Computers, like humans, can only represent a vanishingly tiny proportion of “real numbers”, either by giving specific ones a name, or some kind of symbolic formula by which to compute with them. To do practical computations we need to break everything down into basic arithmetic operations, which ultimately means we have a concrete multiplication/addition(/other operation) table encoded somewhere, e.g. in silicon.

One that I like for doing exact arithmetic on numbers represented by black-box functions is a continued fraction representation, see http://perl.plover.com/yak/cftalk/INFO/gosper.txt

(Note: floating point is successful because trying to use irrational numbers in computations is most of the time a huge waste of computational resources compared to approximating with a binary fraction.)