Question

Can you ever use a calculator to determine if a number is rational or irrational

Answer 1

By definition and according to Google:

"Irrational numbers cannot be represented as terminating or repeating decimals."

So the answer to your question would have to be 'no' as calculators were designed to give us approximate answers when dealing with irrational numbers.

It would be hard to guess whether a large number produced by a calculator is irrational or not given that fact that many rational numbers can be incredibly long.