5 ms·

My understanding (which might be wrong) is that just because PI is infinite and non-repeating, doesn't necessarily mean that every conceivable pattern of digits

by tlonny 4y ago

My understanding (which might be wrong) is that just because PI is infinite and non-repeating, doesn't necessarily mean that every conceivable pattern of digits is present. As a contrived example, consider the pattern: 01 001 0001 00001 etc. This pattern is infinite and never repeats but we will never see two consecutive "1"s next to each other.

mormegil 4y ago
Yes, it doesn't necessarily follow, but it is indeed conjectured that pi is a normal number, meaning all digits appear with the same frequency, but it is not known yet. https://en.wikipedia.org/wiki/Normal_number https://en.wikipedia.org/wiki/Normal_number
- rrobukef 4y ago
  The same frequency does not imply every subsequence appears. Consider the modification which rewrites every sequence of 123 to 132. All digits will have the same frequency but 123 will never appear.
  - sebzim4500 4y ago
    If pi is shown to be normal in every base then every finite sequence must appear in it.
- ithinkso 4y ago
  You haven't read the link you posted though. Every digit appearing with the same frequency means a number is simply normal and it is not enough to get you what you want in this case (as pointed out by sibling comment). Normal number is a number where every possible string of length n has the same frequency of 10^(-n)
  - oreilles 4y ago
    No, you haven't read the link he posted. https://en.wikipedia.org/wiki/Normal_number#Definitions https://en.wikipedia.org/wiki/Normal_number#Definitions: "A disjunctive sequence is a sequence in which every finite string appears. A normal sequence is disjunctive". If Pi is normal, then it is also disjunctive.
- n4r9 4y ago
  When I was at university, one of the senior number theory professors allegedly said during a tutorial that he accepts the normality of pi on the basis of "proof by why the hell wouldn't it be". With tongue in cheek, of course.
terranstyler 4y ago
The difference might be: For your example there is an algorithm to describe the sequence of digits and for Pi there isn't. EDIT + Clarification: There is an algorithm to calculate the digits of your number without calculating all previous digits. But for pi there isn't.
- Scarblac 4y ago
  There is one, how do you think we compute digits of pi.
  - terranstyler 4y ago
    So here's a bet: I give you the 10^100th digit of the above algorithm and you give me the 10^100th digit of pi. Whoever fails owes the other side 10 BTC.
    - ivegotnoaccount 4y ago
      There is an algorithm to get the nth digit of pi. It's just that it does not run in constant time
      - terranstyler 4y ago
        I kind of addressed this here: https://news.ycombinator.com/item?id=31966228 https://news.ycombinator.com/item?id=31966228
    - lolbert2 4y ago
    - mjg59 4y ago
      If your argument is "These algorithms have differing degrees of computational complexity" then that doesn't actually demonstrate that one can't be algorithmically determined
      - 4y ago

by tlonny 4y ago

mormegil 4y ago

Yes, it doesn't necessarily follow, but it is indeed conjectured that pi is a normal number, meaning all digits appear with the same frequency, but it is not known yet. https://en.wikipedia.org/wiki/Normal_number https://en.wikipedia.org/wiki/Normal_number

rrobukef 4y ago
The same frequency does not imply every subsequence appears. Consider the modification which rewrites every sequence of 123 to 132. All digits will have the same frequency but 123 will never appear.
- sebzim4500 4y ago
  If pi is shown to be normal in every base then every finite sequence must appear in it.
ithinkso 4y ago
You haven't read the link you posted though. Every digit appearing with the same frequency means a number is simply normal and it is not enough to get you what you want in this case (as pointed out by sibling comment). Normal number is a number where every possible string of length n has the same frequency of 10^(-n)
- oreilles 4y ago
  No, you haven't read the link he posted. https://en.wikipedia.org/wiki/Normal_number#Definitions https://en.wikipedia.org/wiki/Normal_number#Definitions: "A disjunctive sequence is a sequence in which every finite string appears. A normal sequence is disjunctive". If Pi is normal, then it is also disjunctive.
n4r9 4y ago
When I was at university, one of the senior number theory professors allegedly said during a tutorial that he accepts the normality of pi on the basis of "proof by why the hell wouldn't it be". With tongue in cheek, of course.

terranstyler 4y ago

The difference might be: For your example there is an algorithm to describe the sequence of digits and for Pi there isn't. EDIT + Clarification: There is an algorithm to calculate the digits of your number without calculating all previous digits. But for pi there isn't.

Scarblac 4y ago
There is one, how do you think we compute digits of pi.
- terranstyler 4y ago
  So here's a bet: I give you the 10^100th digit of the above algorithm and you give me the 10^100th digit of pi. Whoever fails owes the other side 10 BTC.
  - ivegotnoaccount 4y ago
    There is an algorithm to get the nth digit of pi. It's just that it does not run in constant time
    - terranstyler 4y ago
      I kind of addressed this here: https://news.ycombinator.com/item?id=31966228 https://news.ycombinator.com/item?id=31966228
  - lolbert2 4y ago
  - mjg59 4y ago
    If your argument is "These algorithms have differing degrees of computational complexity" then that doesn't actually demonstrate that one can't be algorithmically determined
    - 4y ago