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.)