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It seems to me like questions involving decimal vs binary representations of some number are not particularly interesting: for instance $\pi$ or $\sqrt{2}$ are conjectured to be normal in every base, and as far as I know this is open for any particular base.

On the other hand, in calculating entropy there is again a choice of basis. Further, this gives us certain 'distinguished' real numbers: e.g. the entropy of the Gauss-Kuzmin distribution is $3.432527514776...$ bits, while it is $2.379246769061...$ nats.

Are the properties of these digit strings of the same number 'similar' in any way, or is one 'nicer' in some sense?

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  • $\begingroup$ this is a bit vague. does one of those two numbers you gave look nicer to you in some sense? is that why you asked? $\endgroup$ – kodlu Jul 14 at 7:04
  • $\begingroup$ I also don't understand the question. $\endgroup$ – Kurisuto Asutora Jul 14 at 7:41
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    $\begingroup$ I understand the question as follows: some properties of numbers (such as normality) are basis-independent, others are basis-dependent (such as entropy). In the latter case, has the mathematical community come up with arguments in favor of some preferred basis in which to work? $\endgroup$ – Alex M. Jul 16 at 5:38

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