alt Hacker NewsImaginary numbers are strange, basically i * i = -1. So it's a square root of negative one. It's imaginary because well, you need some imagination to come to terms with this. But they are useful to show things on a 2d plane, one axis is the real numbers -1 to 1, and the other -i to i. And then multiplying by number i will rotate in circles: i × i = -1, -1 × i = -i, -i × i = 1, 1 × i = i. And then there is this wonderful property that e ^ iπ = -1, which somehow combines the euler constant, number pi and the imaginary number, and it somehow works. And then also the related formula e^ix=cosx+i sinx, and so to rotate by x you just multiply with e^ix, where x = 2π × frequency. It somehow all fits in neatly, even though none of it is essential for the mechanism described. At least that's my uneducated understanding (my math background is also not that great, that's why I tried to explain this to myself with a more intuition based approach).
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...
Am I still on the right track?