I'm very curious about the relationship between $\pi$ and irrational numbers in general. Whether $\pi$ contains all possible number combinations is unknown, but expected to be true.

What "kind" of irrationals numbers does $\pi$ contain? Consider the set $$ S=\{\alpha\pi+\beta\,\mid\,\alpha,\beta\in\mathbb{Q},\,\alpha\neq 0\}. $$ Since $\pi$ contains only countably many infinite digit strings, we have that $S\subsetneq\mathbb{R}\backslash\mathbb{Q}$. What else can we say about $S$?

For example, one may ask the following questions:

  • Is it true that $\sqrt{2}\in S$? If not, what prevents it from being true?
  • What about $e$? Are $S$ and transcendental numbers relatable somehow?

Any deeper insights are appreciated.

sam wolfe
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    $\sqrt2\notin S$, since that would mean that $\pi$ would be algebraic. The answer to your second question is much much harder to show, but it is also false. – Rushabh Mehta Sep 13 '19 at 15:45
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    It seems like it would be hard to find any special properties of $S$ that don’t also apply to other sets containing all numbers in the form $a\alpha+b$ for some irrational $\alpha$. – Franklin Pezzuti Dyer Sep 13 '19 at 15:47
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    @FranklinPezzutiDyer In fact, if we defined $S$ with (fore example) $\sqrt 2$ instead of $\pi$, it would be a subfield of $\Bbb R$, i.e., something much more specific. The fact that $\pi $ is transcendental over $\Bbb Q$ means quite precisely that we can indeed view $\pi$ as a generic variable $X$ here – Hagen von Eitzen Sep 13 '19 at 15:56
  • You might find it interesting to consider and compare the continued fraction representations of various irrationals. For example, for non-zero $\alpha$, $\alpha\sqrt{2}+\beta$ is an irrational solution of a quadratic equation and so its continued fraction representation is periodic, whereas that of $e$ exhibits a curious pattern, etc. –  Sep 13 '19 at 17:22

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