Re: [math-fun] The value of PI
----- Original Message ---- From: Henry Baker <hbaker1@pipeline.com> To: Dan Asimov <dasimov@earthlink.net>; math-fun <math-fun@mailman.xmission.com> Sent: Thursday, May 1, 2008 11:09:40 AM Subject: Re: [math-fun] The value of PI Even more bizarre, IMHO, is the fact that in the volume formula for n dimensions, it isn't pi itself, but sqrt(pi), that is important. I don't think that the Greeks figured that part out. If you go through all the formulae in a large book -- e.g., Stegun, et al -- I think that "2pi" shows up more often than "pi". Sooner or later, we're bound to run up against a civilization that chose differently which to commemorate with a name. ------------------------------ But the sqrt(pi) is never explicit. The volume is (n/2)! pi^(n/2) R^n, and since (-1/2)! = sqrt(pi), only integer powers of pi appear. Gene ____________________________________________________________________________________ Be a better friend, newshound, and know-it-all with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ
Did the ancient Greeks wonder about the density of integers of a certain kind? After all they are always talking about proportions, and certainly Archimedes had some idea about the meaning of a limit. However I don't think that any of them could have imagined that the square of the ratio of the circumference of a circle to its diameter could have been simply related to the ratio of the set of square free integers to all integers. Victor On Thu, May 1, 2008 at 2:51 PM, Eugene Salamin <gene_salamin@yahoo.com> wrote:
----- Original Message ---- From: Henry Baker <hbaker1@pipeline.com> To: Dan Asimov <dasimov@earthlink.net>; math-fun < math-fun@mailman.xmission.com> Sent: Thursday, May 1, 2008 11:09:40 AM Subject: Re: [math-fun] The value of PI
Even more bizarre, IMHO, is the fact that in the volume formula for n dimensions, it isn't pi itself, but sqrt(pi), that is important. I don't think that the Greeks figured that part out.
If you go through all the formulae in a large book -- e.g., Stegun, et al -- I think that "2pi" shows up more often than "pi". Sooner or later, we're bound to run up against a civilization that chose differently which to commemorate with a name.
------------------------------ But the sqrt(pi) is never explicit. The volume is (n/2)! pi^(n/2) R^n, and since (-1/2)! = sqrt(pi), only integer powers of pi appear. Gene
____________________________________________________________________________________ Be a better friend, newshound, and know-it-all with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The volume formula was just an example. sqrt(pi) shows up more explicitly & forcefully in probability (erf's) and various kinds of transforms. I noticed that high precision sqrt(pi) is required in various software routines. If we had it to do all over again, having a pi-squared constant instead of a pi constant might not be the end of the world. It might simplify a number of formulae. At 11:51 AM 5/1/2008, Eugene Salamin wrote:
----- Original Message ---- From: Henry Baker <hbaker1@pipeline.com> To: Dan Asimov <dasimov@earthlink.net>; math-fun <math-fun@mailman.xmission.com> Sent: Thursday, May 1, 2008 11:09:40 AM Subject: Re: [math-fun] The value of PI
Even more bizarre, IMHO, is the fact that in the volume formula for n dimensions, it isn't pi itself, but sqrt(pi), that is important. I don't think that the Greeks figured that part out.
If you go through all the formulae in a large book -- e.g., Stegun, et al -- I think that "2pi" shows up more often than "pi". Sooner or later, we're bound to run up against a civilization that chose differently which to commemorate with a name.
------------------------------ But the sqrt(pi) is never explicit. The volume is (n/2)! pi^(n/2) R^n, and since (-1/2)! = sqrt(pi), only integer powers of pi appear. Gene
participants (3)
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Eugene Salamin -
Henry Baker -
victor miller