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Comment by oneshtein | Hacker News Reader

> One of the properties that π is conjectured to have is that it is normal, which is to say that its digits are all distributed evenly

.(0123456789) has this property too.

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Being normal requires not only the correct frequencies of single digit substrings, but of k-digit substrings as well, for all k. One number that is normal to base 10 is Champernowne's number [1].

[1] https://en.wikipedia.org/wiki/Champernowne_constant

.(0123456789) is rational, though. I think the implication is that if pi is normal in addition to its other properties — specifically irrationality — that it has to contain all possible digit sequences.

Edit: oops, I forgot about n-length sequences of digits, .(0123456789) is definitely not normal, this is why I’m not a mathematician.

Pi may contain any sequence of digits but it's not proven that it contains all sequences of digits. IMHO, it's possible to construct a sequence of digits which will not be in Pi. It's easy to construct such sequence for first N digits of Pi of any length (for example, N zeroes).

Constructing things for N digit subsets can be misleadingly easy.

It's easy to construct a sequence of digits that is larger than any N digit number. But it's obviously impossible for any such number to be the largest number.

We’re talking about a hypothetical scenario in which pi is normal, though.

This reasoning is true of every real number, yet it's been proven that almost all real numbers are absolutely normal and therefore contains every finite sequence of digits

It's a slight mistatement of what a normal number is that changes it significantly. It isn't that each digit is evenly distributed, but that for every n, every string of digits n long is evenly distributed.

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