Re: Pi still again

From: Stein A. StrÝmme (stromme@mi.uib.no)
Date: Thu Aug 02 2001 - 05:37:17 EDT

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    [John W. Burgeson]

    | I pulled the following from Pickover's LISTSERV -- now I am really over
    | my head!
    | -----------------------------
    | What information can't pi contain in its digit stream?
    |
    | Say that for example you have 3 atoms arranged in an equilateral triangle
    | formation. That means that at least one of the coordinates is equal to
    | the square root of 3, or 1.732etc. The three coordinates of the atoms
    | maybe could be (0,0), (1,0), and (1, 1.732..).
    |
    | Because the square root of three is irrational, its expansion as a real
    | number will continue forever without a pattern. Thus, if pi is to
    | "contain" the sqrt(3) that means that at one point, the digits of pi
    | would have to revert to the digits of sqrt(3), and continue that way for
    | infinity. Which would mean that pi is not transcendental. Pi has been
    | proven to be transcendental, so an infinite expansiion of sqrt(3), or any
    | other root for that matter, is impossible.
    | --------------------------------

    | Ignoring the question of the proper definition of "transcendental,"
    | I still don't see this follows. The infinite expansion of pi ought
    | to be able to contain a finite number of infinite expansions. Maybe.

    It all depends on how you interpret the word "contain".

    For any increasing sequence of natural numbers 0 < n_1 < n_2 <n_3 < ...,
    there is a real number A with decimal expansion given by the rule

                 i-th decimal of A = n_i-th decimal of pi.

    There are uncountably many such sequences, in fact, even uncountably
    many such sequences with the property that 1 <= n_(i+1) - n_i <= 2 for
    all i. The quote from Pickover's listserv deals with a very strict
    subclass of sequences, namely those which are of the form n_i = N + i
    for some number N.

    -- 
    Stein Arild StrÝmme  <http://www.mi.uib.no/~stromme>
    



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