Here is a little demonstration of arbitrary precision floating-point (actually fixed-point) arithmetic in Haskell, using Data.Numbers.Fixed in the numbers package. Using dynamicEps, we calculate sqrt(1/x) to arbitrary precision, outputting the number in a user-specified base.
Performance is not so great; the package implements Floating functions using elegant but not necessarily high performance algorithms based on continued fractions (Gosper, HAKMEM). The hmpfr package might be better. (For square root, it might also be easy to roll one's own implementation of Newton's method on Rational.)
Radix conversion of a fractional number is implemented as an unfold, similar to radix conversion of an integer.