RE: Dare to interpret?
February 3, 2015 at 10:51 am
(This post was last modified: February 3, 2015 at 11:10 am by Alex K.)
The point is that we have a formula which gives pi as a sum with powers of 1/16, so the first four terms for example come with pre-factors 1/16^k =
k=0: 1
k=1: 1/16
k=2: 1/256
K=3: 1/4096 ....
Since this becomes small so quickly, the formula yields an impressive precision using very few of these terms. Taking the sum to k=3 only, we already obtain seven digit precision,
47/15 + 53/6552 + 829/5026560 + 79/15590400 = 3.1415924....
as opposed to pi= 3.1415926....
Edit: description has slightly changed
More importantly, since this pre-factor becomes smaller in this very particular fashion, one can reverse engineer how these terms in the sum contribute to any particular arbitrary digit of pi. This allows us to calculate the, say, quadrillion and oneth digit of pi without having to calculate the quadrillion digits preceding it, with minimal calculational effort of a tiny fraction of the computing time one would otherwise need. Edit: finding the quadrillionth digit is still a huge computing problem using this method, but would be completely unreachable otherwise.
https://www.math.hmc.edu/funfacts/ffiles/20010.5.shtml
http://www.ams.org/samplings/feature-column/fcarc-pi
k=0: 1
k=1: 1/16
k=2: 1/256
K=3: 1/4096 ....
Since this becomes small so quickly, the formula yields an impressive precision using very few of these terms. Taking the sum to k=3 only, we already obtain seven digit precision,
47/15 + 53/6552 + 829/5026560 + 79/15590400 = 3.1415924....
as opposed to pi= 3.1415926....
Edit: description has slightly changed
More importantly, since this pre-factor becomes smaller in this very particular fashion, one can reverse engineer how these terms in the sum contribute to any particular arbitrary digit of pi. This allows us to calculate the, say, quadrillion and oneth digit of pi without having to calculate the quadrillion digits preceding it, with minimal calculational effort of a tiny fraction of the computing time one would otherwise need. Edit: finding the quadrillionth digit is still a huge computing problem using this method, but would be completely unreachable otherwise.
https://www.math.hmc.edu/funfacts/ffiles/20010.5.shtml
http://www.ams.org/samplings/feature-column/fcarc-pi
The fool hath said in his heart, There is a God. They are corrupt, they have done abominable works, there is none that doeth good.
Psalm 14, KJV revised edition
