Points on a Plane: Representation with 1 or 2 Real Numbers?

[jeremy22511]

If there is a bijection between R and R², then why can't a point on a plane be represented by one real number instead of two?

 

[RUber]

Do you know of a bijection between R and R^2?

 

[mfb]

It can.
All those bijections are highly impractical for actual applications, using two values is much easier. If you are worried about memory: getting the same precision with a single number means you have to store (at least) twice the number of digits, so you don't gain anything.

 

[micromass]

If there is a bijection between R and R², then why can't a point on a plane be represented by one real number instead of two?

The reason why we don't do this is because we often want that representation to satisfy some other properties. The bijections between $\mathbb{R}$ and $\mathbb{R}^2$ do not satisfy many other nice properties. Some properties that they can have are addition preserving, so it can be a group isomorphism. If you don't require injectivity, then it can be continuous. But that's basically where it ends. You can't make it be smooth, or linear. So this means that the bijections are not very geometrical and thus not very useful.