Bijection between computable reals and rationals?

Question

This wikipedia article

http://en.m.wikipedia.org/wiki/Computable_number#Properties

suggests that there is such a bijection. How does it look like? And how to map computable transcedentals like pi to a rational?

Answer 1

The only construction I know is really a brute-force sort of thing: I'm pretty sure anything that can be proved to be such a bijection cannot be computable.

Every computable real number has at least one program that computes it.

Every program can be expressed as a pattern of bits.

Every pattern of bits corresponds to a natural number.

Thus, there exists an injective function that maps computable real numbers to natural numbers.

The numbers in the image can be listed in order, which amounts to putting them in bijection with the set of all natural numbers.

There is a bijection between the set of all natural numbers with the set of rational numbers.

Thus, combining everything together shows the existence of a bijective map from computable reals