Do kids not learn this in 8th grade any more? I'm seriously not trying to be snarky, I just can't think of a way to write that question that sounds unsnarky. I thought everyone learned about the polar representation of complex numbers.
I learned the polar representation, yep, but didn't connect as i as an "rotation operation" that took you from one dimension to the other. For all I knew, having i be 2-dimensional was like saying "We can represent condiments by having an axis for ketchup and and an axis for mustard".
It was 2 random quantities represented as an (x,y) pair, without the notion that they were deeply connected.
My town's school system was the second in the entire state, and they taught Algebra I (ie, factoring/FOIL) in 8th grade. I went to a well-respected private school, and they taught complex numbers in 10th grade, and polar coordinates only in the honors section of 10th/11th grade classes. For those, they taught matrix multiplication and determinants for 2x2 matrices only (ie, not 3x3 or up). This means that any real linear algebra was completely out of the picture.
This was well more than enough to score a very good score on the SAT II Math IIC exam (is this still even around?)
I was fortunate that I was able to teach myself outside of my school's curriculum (and my school also let me place out of certain classes), but overall, the standards are far below what you might hope.