# 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 '19 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 '19 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 '19 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 '19 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 '19 at 2:51

## 1 Answer

This is a meaningless question. Dimensionless physical constants such as the fine-structure constant or the muon-electron mass ratio can be measured only to some limited precision. To know whether a number is rational or irrational you have to know it to infinite precision.

For example, the first hundred digits might repeat every five digits, making you think it was rational, but then the 101st digit breaks the pattern. Does this mean it isn’t rational? No, it might repeat every 101 digits, or every 237,765 digits. Or it might never repeat and be irrational.

Of course, we might someday have an accepted theory that predicts a rational or irrational value for each known constant. But we will never tell by experiment which they are.

• There is a pattern starting to show up in poster's questions that they need to be aware of. In effect, physics is NOT math. Mathematical models are used to represent particular physics phenomena such that the models approximate reality to a greater or lesser degree. However, concepts such as the rationality of physics constants never come up in physics precisely because physics is not math. Such a question doesn't make any more sense than turning that question around, and asking if algebra (or calculus) conserves momentum or energy. – David White Aug 16 '19 at 4:35
• Based on our current theories as well, we do know if some of the constants should be rational or irrational, right? Of course, we can never verify the rationality or irrationality in a direct experimental probe. – Dvij D.C. Aug 16 '19 at 6:11
• I think the question may be similar to asking if the energy levels of the electrons in an isolated atom have infinitesimally thin ranges-- or instead have some degree of "absolute" uncertainty. The experiments I can imagine to measure that range would take an infinite amount of time. By analogy, it would take forever to sum all positive integer powers of 1/2 one at a time, but we have no problem with saying immediately that the sum equals exactly 1. I'm wondering if there might be analogous experimental methods that, sans an infinite number of measurements, says "rational" vs "irrational" – S. McGrew Aug 16 '19 at 14:06
• @gsmith, please explain why "we will never tell by experiment which they are [rational or irrational]". I suspect you are right; but I haven't seen a proof of that statement, except based on the assumption that all measurements are inherently statistical: that no measurement can provide absolute certainty that a given assertion in physics is true or not true. – S. McGrew Oct 11 '19 at 13:55
• @S.McGrew The first two paragraphs of my answer explain why. To know whether a number is rational or irrational you have to know it to infinite precision. No experiment can produce an infinitely precise measurement. – G. Smith Oct 11 '19 at 15:52