Wednesday, March 14, 2007
Abe's Paean to Pi: Finding Pi's 'Nth' Decimal via Newton
Welcome all to Abe Linkum's salute to today's honoree, the transcendental constant Pi. Most people don't know that every March 14th is devoted to Pi since 3.14 is a rough estimate of its value. It's Pi Day!Some people know that the number of non-repeating decimals in Pi is infinite and that many geeks try to out match each other this time of year by either attempting to recite sequentially the most decimals of Pi, or try to become the one person with the supercomputing access needed to set the record for the longest string of computed decimals -- currently set at a little over one-trillion.
Abe has done a little investigation of Pi and has concluded that one could employ Newton's Method for estimating roots of equations, such as y = sin x, to calculate the nth decimal of Pi. And by using this simple technique, huge additional quantites (exponential?) of Pi's decimals will spew out of your computational apparatus, be it a Cray supercomputer or a Crayola crayon, with each successive estimation.
Abe has drawn a rough graph of y = sin x along with three iterations of Newton's Method, demonstrating that each successive iteration comes nearer to the root of y = sin x. (X1, X2 and X3, respectively.)
The equation for Newton's Method: xn+1 = xn - f(xn)/f'(xn), also serves as a handy algorithm for you, the calculator.
And that, since the curvature of the graph y = sin x in nearly nil as it approaches its root, Pi, the amount of decimals added by, say, the estmation Xn-1 versus Xn-2 will be astronomical!Enjoy, all ye Geekazoids. Happy calculating! Don't forget to include Ole Abe when you go to accept your Fields Medal.
Next time in scence news at Abe Linkum: Abe explains the accelerating universe!
Labels: mathematics, science news
