It’s a common thinɡ to call the golden ratio (√5 + 1)/2 “the most irrational number” (corresponds to YES). Howeveɾ, it can probably be called “the most rational irrational number” (corresponds to NO).
There is at least an argument that a rational is Diophantine-approximated by other rationals even more badly than irrationals are (so the baddest of them, golden ratio and its equivalents, are the most rational). I tried to restate that in terms of continued fractions and then it seemed the golden ratio is not too rational: rational numbers’ partial quotients end up an infinite sequence of ∞, when it is goodly-approximable irrationals that do have large partial quotients (you can get a nice approximation if you trim a continued fraction right before a large partial quotient).
So you decide, maybe you have good arguments for NO! Or even good arguments for YES, who knows… 🤔🙄 I’ll consider your thoughts and references, though arguments from tradition (“it have been always called this, why change”) obviously don’t count. I’ll try to make a decision which of YES or NO sides’ arguments are more to the point. If I end up thinking both notions are quite bad for intuition after all, then I resolve N/A.
Thoughts about other (non-Diophantine) ways to approximate by rationals are very welcome if nontrivial.
Clarifications:
Don’t consider this notion (if you bet on YES or NO and don’t expect N/A) to be readily helpful to math beginners which aren’t yet firm on the ground about rationals/irrationals. When in doubt, expect there to be a comment about equivalence: that “most”/“least” is applied to the class of numbers with continued fraction tail 1, 1, 1… (Likewise, “the” second most/least number will be any of those with tail 2, 2, 2…, and the third—any with tail 1, 1, 2, 2, 1, 1, 2, 2… and so on in a non-obvious fashion which is outside of the scope of this market description.)
Related questions
So if I'm interpreting your equivalence statement correctly, you're looking for a preorder on irrational numbers that meaningfully relates to the irrationality of those numbers and have a greatest/least set of equivalent elements? Assuming you found a satisfactory one, the market would resolve to whether the golden ratio is part of either equivalence class?
@Imuli Yep. Moreso I have a candidate preorder, induced by corresponding Lagrange numbers: https://en.wikipedia.org/wiki/Lagrange_number — it’s just that it’s not obvious if Dedekind approximation to which it is related, is the way (@MichaelLucy’s point 3 in past comments).
@TexanElite Not necessarily misunderstood. Liouville numbers are both transcendental and the best Diophantine-approximable. And quadratic irrationals which are the worst as a group, are simplest algebraics. Looks sus to me.
Three independent arguments for No, to be considered independently instead of together:
"Rational" vs. "irrational" is a binary, not a scale. A number is either rational or not. A more precise thing you could say is that the golden ratio is "the least well-approximable irrational number". Calling the golden ratio the most-irrational number confuses people who are already having trouble separating the formal definition of rational vs. irrational from their intuitive sense of what the words mean, by making irrationality seem like a continuum rather than a binary.
The golden ratio is not the most irrational number. There are an infinite number of equally-irrational numbers, such as the golden ratio minus one. Calling the golden ratio the most irrational number makes it seem like it's special or a unique global maximum, when it's not.
The notion of well-approximability that is used to justify calling the golden ratio the most irrational number is sort of arbitrary. (I'm using the version in https://web.archive.org/web/20181008125136/https://math.arizona.edu/~jaytaylor/files/york/FurtherNumberTheory.pdf ; if there's a more strongly-justified reason for that notion of approximability, or a different notion that gives the same answer, I'd be interested in hearing it.) There are plenty of other plausible notions of what it means to be approximable, or what it means to be the "most irrational". Calling the golden ratio the most irrational number makes it seem like this notion of approximability is intrinsic to the definition of an irrational number, when it is in fact a separate additional thing.
e.g. you could have a notion of "most irrational" based on the idea that all rational numbers have terminating or repeating decimal expansions, and say the most irrational numbers are the ones that have the lowest expected internal shared substring length as a function of number of digits computed.
@MichaelLucy Thanks, these are pretty good! (For the investigation, not for this market. 😁)
(1) and (2) can be objected to or accepted depending on what level one would want to use the soft notion of “most/least irrational”: for general public, I would expect harm too, but for a bit more learnedness I’d guess it to be okay (treat “most” modulo equivalence like category theorists treat “the” modulo natural isomorphism). And I see now I didn’t clarify which level we intend to be on. I’ll think about that.
Usually I think about kind of sophisticated-intermediate-leveled people, as thinking about beginner level also requires considering pedagogy and not just what looks helpful for intuition. But when a person is sufficiently math-aware, so to speak, succumbs to playing the game and doing it carefully, then our hands are freer. It’s not a foolproof tactic of course, as demonstrated by “monads are monoids in the category of endofunctors”: there is good intuition in this but it’s not available on time, ever. (Probably? CMV.) So I need to decide how to frame this, for which level do we need to consider this allegory to be (questionably) useful in the first place. Right now I don’t know in which words.
@wadimiusz As usual with these, I’ll read the suggestions for YES and NO and try to make a decision which is more to the point. If I end up thinking both notions are quite bad for intuition after all, then I resolve N/A. I’ll add this to the description now.
@wadimiusz Why I take. For me this question has philosophical value, it’s about a soft notion which could be. Given that I’m also on a fence about trying to get a use from golden ratio and related numbers as musical intervals, or letting them go and pursuing other ideas, it has value if an obscure one.
Knowing you’re accustomed to my writings on topics like that I’m a bit surprised if that was really said seriously. 😀 If not this reply might be useful to others, I haven’t considered that the question might be weird enough.