Generally you can just stick to storing 2-coordinate vectors and using vector operations.
The places where you might want to convert to a 1-number representation are when you have a lot of numbers you want to store or transmit somewhere. Using the stereographic projection (half-angle tangent) instead of angle measure works better with the floating point number format and uses only rational arithmetic instead of transcendental functions.
Comments
Generally you can just stick to storing 2-coordinate vectors and using vector operations.
The places where you might want to convert to a 1-number representation are when you have a lot of numbers you want to store or transmit somewhere. Using the stereographic projection (half-angle tangent) instead of angle measure works better with the floating point number format and uses only rational arithmetic instead of transcendental functions.