Why is φ\varphi called “the most irrational number”?

I have heard φ called the most irrational number. Numbers are either irrational or not though, one cannot be more “irrational” in the sense of a number that can not be represented as a ratio of integers. What is meant by most irrational? Define what we mean by saying one number is more irrational than another, and then prove that there is no x such that x is more irrational than φ.

Note: I have heard about defining irrationality by how well the number can be approximated by rational ones, but that would need to formalized.

Answer

How good can a number α be approximated by rationals?
Trivially, we can find infinitely many pq with |αpq|<1q, so something better is needed to talk about a good approximation.
For example, if d>1, c>0 and there are infinitely many pq with |αpq|<cqd, then we can say that α can be approximated better than another number if it allows a higher d than that other number. Or for equal values of d, if it allows a smaller c.

Intriguingly, numbers that can be approximated exceptionally well by rationals are transcendental (and at the other end of the spectrum, rationals can be approximated exceptionally bad - if one ignores the exact approximation by the number itself). On the other hand, for every irrational α, there exists c>0 so that for infinitely many rationals pq we have |αpq|<cq2. The infimum of allowed c may differ among irrationals and it turns out that it depends on the continued fraction expansion of α.
Especially, terms 2 in the continued fraction correspond to better approximations than those for terms =1. Therefore, any number with infinitely many terms 2 allows a smaller c than a number with only finitely many terms 2 in the continued fraction. But if all but finitely many of the terms are 1, then α is simply a rational transform of ϕ, i.e. α=a+bϕ with aQ,bQ×.

Attribution
Source : Link , Question Author : PyRulez , Answer Author : Hagen von Eitzen

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