There's a fair point there, which is that some computations may look hard simply because we picked a difficult representation, but I'm not sure that applies her

There's a fair point there, which is that some computations may look hard simply because we picked a difficult representation, but I'm not sure that applies here. There's finite, exact representation for 1/3 in whole numbers, namely itself, but as far as I know (?) there isn't one for sqrt(2) unless, say, your choice of representation is the root of some polynomial. Is there a bounded procedure that shows whether any two polynomials represent the same set of roots? Getting back to the original etymological question, my point is that irrational in the sense of 'can't be reckoned, computed' is very close to how we would see it. The Greeks simply had a different conception of computation, one grounded in finitism and constructing things geometrically.