# Is there a way to test whether dimensionless physical constants are rational?

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.

• I suppose that depends on what you mean by "observed". "pi" exists, and is irrational - but if we try to measure "pi" by counting, we can only come up with a rational value. – S. McGrew Aug 16 at 5:59
• @safesphere, we do know the value of * π*. We just can't write it out in our inherently rational decimal notation -- because it is irrational. – S. McGrew Aug 16 at 13:55
• Depends what "defined" means. "The ratio of the circumference to the diameter" doesn't refer to infinity, but all the ways people have developed to quantify that ratio- or even write it down in rational notation- require an infinite number of steps. There's no doubt it's possible to find mathematicians, philosophers, or even physicists who are willing to argue that irrational numbers don't exist. I think there are an overwhelmingly larger number of those who believe "pi", "e", and the diagonal of an arbitrary rectangle with integer sides really do exist. – S. McGrew Aug 17 at 0:34
• And note that if we (for some odd reason) used π as the base of our number system there would almost certainly be some who would insist that rational numbers, which could only be represented as an infinite series, don't exist. – S. McGrew Aug 17 at 2:43
• The integer \$10^100 doesn't exist because it can't be observed or measured? It can certainly be meaningfully defined, which I think means it can be meaningfully quantified-- unless "quantified" is taken to mean "expressed as a base-10 decimal number with a finite number of digits". – S. McGrew Aug 17 at 2:51