I often think that a lot of math would work out more easily if we _only_ used polar coordinates and regarded the negative numbers as a "separate number line" ra

I often think that a lot of math would work out more easily if we _only_ used polar coordinates and regarded the negative numbers as a "separate number line" rather than a continuation of the positive numbers. (In particular, if you know about delta functions... a lot of weirdness around x=0 goes away if you write everything in terms of r \sgn (r) and take the derivatives of both terms. e.g. This gives "for free" the fact that the divergence of 1/r^2 is 4 pi delta(r).) I have heard of systems in which one sticks more lines out from 0 than just the positive and negative numbers. At some level that's what R^2 is, with four copies of the positive number line, but I don't see a strong reason why in principle you couldn't have an odd number of lines, which would correspond to... uh... R^1.5. But you have to define how these lines rotate into each other, and it is gonna be weird.