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.
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 did find a possible motivation for Ramanujan's approximation to pi. It would follow from the obvious rational approximation to zeta(4) = 1.08232323+ On 10/13/2012 4:32 PM, Eugene Salamin wrote:
A much more interesting approximation is Ramanujan's π = (2143/22)^(1/4), good to 9 places.
My favorite pi approximation is sqrt 2 + sqrt 3. This value arises in the derivation of the count of canonical sequences of length n for the Rubik's cube. On Oct 14, 2012 9:06 AM, "David Wilson" <davidwwilson@comcast.net> wrote:
I did find a possible motivation for Ramanujan's approximation to pi. It would follow from the obvious rational approximation to
zeta(4) = 1.08232323+
On 10/13/2012 4:32 PM, Eugene Salamin wrote:
A much more interesting approximation is Ramanujan's π = (2143/22)^(1/4), good to 9 places.
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