Re: [math-fun] Another approximation of pi
And all of them pale in comparison with Ramanujan's ln(640320^3+744)/sqrt(163). (Actually, there's some sum which approximates pi/8 to over 40 digits of accuracy. But that's cheating slightly, using an infinite series instead of a closed form.) Sincerely, Adam P. Goucher http://cp4space.wordpress.com
----- Original Message ----- From: Eugene Salamin Sent: 10/13/12 09:32 PM To: math-fun Subject: Re: [math-fun] Another approximation of pi
A much more interesting approximation is Ramanujan's π = (2143/22)^(1/4), good to 9 places.
-- Gene
________________________________ From: David Wilson <davidwwilson@comcast.net> To: Math Fun <math-fun@mailman.xmission.com> Sent: Saturday, October 13, 2012 11:43 AM Subject: Re: [math-fun] Another approximation of pi
On 10/13/2012 3:19 AM, Peter Luschny wrote:
Alonso del Arte:
As far as I know, our own Daniel Forgues is the first to notice that sqrt(9.87654321) = 3.1426968... which is about as good an approximation of pi as 22/7. For some Sequences of the Day in September, to suggest that a keyword:cons sequence ought to be chosen, I put in the number 9.87654321 purely as a placeholder. To my pleasant surprise, Dan added his observation to the September 30 entry. PL> Well, the problem here is that you did not stop your placeholder at 9.87. PL> This would have given Daniel the chance to notice a much better PL> approximation of Pi than 22/7
DW> Hence pi ~= sqrt(9.87654321) ~= sqrt(200/81) = 20 sqrt(2) / 9. sqrt(200/81) should have been sqrt(800/81), sorry.
Now what's the point of this?
20*sqrt(2)/9 - Pi = .0011041516837513144 sqrt(9.87) - Pi = .0000629608912084013
Peter
What's the point of any of it? Clearly sqrt(9.87) beats sqrt(9.87654321) in accuracy, but there's nothing particularly exciting about sqrt(9.87), If I was concerned only with accuracy, I would proffer the yet more accurate and uninteresting sqrt(9.8696). If any of these expressions had a practical use, it would be in reducing manual computations, but here sqrt(9.87) is inferior to 355/133 or even 3.1416 in both accuracy and computation facility. Now that computers have largely obsoleted manual computations, the only value of this sort of approximation is curiosity, and I found it curious that 9.87654321 ~= 800/81 led to an expression in which so many squares dropped out of the radical.
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I would tend to group approximations into two general categories: Definition-based approximations, which follow mathematical properties, for example π ~= 3 π ~= ln(640320^3+744)/sqrt(163) The first follows from the approximate equality of the perimeter of a circle and its circumscribed hexagon, the second follows from a truncated series expansion in class theory. There are reasons for these approximations. Value-based approximations have no apparent direct relationship to mathematical definitions. For example, π ~= 22/7 π ~= 355/113 π ~= sqrt(9.87) π ~= cbrt(10) π^4 + π^4 ~= e^6 e^π - π ~= 20 all seem to be value-based. These relationships are commonly found by observation or numerical analysis of the numerical values, not derived from mathematical properties. This is clearly a subjective distinction. Tomorrow we may find reasons for approximations that baffle us today. I am curious about Ramanujan's π ~= (2143/22)^(1/4) Was this an observation, or a result? On 10/14/2012 8:19 AM, Adam P. Goucher wrote:
And all of them pale in comparison with Ramanujan's ln(640320^3+744)/sqrt(163).
(Actually, there's some sum which approximates pi/8 to over 40 digits of accuracy. But that's cheating slightly, using an infinite series instead of a closed form.)
Sincerely,
Adam P. Goucher
----- Original Message ----- From: Eugene Salamin Sent: 10/13/12 09:32 PM To: math-fun Subject: Re: [math-fun] Another approximation of pi
A much more interesting approximation is Ramanujan's π = (2143/22)^(1/4), good to 9 places.
-- Gene
* David Wilson <davidwwilson@comcast.net> [Oct 14. 2012 18:44]:
I would tend to group approximations into two general categories:
[...]
(I share this distinction of approximations.)
π ~= (2143/22)^(1/4)
Was this an observation, or a result?
? contfrac(Pi^4) [97, 2, 2, 3, 1, 16539, 1, 6, 7, 6, ...] Regards, jj
[...]
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Adam P. Goucher -
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Joerg Arndt