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> As pi never repeats itself, that also means that every piece of conceivable information (music, movies, texts) is in there, encoded. This is true for normal

by mkl 2y ago

> As pi never repeats itself, that also means that every piece of conceivable information (music, movies, texts) is in there, encoded. This is true for normal numbers [1], but is definitely not true for all non-repeating (irrational) numbers. Pi has not been proven to be normal. There are many non-repeating numbers that are not normal, for example 0.101001000100001... Storing the index into pi for a file would usually take something like as much space as just storing the file, and storing or calculating enough digits to use that index would be impossible with the technology of today (or even probably the next century). [1] https://en.wikipedia.org/wiki/Normal_number https://en.wikipedia.org/wiki/Normal_number

tombert 2y ago
It's conjectured to be normal isn't it? I know it hasn't been proven yet, and I cannot seem to find where I read this, but I thought there was at least statistical evidence indicating that it's probably normal.
- adgjlsfhk1 2y ago
  100% of real numbers are normal, so that's pretty strong statistical evidence
  - hn_throwaway_99 2y ago
    What? No they're not, e.g. no rational numbers are normal, and they are real.
    - GraphEnthusiast 2y ago
      The rational numbers make up "zero percent" of the real numbers. It's a little hard to properly explain without assuming a degree in math, since the proper way to treat this requires measure theoretic probability (formally, the rationals have measure zero in the reals for the "standard" measure). The short version is that the size of the reals is a "bigger infinity" than the size of the rationals, so they effectively have 'zero weight'. Reference (very technical): https://math.stackexchange.com/questions/508217/showing-that-rationals-have-lebesgue-measure-zero#508225 https://math.stackexchange.com/questions/508217/showing-that...
      - hn_throwaway_99 2y ago
        But then the original implication, "100% of real numbers are normal, so that's pretty strong statistical evidence", still doesn't make any sense, as it's essentially using "100%" to imply "strong statistical evidence" that the rationals don't exist, which obviously doesn't follow.
        staunton 2y ago
        > still doesn't make any sense Right. I'm pretty sure actually that it was a joke...
        adgjlsfhk1 2y ago
        it was about half a joke. statistical evidence doesn't really exist for the type of problem since polynomialy computable numbers are countably infinite so you can't define a uniform distribution over then
        mhink 2y ago
        I got the impression that the comment was a bit tongue-in-cheek. The joke lies in the fact that saying "100% of real numbers" isn't *technically* the same thing as saying "all real numbers", because there's not really a good way to define a meaning for "100%" that lets you exclude rational numbers (or any other countable subset of the reals) and get something other than 100%.

by mkl 2y ago

tombert 2y ago

It's conjectured to be normal isn't it? I know it hasn't been proven yet, and I cannot seem to find where I read this, but I thought there was at least statistical evidence indicating that it's probably normal.

adgjlsfhk1 2y ago
100% of real numbers are normal, so that's pretty strong statistical evidence
- hn_throwaway_99 2y ago
  What? No they're not, e.g. no rational numbers are normal, and they are real.
  - GraphEnthusiast 2y ago
    The rational numbers make up "zero percent" of the real numbers. It's a little hard to properly explain without assuming a degree in math, since the proper way to treat this requires measure theoretic probability (formally, the rationals have measure zero in the reals for the "standard" measure). The short version is that the size of the reals is a "bigger infinity" than the size of the rationals, so they effectively have 'zero weight'. Reference (very technical): https://math.stackexchange.com/questions/508217/showing-that-rationals-have-lebesgue-measure-zero#508225 https://math.stackexchange.com/questions/508217/showing-that...
    - hn_throwaway_99 2y ago
      But then the original implication, "100% of real numbers are normal, so that's pretty strong statistical evidence", still doesn't make any sense, as it's essentially using "100%" to imply "strong statistical evidence" that the rationals don't exist, which obviously doesn't follow.
      - staunton 2y ago
        > still doesn't make any sense Right. I'm pretty sure actually that it was a joke...
        adgjlsfhk1 2y ago
        it was about half a joke. statistical evidence doesn't really exist for the type of problem since polynomialy computable numbers are countably infinite so you can't define a uniform distribution over then
      - mhink 2y ago
        I got the impression that the comment was a bit tongue-in-cheek. The joke lies in the fact that saying "100% of real numbers" isn't *technically* the same thing as saying "all real numbers", because there's not really a good way to define a meaning for "100%" that lets you exclude rational numbers (or any other countable subset of the reals) and get something other than 100%.