Let be an irrational number. As long as there exists an algorithm the can decide whether or for any given rational , you can obtain arbitrarily good rational approximations for . Especially, you can find upper and lower bounds good enough to uniquely determine any desired number of decimals.
For , the decision algorithm is quit simple: If with, then .
Suppose that for every irrational number there were an algorithm that takes a natural as input and returns the -th digit of . The possible algorithms are countable, all the irrationals are not, hence it is not possible to have such algorithms for every irrational.
However, such algorithms do exist for the so-called computable number.