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How to find 'self locating digits' in irrational numbers

  1. Aug 3, 2012 #1
    Let us take the most mainstream irrational out there, (Pi).

    Now write (Pi) as:

    3.
    14159265...

    Let us number the decimals of Pi.

    0 gets paired with 1
    1 gets paired with 4
    2 gets paired with 1
    .
    .
    .
    6 gets paired with 6

    Thus 6 is a self locating digit.

    My question is then how do we devise a method to find these self locating digits in a fast way.

    This is how I've gone about it.

    consider λ=(Pi)-3=0.14159265...

    now consider digit number n, that is, the digit that is n places along:

    0 gets paired with 1
    1 gets paired with 4
    2 gets paired with 1
    .
    .
    .
    6 gets paired with 6
    .
    .
    .
    n gets paired with x

    We need an algorithm for finding out what x is without writing the whole of λ out.

    Consider a new rational number, ρ.

    Let ρ_n be the number which terminates at digit n.

    Then ρ_n=0.1415926...x
    ρ_(n-1)=0.1415926...w where w is the (n-1)th digit etc

    Now consider (ρ_n)*10^(n+1) this is equal to 1415926...x. Let us call this new number β.

    We can then find what x is by subtracting ρ_(n-1)*10^(n) from β.

    Now if the x = n we have a self locating digit.

    This method isn't terribly practical as we still have to basically know what ρ_n is.

    Maybe I'll come up with an improvement after some thought but in the mean time I'd love to see what you guys come up with :)
     
  2. jcsd
  3. Aug 3, 2012 #2
    Right, the problem is that we have no formula for the n-th digit of pi that doesn't involve calculating all the preceding digits.

    But it turns out that someone did find such a formula. It only works in base 16, but it's still amazing that such an algorithm exists.

    http://en.wikipedia.org/wiki/Bailey–Borwein–Plouffe_formula
     
  4. Aug 3, 2012 #3
  5. Aug 5, 2012 #4

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    It is a formula for all digits. But if you want to calculate a specific one, you can calculate it quickly, the wikipedia page explains how.

    By the way, what happens to the 11th digit in base 10, for example? A digit cannot be "11" there.
     
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