There is a huge literature now on the subject: Computing inaccessible digits of Pi^2 and G (http://www.carma.newcastle.edu.au/jon/bbp-bluegene.pdf) will answer all these questions. It will eventually appear in the Notices of the AMS.

http://www.carma.newcastle.edu.au/jon/bbp-bluegene.pdf

Do u have a precise reference? I dont think its a formula, more an algorithm based on

a formula of Ramanujan..

There is a huge literature now on the subject: Computing inaccessible digits of Pi^2 and G (http://www.carma.newcastle.edu.au/jon/bbp-bluegene.pdf) will answer all these questions. It will eventually appear in the Notices of the AMS.

http://www.carma.newcastle.edu.au/jon/bbp-bluegene.pdf

Thank you Jon for the references.

@Jon. I thoroughly enjoyed reading that. Nice paper! I heard you do a talk about 10 years ago and found it fascinating....so when I said "base 16" in my question, above, it was from my distant memory. After now reading the paper, I note it can be base 2 or 16. Also the paper suggests all sorts of other bases for other transcendentals.

My question is do you have any intuitive picture why certain bases work for pi? Or is it a mystery? Is there an intuitive argument why base 10 cannot work?

Interesting that there is no formula for e. What about e^2 ?

Hi Jon, I had the chance to meet u in Vancouver in 1995...Will u visit KAU in Jeddah this year?Ur paper is killling my SIM card..must be very large...

OK I managed. Thanks for the reference Jon!

The standard literature on this topic under values and misstates Greek and Egyptian rational number contributions to the pi topic. Egyptians began with 256/81. nice round numbers, but quickly improved to 22/7. Archimedes improved by inspecting unit fraction aspects of square root calculations ... http://planetmath.org/encyclopedia/EgyptianAndGreekSquareRoot.html and therefore did not need to apply a geometric method, as often reported.

@Milo: what is the connection from square root to Pi?

The finite rational number system in which pi and irrational square roots must be approximated offers a numerical domain that Egyptians and Greeks shared from 2050 BCE to 800 AD. That is, I'd be happy to write Archimedes'

223/71 < pi

oops. Archimedes may have trapped pi in another set of limits ...

Archimedes Traps Pi

Pi is defined as the ratio of the circumference of a circles to its diameter.

(pi as a symbol wasn't used until 1706)

How did Archimedes estimate the value of pi?

He drew two polygons around the circle's center -

one outside the circle (circumscribed) so its perimeter was greater than the circle's, and

one inside the circle (inscribed) so its perimeter was less than the circle's.

A Really Bad Example

The more sides the polygons have, the better the estimate of pi.

Archimedes used a polygon with 96 sides to show that the value of pi was between

3 10/70 and 3 10/71 !

But where are the discussions of unit fraction series that represent 10/70 and 10/71 that Archimedes would have written? That is the issue. Sttidy the unit fraction series as Greek square root solved

(164)^1/2 by (12 + 2/3 + 1/15 + 1/24 + 1/32)^2 with a small irrational error.

considering (3 + 10/70) > pi > 10/71 as Archimdes' summed unit fraction series . let's find the final answer's in Egyptian notation, 1/n = n' , then

1. 10/70 = (1/7 - 1/10) = 3/70 - 1/35 + 1/70 MEANT

PI< (3 + 10' + 35' + 70') or (3 + 7')

as well as alternative series

2. 10/71 = (10/71 - 1/8) = (8 + 1)/568 MEANT

PI > (3 + 8' + 71' + 568')

pretty easy to find. Therefore. no geometry was needed by any Greek or Egyptian!

Q.E.D.

..

@ Milo: So polygons are not geometry for you?

Patrick, I love geometry of all forms, plane, solids, projective and non-Euclidean.. In college 50 years ago I suffered through engineering problems that stressed the Pythagorean Theorem o solve certain problems. Today, algebra and theoretical arithmetic are allowed.in the same problems.

That is, math models and solves real world problems with rational numbers ... extended by Gauss to complex numbers in 1796. Irrationals and general solutions of the fundamental theorems of algebra and arithmetic were first solved by Gauss in 1796 and 1801.

However, irrationals were well known to Greeks and surprising even to me to Egyptians. I am still working on the square root method that clearly began with a rational error E1 4/25 that estimated (164)^1/2 = (12 + 4/5)^2 and ended with (12 + 2/3 + 1/15 + 1/24 + 1/32)^2 with an irrational root of (160)^2. Rational numbers were a powerful notation ... used from 2050 BCE to 800 AD in a unit fraction context .. only decoded in the last 10 years..

You might be interested in some lessons I did in my math circle on computing pi. There are some notes in the math circle's Facebook page.

https://www.facebook.com/LSSMathCircle

I also find this Ramanujan's formula for 1/ Pi interesting. See attached.

