This question is related to a question asked 3 years ago on SE (I was not the OP), but not quite the same. I would like to know if there is a way to test whether a dimensionless physical constant is rational or irrational. I suspect the answer is "no"; and that it's only possible to determine that if such a constant is rational its smallest possible denominator must be larger than a value obtained by experiment.
Edit: I think we are quite firmly convinced that, for example, the number of electrons divided by the number of protons in any system is indeed a rational number. We are firmly convinced that charge comes in an integer number of packets, each with exactly the same amount of charge. Some highly respected physicists have proposed that physical quantities like volume, distance, and time come in discrete units analogous to the Planck length- which (I think) would force many dimensionless values depending only on such quantities to be rational.
This is not a mathematical question; it is a question about what is possible to test by experiment. We can count the number of teeth on a gear; we can count the number of electrons in an atom; and in each case we will know that the answer will be an integer.
I suspect it would be silly to write equations describing atomic structure using forms that allow for non-integer numbers of electrons, or irrational ratios of charge to e, but I also suspect that the main reason it would be silly is not that we have proven with absolute certainty that such values cannot exist, but rather that the assumption that numbers of electrons are always integers and charge is always an integer multiple of e has never (yet) led to contradictions with experimental results.