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Pi odd sequance of digits

  1. Apr 4, 2003 #1
    I was looking at the number Pi yesterday, with its odd sequance of digits, and an plausible idea popped into my head: could any possible sequance of digits be found as a substring of the string of digits that make up pi? (sure seems so)
     
    Last edited by a moderator: Feb 4, 2013
  2. jcsd
  3. Apr 4, 2003 #2

    Hurkyl

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    The vast majority of real numbers have that property. As to whether &pi does, I don't think anyone knows.

    Hurkyl
     
  4. Apr 5, 2003 #3

    selfAdjoint

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    Hey, I noticed that in your post &pi doesn't come out as [pi]. Are they kidding us?
     
  5. Apr 5, 2003 #4

    Hurkyl

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    Aha, so [pi] is made the same way on this forum!

    I guess that we're faced with a tough decision...

    Do we use &pi that fits in nicely with the font but is unrecognizable if you don't know what it's supposed to be, or [pi] which looks nice but is a frusterating half a line above the rest of the equation

    I dug out Windows' character map, it seems that the culprit is the (default) Verdana font. (For those lucky ones without Verdana, you get arial as default!)

    Arial looks like this: &pi.
    Times New Roman looks like this: &pi
    Courier looks like this: &pi
    Century Gothic looks like this: &pi

    Looks like I might be doing my posting in TNR from now on.

    Hurkyl
     
    Last edited: Apr 5, 2003
  6. Apr 5, 2003 #5
    Nope. I don't think there is any sequence which can describe the randomness of the successive digits of pi.
     
  7. Apr 5, 2003 #6
    No, not quite what I meant. I'm wondering whether given any natural number x, you could find the x embedded somewhere within the digits of pi (i still have to looked up how to do this math board notation).

    Say for instance I chose "322", could I find "322" inside the digits of pi. What about any other natural number?
     
  8. Apr 5, 2003 #7
    That's an interesting idea. It seems like since the digits of π are infinite that you should be able to as long as you looked for long enough.
     
    Last edited by a moderator: Apr 5, 2003
  9. Apr 5, 2003 #8

    FZ+

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    Yep. I think that the same is true of any irrational number. If it does not repeat and goes on for infinity, I would think all possible combinations would be exhausted. At least, it seems so.
    I'll be very interested if anyone can find a numerical proof for it though.
     
  10. Apr 5, 2003 #9

    this is certainly not true for every irrational number. just most of them.

    this property is called normality. it is strongly suspected that π is normal, and most other irrational numbers, but so far it is unproved
     
  11. Apr 5, 2003 #10
    Please explain more about this normality. This is really interesting. Are there any irrational numbers which are known not to be normal, are/or are their any irrational numbers that are known for sure to be normal? Could you point me in the direction of some papers on this or anything? Thanks!

    ps: lethe, the times new roman does look better for the math symbols, but I found it works a little better if you increase the size to large too, just makes it easier to read, but for sure the pi looks better in the tnr then it does in the defualt where you can barely distinguish it!
     
  12. Apr 5, 2003 #11
    How s this look? π

    OK, anyway, to answer: firstly, yes there are certainly numbers which are known to not be normal. i m going to show you one in a second.

    secondly, as far as i know, there are no numbers which are known for sure to be normal, although it is experimentally verified for the first many millions of digits of π and √2 and some other irrationals.

    apparently normality depends on what base you re in! so just because π is normal in base 10, does not mean it is normal in binary!

    i think mathworlds page about normal numbers is pretty good, it has some tables showing statistics about digits of π. also, apparently there are websites that will search digits of π for your phone number. if it truly is a normal number, like it appears to be, then your phone number is in there.

    now, lemme show you a nonrepeating irrational (that s redundant, eh?) number which is not normal:

    0.1010010001000010000010000001....

    this number will obviously never repeat itself, so it is not a rational number, and yet you will never find your phone number here (unless you have a pretty funky phone number).


     
  13. Apr 5, 2003 #12

    i should have read the link before i posted. yes, there are few numbers which are known to be normal, as you will see on mathworld, but not very many. two such numbers are the champernowne constant and the copeland-erdos constant which are both basically constructed in such a way as to ensure that they are normal.

    no "naturally occurring" irrational numbers are known to be normal, however.
     
  14. Apr 5, 2003 #13
    do you mean by this statement that the cardinality of the normal numbers is greater than the cardinality of the non-normal numbers? i.e. that nonnormal numbers are countable?
     
  15. Apr 5, 2003 #14
    For these numbers which are known to be normal, do you know if 1/the number is also normal? It seems like if that n is normal that 1/n should be too but I don't know much about number theory so...
     
    Last edited by a moderator: Apr 5, 2003
  16. Apr 5, 2003 #15

    Hurkyl

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    Well, non-normal numbers can't be countable; the set of all decimal numbers without any 8's in their representation is clearly nonnormal, but the set is uncountable.


    http://mathworld.wolfram.com/AbsolutelyNormal.html

    "Almost all" numbers in the set [0, 1) are absolutely normal, a stronger condition than normality.

    Almost all is a technical term, though it apparently has more than one definition since the definition at mathworld at http://mathworld.wolfram.com/AlmostAll.html is different from the definition I have in my Measure Theory text, but they both mean roughly the same thing; the ratio of nonnormal numbers to normal numbers is zero.

    Hurky
     
  17. Apr 5, 2003 #16
    the definition of "almost all" that i learned in school is "on all but a set of measure zero". is that what you learned too?

    OK, so when you said "vast majority" you were not referring to cardinalities, but to measure.

    got it. thanks.
     
  18. Apr 5, 2003 #17
  19. Apr 5, 2003 #18
    well, it seems like this statement should be true. beyond that i can t comment.
     
  20. Apr 5, 2003 #19

    Hurkyl

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    Good call! It's "almost everywhere" that I got from my measure theory text (yes, it was defined as the set of counterexamples having measure 0).

    Hurkyl
     
  21. Apr 5, 2003 #20
    Anyone else have any thoughts on my question about if n is normal is 1/n normal too?
     
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