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I'm currently researching the question 'Are some irrational questions more irrational than others?' for an extended project qualification I aim to write. My research has led me to seeing how spokes on a virtual flower can show the golden ratio as the most irrational number, as it is the hardest to approximate. But further research has led me to discovering the liouville-roth irrationality measure, which finds pi to be the most irrational. As an A-level maths and further maths student I am fascinated by this topic but find some of the content available online a little hard to understand. I was wondering if anybody had any opinions or explanations that could help me.

eamdin
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  • No, $\pi$ is not the "most irrational". Its irrationality measure is believed to be $2$, just like $e$ and $\phi$. It's just that nobody has been able to prove that: the best bound so far (according to Wolfram MathWorld is 7.10320533 due to Zeilberger and Zudlin. And having a high irrationality measure does not really make a number "more irrational". – Robert Israel May 06 '20 at 16:51
  • The golden ratio is the worst number regarding closeness of approximation by rational numbers (in a certain precisely defined sense I'll skip defining here), but it's very well behaved in other ways, for example being a quadratic irrational (in some ways, the simplest type of irrationality). Numbers that are especially good regarding closeness of approximation by rational numbers, on the other hand, are forced to be transcendental, which is considered strongly irrational in certain ways. This "especially good approximation property" is an often-used method to show (continued) – Dave L. Renfro May 06 '20 at 16:52
  • certain numbers are transcendental when it can be used (some transcendental numbers do NOT have the "especially good approximation property", so other methods have to be used to prove they are transcendental). See Chapters 6 and 7 of Ivan Niven's book Numbers: Rational and Irrational for a very elementary and informative (Niven was very knowledgeable about these matters) treatment of the type of approximation by rationals under consideration here. – Dave L. Renfro May 06 '20 at 16:58
  • Incidentally, regarding the connection between normal numbers (which are irrational in a certain regular way) and irrationality measure, see this answer (which is a bit technical, and mostly mentioned here for others here who might be interested). – Dave L. Renfro May 06 '20 at 17:05

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