Some interesting close approximations to Pi can be
found at the following site:


http://pi.lacim.uqam.ca/eng/approx_en.html
----Brian Trial

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Hello!

What do you like this:

2^248 / 213^31 = 299 + contfract(48836;3;1;5;...)

19^64 / 14^69 = 2*2*11*13 - contfract(25641;...)      19+64=14+69

Hmm (69)..:

115^60 / 116^59 = 69 - contfract(185619;...)

Juha Saukkola
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Ed,
 
Inspired by your example of 25AUG2003 I've just found the following using a calculator:
(10 * 13^(34/13)) ^ (1/13) = 2.000000128...
(1732/225)^(7/3) = 117.000000156...
 
Geoff Morley

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Hi Ed,

I came across a close approximation the other day and immediately thought
of the mathpuzzle page. I doubt it is a rare gem, but I thought I'd pass
it on anyway...

pi =~ 16 ArcTan[1/5] - 4 ArcTan[1/239]

(correct for the first 100,000 decimal points apparently!)

Kind regards,

Josh Downer

[Ed -- actually, it's exact.  Another exact representation is 
88 ArcTan[1/28] + 8 ArcTan[1/443] - 20 ArcTan[1/1393] - 40 ArcTan[1/11018]