[Fargren]

That depends on how you define your representation. It is useful for some problems to represent an interval with it's first bits, e.g. to take 0.101 as representing all numbers in the [0.101, 0.110) interval.

[konstruktor]

You are correct. What I meant to say was: You cannot represent all the distinct real numbers in a nonempty interval using a finite number of bits.

[mikeash]

Why is that relevant?