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A real number is said to be Decimal Alexandrian if its decimal representation contains every possible finite decimal sequence. It is a popular question whether $\pi$ is Decimal Alexandrian, or even in other bases. It is suspected to, but a proof has not been found yet. My question is whether any familiar number is. By familiar I mean Something like $0, 5, -8, \pi, e, \sqrt5, \phi, \gamma$, or a billion.

It is easy to generate a number just by picking its expansion, for example the number 0. 00 01 02 10 11 12 20 21 22 100 101 110 111 200 201 210 211 .... is Ternary Alexandrian. But are there any proofs that a well-known real is Alexandrian?

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  • $\begingroup$ Every normal number meets this criterion. This is almost every real number, but almost no familiar number. For example, rationals are neither normal nor Decimal Alexandrian. $\endgroup$
    – vadim123
    Oct 9, 2013 at 21:12
  • $\begingroup$ I'm not familiar with normal numbers. Does every normal number contain every finite sequence, or do they just do so "on average"? $\endgroup$
    – Daron
    Oct 9, 2013 at 21:15
  • $\begingroup$ Where did you see the term "Decimal Alexandrian"? There isn't a well-known standard term for this notion at this time, as far as I know, but "disjunctive in base $10$" has been used in several papers. Related questions: Irrational Numbers Containing Other Irrational Numbers and Prove there are no hidden messages in Pi. $\endgroup$
    – Dave L. Renfro
    Oct 9, 2013 at 21:20
  • $\begingroup$ Daron, for a (hopefully) fairly down-to-Earth proof that every normal number contains every finite sequence, see this 23 October 2000 post of mine, specifically the paragraph that beings with: A normal number is one in which each block of digits occurs $\endgroup$
    – Dave L. Renfro
    Oct 9, 2013 at 21:44
  • $\begingroup$ It came from my head: I thought Alexandrian was a good name because an Alexandrian number contains every book ever written. $\endgroup$
    – Daron
    Oct 9, 2013 at 22:04

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(Added in response to comment) According to Wikipedia, no number that arises naturally from algebra or analysis is known to be normal. All known examples are contrived by concatenation of decimal (or other-base) digits.

The Champernowne constant 0.123456789101112131415161718192021222324...., is pretty well known. By construction, it has every finite string of decimal digits not prefixed by zeros. To see that it also has those prefixed by zeros, for example 000000000849102276, just note that such a string is a substring of the same string prefixed by 1: 100000000084912276.

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  • $\begingroup$ What I am really interested in is what a proof of normality/Alexandrianity/disjunctivity of a number defined in an algebraic/analytic manner -- without referring to its decimal expansion, say $\sqrt 5$, is -- would even look like, $\endgroup$
    – Daron
    Dec 14, 2013 at 0:41
  • $\begingroup$ @Daron: I added to my answer. $\endgroup$
    – John Bentin
    Dec 14, 2013 at 9:34

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