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The post A question about decimal representation of irrational numbers. asked: "Is this true that any finite word of the alphabet $\mathcal{A_9}=\{0,1,2, \ldots,8,9\}$ appears somewhere in the decimal representation of $\sqrt{2}$ ?" (A)

The given answers say it's a very similar proposition to saying "$\sqrt{2}$ is a normal number." (B)

The phrase "similar" confuses me. It is not equal (A $\iff$ B). So can A be true if $\sqrt{2}$ is an irrational number?

(The start of the sentance in A "Is this true..." suggest that there is probably a source to this question, but I couldn't find anything.)

Mount-Chiliad
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    Normal numbers are irrational, always. (good exercise) Normality is a stronger property than the simple occurrence of arbitrary strings since it also specifies that each string of a given length is equally probable. – lulu Feb 19 '22 at 12:05

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No, numbers in which every finite sequence of digits occurs are called disjunctive, disjunctivity is weaker then normality, in the sense being normal implies being disjunctive but the converse is not true.

Take a look at this https://oeis.org/wiki/Disjunctive_numbers for more info.

As explained in the link $0.10200300000040000000000005...$ is disjunctive in Base $10$ but not normal.

Vivaan Daga
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  • So this property is called disjunctive, thank you. I'm just not 100% clear on the "disjunctive in base n" part.

    Does it go in the direction of larger bases can't fit into smaller? Would the statement, "If it is disjunctive in base n, it would be disjunctive in all bases k with k < n" be true?

     – Mount-Chiliad Feb 19 '22 at 13:46
  • Which are irrational numbers are known to be disjunctive? – Mount-Chiliad Feb 19 '22 at 14:15
  • https://oeis.org/wiki/Champernowne_constant it’s even transcendental. – Vivaan Daga Feb 19 '22 at 14:16
  • Do you know any "non-trivial" that are "like" Pi or Sqrt(2). Or are these kind of numbers unknown to be disjunctive and is this "unknown" an "it is assumed that they are but not proven"? – Mount-Chiliad Feb 19 '22 at 14:22
  • Yes, it’s assumed they are disjunctive but not proven. – Vivaan Daga Feb 19 '22 at 14:23