- --- 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/

I'm unclear as to what the formula actually is, due to the limited

> (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

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 mathforfun@yahoogroups.com, slim_the_dude <no_reply@y...> wrote:
>

In this formula perform the divisions before the subtractions. So far

> 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).

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 - --- 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))

Ivars Peterson's article (linked above) contains a (broken:) link to

> 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?

>

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