Question

What is the difference between a rectangular coordinate system and a polar coordinate system?

Answer 1

One of the most interesting differences is that every point in the plane has exactly one representation as a pair of coordinates in the rectangular (or any other parallelogram) coordinate system, but has infinitely many representations in polar coordinates.

Example:

The point whose rectangular coordinates are `(1,1)` corresponds to polar coordinates:
`(sqrt2, pi/4)` and also `(sqrt2, (9 pi)/4)` and `(sqrt2, (-7 pi)/4)` and `(-sqrt2, ( 5 pi)/4)` and infinitely many others.