How Many Addresses Do You Have?
In rectangular coordinates, houses only have one address. For example, if you live two blocks west and 3 blocks north of the town center your address is (-2, 3). Any other combination of numbers is the address for somewhere else.
What happens if we ask the polar bear to face the 30° meridian and then walk backwards two blocks? We've asked him to go to -2\(\angle\)30°. This is exactly the same place as 2\(\angle\)210°! Every point on the polar bear's map has two addresses.
But wait, there's more; not sold in any store. The polar bear has always measured counterclockwise to find the direction he needs to travel. What if we give him a negative angle? Just like negative numbers go the other way on the number line, negative angles turn the other direction. Let's send him to 2\(\angle\)-150°. Turning clockwise 150° is the same direction as turning counter-clockwise 210°; once again he's arrived at 2\(\angle\)210°. Now we're up to four address for every point on the map.
Order before midnight and we'll include this amazing free gift. If we ask the polar bear to measure more than 360° he keeps turning in circles until he gets to the required direction. To get to 570° he has to turn a full circle plus 210° so 2\(\angle\)570° is another address for 2\(\angle\)210°. Add or subtract as many multiples of 360° as you want to the angle in an address and the polar bear gets to the same spot. Every point on the polar map has an infinite number of addresses.
The polar bear doesn't like getting dizzy so he figured out that he can take any big angle and divide by 360° and use the remainder to determine how far he needs to turn. For instance 1234° / 360° = 3 R 154° so he can turn 154° and stop instead of spinning around nearly 3 1/2 times.
Converting Between Coordinate Systems