Careful -- because R is uncountable, there is no system for representing arbitrary irrational numbers with finite strings. All irrational numbers with periodic continued fractions are quadratic irrationals, i.e. they can be written as A + B sqrt(C) where A, B, C are all rational. And of course, that formula immediately provides a much more straightforward way to represent such irrational numbers by finite strings!

Yes, I avoided the word 'arbitrary'. The references make it clear that what they're dealing with is the computable reals - only a countably infinite subset of R. Other representations get used too (like streams of dyadic rationals), and the computable reals contain more than just quadratic roots - eg: pi - but the computable reals are all you get, and this means there's some rough edges - http://en.wikipedia.org/wiki/Specker_sequence.

Still fascinating though.

If you are interested, I have been playing with some of these ideas here: https://github.com/mvr/cf