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

Mar 26, 2015

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 $\left(1 , 1\right)$ corresponds to polar coordinates:
$\left(\sqrt{2} , \frac{\pi}{4}\right)$ and also $\left(\sqrt{2} , \frac{9 \pi}{4}\right)$ and $\left(\sqrt{2} , \frac{- 7 \pi}{4}\right)$ and $\left(- \sqrt{2} , \frac{5 \pi}{4}\right)$ and infinitely many others.