the real thing, but the description is more "abstract" in that the words themselves (i.e., squiggles on a page) don't resemble the real thing.

Circles are good at math, but what to do if you not have circle shape? Easy, redefine problem until you have numbers that look like the numbers the circle shape uses. Now we can use circle math on and solve problems about non-circles!