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.