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Pi Calculated To Record 2.5 Trillion Digits
Joshua writes "Researchers from Japan have calculated Pi to over 2.5 trillion decimals using the T2K Open Supercomputer (which is currently ranked 47th in the world according to a June, 2009 report from Top500.org). This new number more than doubles the previous record of about 1.2 trillion decimals set in 2002 by another Japanese research team. Unfortunately, there still seems to be no pattern."
Of course there's a pattern. In fact, an infinite number of them. My favourite is the one in the generalised continued fraction expansion of pi.
entropy happens
Since Pi is irrational, does that mean that a "perfect" circle cannot actually exist? If you don't understand my question, think about it like this. Let's say I want to construct a circle of radius R. To create a "perfect" circle, it seems like I would need a length of material to build the circle out of that was exactly 2*Pi*R, but since Pi is irrational, it seems that you could never actually get any length which is an exact multiple of Pi? If Pi really expands out infinitely, even a circle with a radius the size of a galaxy, or a cluster of galaxies, could never be *exactly* the right length?
You're absolutely right: pi is irrational, and as such, there won't be any repeats. However, that doesn't mean there isn't a pattern. For example, 0.12112111211112111112... is irrational, but there's a clear pattern that you could extend to an infinite number of digits. Does such a pattern exist once you get to a certain number of digits in pi? We don't know.
Of course there's a pattern, even a simple and elegant one. It's equal to:
4 * (1 -1/3 + 1/5 -1/7 +1/9 -1/11 +1/13 -1/15 etc., etc., etc.)
Just because the pattern doesn't come out pretty in a decimal representation doesn't mean it's not elegant or not a pattern.
I always found the Basel problem to be the most elegant converging series involving pi (being the square root of six times the sum of the reciprocals of the squares), probably because there are so many (elegant) proofs of this (pdf), because it's so simple to understand yet not so simple to prove on a cursory inspection, and because it's the specific case that generalized to one of the most important unsolved problems in mathematics.
Well, I'm not a mathematician, but it seems to me that's precisely why there isn't a repetitive pattern in the numerical representation. If there was, that would mean the ratio can be exactly defined by a finite amount of information. It seems to me that asking for a finite decimal represensation of pi is similar to asking someone to exactly represent a circle out of line segments (or to exactly define a circle using a finite set of points). The circumference of the circle is the sum of the length of line segments delineating the circle. The problem is that you need infinitely many of them to exactly define the circle.
The pattern just isn't in base 10. It's in base e. Why does anyone expect to see a numerical pattern in an arbitrary number base like 10? Just because we have 10 fingers doesn't make it the "correct" base for anything.
There's a good argument that the choice of pi = (circumference / diameter) was unfortunate; it should have been (circumference / radius). In the light of modern mathematics it seems clear that the radius is more "fundamental" than the diameter; choosing pi = (circumference / radius) = 6.28... gives a number of nice things like:
A = (1/2)pi r^2, just as E = (1/2)m v^2 or d = (1/2)a t^2, and for the same reason.
In general, in the current convention, 2pi seems to show up a lot more than pi, e.g. there are 2pi radians in a circle, sin(x) has period 2pi, etc. All these would become simply pi with the (circumference / radius) convention
\frac{1}{2^6}\sum_{n=0}^\infty \frac{(-1)^n}{2^{10n}} \left( -frac{2^5}{4n+1} - \frac{1}{4n+3} + \frac{2^8}{10n+1} - \frac{2^6}{10n+3} - \frac{2^2}{10n+5} - \frac{2^2}{10n+7} + \frac{1}{10n+9} \right)
This gives pi in binary, and there is a definite pattern there.
it's useless to know pi to more than 50 digits in this universe
I think you are confused, repeat after me ... "This is SsLlAaSsHhDdOoTt, universe has nothing to do with it"
BAIN http://www.devslashzero.com
Actually, the program itself is a perfectly fine way of representing pi.
So... random honest question. How do they know the program (or its output) is correct? Is it possible to create a proof that the program will generate correct output?
I mean, sure, we can look at the first nine digits and say "yeah, that looks right". But does anyone really know if digits 1.2 trillion through 2.5 trillion in the output are correct?
Yeah. Pi acts like Infinite Monkeys. All _we_ have to do is to point to the monkey that actually does write Shakespeare, i.e.: the index of Pi which actually represents Kill Bill Complete in AVI format.
The only problem is the size of that index, but hey, if you zip that number and take its MD5, you have achieved something similar to this.
"We can confirm that Debian does *not* ship the version with the trojan horse. Our version predates it." [CA-2002-28]
Every possible pattern, interesting or not, occurs in the digits of Pi because they go on forever and do not repeat
Your conclusion does not follow from your premise.
Liouville's constant is trancendental. It goes on forever, it does not repeat, and it consists almost entirely of zeros with an occasional 1 and no other digits at all.
http://en.wikipedia.org/wiki/Liouville_number#Liouville_constant
Tim.
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
Now if only God provided source code! Instead we get these damn executables...