If you can compute a number's digits (i.e. approximate it), then you can see if they start repeating or not. If you compute a billion digits and they don't repeat yet, and you have no reason to expect the number to be a fraction with a huge denominator, then a good conjecture is that the number is irrational and vice versa when there is repetition. You'll be right more often than not.

(Of course, like many heuristics this one is easily exploitable by an adversary trying to make you give the wrong answer. And clearly this is not a proof, just a rule of thumb where you expect to be wrong now and then but not usually.)