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Re: pi in hexadecimal

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  • slim_the_dude
    ... I m unclear as to what the formula actually is, due to the limited amount of parentheses. i.e. the order of operations between the minuses and slashes.
    Message 1 of 4 , Aug 31, 2005
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      --- In mathforfun@yahoogroups.com, "cooperpuzzles"
      <cooperpuzzles@y...> wrote:
      > pi=sum(n=0 to infinity) of (1/(16^n))(4/(8n+1)-2/(8n+4)-1/(8n+5)-1/
      > (8n+6))
      > Bailey, Borwein, and Plouffe
      >
      > The above formula is a relatively new way to calculate pi. Allegedly
      > one can use it to calculate any digit of pi in hexadecimal without
      > calculating all the previous digits. See
      > http://www.maa.org/mathland/mathtrek_3_2_98.html.
      >
      > I don't see how. Can someone calculate the tenth hexadecimal digit of
      > pi using the above formula without calculating the previous nine
      > digits?
      >
      > John

      I'm unclear as to what the formula actually is, due to the limited
      amount of parentheses. i.e. the order of operations between the
      minuses and slashes. One school of thought says to do all divides
      before all minuses. Another school of thought says to go in order
      from left to right (after first evaluating the explicitly
      parenthesized expressions).
    • cooperpuzzles
      ... In this formula perform the divisions before the subtractions. So far as I know this is always the correct order of operations (unless parentheses indicate
      Message 2 of 4 , Sep 1, 2005
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        --- In mathforfun@yahoogroups.com, slim_the_dude <no_reply@y...> wrote:
        >
        > I'm unclear as to what the formula actually is, due to the limited
        > amount of parentheses. i.e. the order of operations between the
        > minuses and slashes. One school of thought says to do all divides
        > before all minuses. Another school of thought says to go in order
        > from left to right (after first evaluating the explicitly
        > parenthesized expressions).

        In this formula perform the divisions before the subtractions. So far
        as I know this is always the correct order of operations (unless
        parentheses indicate otherwise.) (When division is indicated by a
        horizontal line, parentheses are implied around the expressions above
        and below the line.)

        John
      • adh_math
        ... Ivars Peterson s article (linked above) contains a (broken:) link to the Bailey-Borwein-Plouffe preprint, which is quite readable, relying mostly on
        Message 3 of 4 , Sep 6, 2005
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          --- In mathforfun@yahoogroups.com, "cooperpuzzles"
          <cooperpuzzles@y...> wrote:
          > pi=sum(n=0 to oo) (1/(16^n))(4/(8n+1)-2/(8n+4)-1/(8n+5)-1/(8n+6))
          > Bailey, Borwein, and Plouffe
          >
          > The above formula is a relatively new way to calculate pi.
          > Allegedly one can use it to calculate any digit of pi in
          > hexadecimal without calculating all the previous digits. See
          > http://www.maa.org/mathland/mathtrek_3_2_98.html
          >
          > I don't see how. Can someone calculate the tenth hexadecimal
          > digit of pi using the above formula without calculating the
          > previous nine digits?
          >

          Ivars Peterson's article (linked above) contains a (broken:) link to
          the Bailey-Borwein-Plouffe preprint, which is quite readable, relying
          mostly on calculus and modulo arithmetic:

          http://www.lacim.uqam.ca/~plouffe/


          Roughly, the digits of a reciprocal 1/k can be computed "easily" in
          base b. For example, in hex, if

          16^n = a (mod k),

          then

          (16^n/k) - (a/k)

          is an integer, so the fractional part of (a/k) gives the hex digits of
          1/k starting with the nth. (Is it not nifty?)

          In the formula for pi above (which comes from calculus), the terms in
          parentheses sum to a number less than 1 (fairly clear:) so the dth hex
          digit depends only on summands with n < d (because of the 16^{-n}).
          Since digits of reciprocals can be calculated, it's possible (both in
          principle, and with computational feasibility) to find the dth hex
          digit of pi without calculating the previous digits. To quote
          Peterson, however:

          "A year after the discovery of the formula, Fabrice Bellard [a
          student]...used it to calculate the 100 billionth hexadecimal
          digit of pi: 9, followed by C381872D2...

          "Last September, he computed the trillionth digit: 8, followed by
          7F72B1DC... The main calculation required a month of computation
          on more than 20 workstations and personal computers."

          Presumably, that's fast compared to complete computation of the first
          trillion digits.

          Regards,
          adh
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