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