The Borwein manuscript (1) for the square root is very similar to the continued fraction Bombelli (p64 Brezinski) gave in 1572. sqrt(q) = sqrt{a^2 + r} = a + x; a^2 + r = a^2 + 2ax + x^2

Neglect x^2 then r=2ax. sqrt(q) = a + r/(2a + r/(2a ....... as a continued fraction.

The first line of (1) is identical for n=0, and I expect that a would be a whole number so that r can be a decimal fraction that would be easy to compute as opposed to the second line of (1).

Or perhaps it is more similar to Cataldi (p68)

What about the Babylonians who seemed to know this? (p12)

See p12 p64 and p68 Brezinski, History of Continued Fractions and Pade Approximants.

I've found a formula in base 2.

http://en.wikipedia.org/wiki/Bellard%27s_formula

Leonardo.

The Chinese found much earlier a far better value for pi namely :

355/113 = 3.14159192 pi = 3.1415926535....

So this is not so bad.

yes, the Chinese version was better than averaging Archimedes' guess

22/7 < pi < 223/71

anyone know how to solve pi from R, in A = pi(R^2), such that Egyptian and Greek square root method can be applied in that manner?

Bet some nice solutions came from that ... like 22/7 and better ...

thanks..

My Proposed Equation for X=pi:

X^4 + [4^2]*X^2 - 4^4 = 0

Ref :

http://www.stefanides.gr/pdf/2012_Oct/PHOTO_12.pdf

http://www.stefanides.gr/Html/piquad.htm

http://www.stefanides.gr/Html/QuadCirc.htm

http://www.stefanides.gr/Html/gmr.htm

Regards from Athens,

Panagiotis Stefanides

http://www.stefanides.gr

If I substitute X=pi into the above equation I get a number >200, which is nowhere near zero. Am I misunderstanding something?

Dear Derek,

I thank you.

I imply here the solution of this biquadratic equation gives for X = 4/ [SQUARE ROOT OF THE GOLDEN RATIO] = 3.144605511.... , as proposed value of pi.

i.e [ 3.144605511....]^4 + 16* [ 3.144605511....]^2 - 256 =0

Panagiotis Stefanides

http://www.stefanides.gr/pdf/BOOK%20_GRSOGF.pdf

Dear Milo,

I attach my "Tracing Archimedes Method of Polygons for Circle Measurement" under the link:

http://www.stefanides.gr/Html/Classical_Problems_et_Alii_with_Web_Links.htm

Regards from Athens,

Panagiotis Stefanides

http://www.stefanides.ge

P.S.

However I propose my work under:

http://www.stefanides.gr/pdf/BOOK%20_GRSOGF.pdf

Panagotis, yoir paper assumes that Archimedes usually thought in 1/4th geometric series and solved the area of a conic section by the method of exhaustion 4A/3 += A + A/4 +A/16 + 1/64 + A/64 + ...

EQUALLY OR MORE LIKELY IS THAT ARCHIMEDES FOLLOWEDE EUDOXUS IN APPLYING 1/4TH GEOMETRIC SERIES AS A STATEMENT OF A PROBLBEM ... SOLVED BY FINDING A FINITE SERIES ... IN THIS CASE 4A/3 = A + A/4 + A/12 ... Q.E.D. AS HIEBERG PULISHED IN 1906.

Note that archimedes and greeks solved square root outside of geometry and nside of arithmetic and algebra by: http://mathforum.org/kb/thread.jspa?threadid=2421477.

Yes, archimedes' sqaure of 3 problem was solved in the same manner

22*X^4 = 2143 is a better value then the quartic above.

X = (2143/22)^(1/2) X = 3.14159265258 is not so bad.

pi = 3.1415926535....

Archimedes algebraic looked like this http://planetmath.org/encyclopedia/EgyptianAndGreekSquareRoot.html ... still need raw fraction data to complete this project

Pi is popularly known to be 22/7 = 3.14159265258

22/7 = 3.142857143 and a very bad one.

I emailed one of the authores of the base 16 forumla paper once, to ask about other bases. Whether such a formula exists for base p^n depends on certain properties of the prime p that I can't remember right now. In particular, however, there's no similar formula for base 3^n, a fact which I found disappointing. I have a web page allowing people to search for their name in the base 27 digits of pi, and thought that a Borwein-Plouufe-like formula for the base 27 digits might be useful.

Egyptian and greek square root is needed to complete this paper http://www.stefanides.gr/pdf/book%20_grsogf.pdf ,, why not add it?

Algebraic arithmetic offers ancient square root needed to this paper ...http://www.stefanides.gr/pdf/book%20_grsogf.pdf