Hmm.. Imaginary numbers are indeed a bit confusing.
I'm trying to imagine a 2d surface where the X-axis coordinates are all the real numbers, and the y axis are all the imaginary numbers. That makes them orthogonal, and that seemed to add up with your explanation, up until ixi=-1.
The only way I can get that to add up is if I instead imagine a arbitrary coordinate system, where x and y are not necessarily perpendicular, and i describes the angle between x and y.
I've only just finished my first cup of coffee for the day, so I haven't quite decided yet if that makes any sense whatsoever, but it's the only way I can intuit about it that includes a circular motion like the one you describe..
In this case you could almost describe i as the square root of 180°, which... Yeah it's kinda weird...
I think so, it's called the complex plane. A complex number has a real and an imaginary component a+bi, so like a vector. The amount of each gives you the coordinates on the plane (a or b can be zero as well on the axes).
I had a chat with gpt to try and clear out some details. It seems that one is supposed to think of real and imaginary as a vector. The rotation part comes in when the imaginary numbers is used as an exponent to the real, in which case you're no longer saying "3 left, two right" but "4 units from origin, at an angle of (imaginary number)"
of course, the math here doesn't work out as using degrees or any other unit of rotation a normal person is used to, but instead, some other unit of rotation I haven't quite wrapped my head around yet (what the hell does atan2(b,a) mean? Is atan(a,b) deprecated or what? ) I didn't know namespace collisions were a thing mathematicians worried about, they should just release maths 2.0 and be rid of the legacy atan at this point!
I think it's because the normal atan receives one argument, eg atan(y/x) and then sometimes you can't divide by zero, and it can’t distinguish quadrants (because we loose info on the sign of Y-coordinate and the X-coordinate). atan2 takes 2 params and knows the signs so it can understand the quadrants and also handle divide by zero. I now realize that the name atan2 probably refers to 2 parameters.
I'm trying to imagine a 2d surface where the X-axis coordinates are all the real numbers, and the y axis are all the imaginary numbers. That makes them orthogonal, and that seemed to add up with your explanation, up until ixi=-1.
The only way I can get that to add up is if I instead imagine a arbitrary coordinate system, where x and y are not necessarily perpendicular, and i describes the angle between x and y.
I've only just finished my first cup of coffee for the day, so I haven't quite decided yet if that makes any sense whatsoever, but it's the only way I can intuit about it that includes a circular motion like the one you describe..
In this case you could almost describe i as the square root of 180°, which... Yeah it's kinda weird...
Am I still on the right track?