## Pi Computed To 10 Trillion Digits 414

Posted
by
samzenpus

from the because-he-can dept.

from the because-he-can dept.

An anonymous reader writes

*"A Japanese programmer that goes by the handle JA0HXV announced that he has computed Pi to 10 trillion digits. This breaks the previous world record of 5 trillion digits. Computation began in October of 2010 and finished yesterday after multiple hard disk problems, he said. Details in English are not fully available yet, but the Japanese page gives further details. JA0HXV has held computation records for Pi in the past."*
## Re:What Does This Mean? (Score:3, Informative)

No, you only need about 50 decimal places to have an accurate enough approximation to calculate the circumference of

the entire universewith less than 1 planck length of error.This is just a "because we can" exercise. (Also, supposedly, to determine if PI is actually infinite or whether it contains a repeating pattern after you get to a certain point)

## Re:What Does This Mean? (Score:5, Informative)

The only practical application I've ever heard of for projects like this is as an integrity check on new supercomputers. They compute the first X digits of pi and then compare it to a known result which someone computed and verified earlier.

On a completely separate note, it's "pi", not "Pi". The Greek letter used is lowercase, and the standard English version is similarly lowercase.

## Re:What Does This Mean? (Score:5, Informative)

(Also, supposedly, to determine if PI is actually infinite or whether it contains a repeating pattern after you get to a certain point)

What? There's a mathematical proof that pi is irrational (in fact, transcendental). Specifically, if it were not, -1 would be irrational (in fact, transcendental) thanks to the Lindemann-Weierstrass theorem [wikipedia.org] and the fact that e^(pi*i) = -1. The digits cannot simply start repeating after a while (in particular, they cannot eventually just become 0, as happens with, for instance, 1/2 = 0.5000... .

## Re:What Does This Mean? (Score:3, Informative)

I imaging that it has applications in astronomy. When you want to precisely compute something over the distance of light years, you may want more than just 10 digits for Pi.

As a professional astronomer I can guarantee that distance scale measurements are a little bit less precise than one part over 10^13. Even for most precise measurements, e.g. gravitational waves experiment, 16 digit suffices!

## Re:Now that is a key! (Score:2, Informative)

A one time pad that can generated perfectly by anyone using simple maths and published techniques? Try worst pad set ever, by telling your adversary the pad is found in the first 10 trillion digits of pi, you just reduced the search space to at worst log2(10*10^12) 45 bits.

## Re:What Does This Mean? (Score:5, Informative)

The radius of the part of the universe visible to us is about 46 billion light years [wikimedia.org] or about 4*10^26 meters. The planck length, assumed to be the shortest length there is, is about 1.6*10^-35 meters. That is, the radius of the known universe is 2.7*10^61 planck lengths. Thus with just 62 digits of pi you are as accurate as the laws of physics allow. In practice you'll never need even that. Indeed, you'll not even measure cosmic distances to the meter (27 digits), or even to the kilometer (24 digits). Even measuring to the light year (12 digits) is probably impossible for objects that far out.

## Re:What Does This Mean? (Score:4, Informative)

Roughly how many digits is that?

No need to google it... here you go: 3.14159265358979323846264338327950

## Re:What's the message? (Score:5, Informative)

To decipher the math-speak on that page for the less mathematically inclined, here's my explanation of what a normal number is, geared towards a programmer.

Say you generated a number by randomly picking digits 0-9. After generating 100 digits, you'd expect close to 10 of them to be "7" (1/10). After generating 1000 digits, you'd expect about 100 to be "7" (1/10 again), but you'd expect only about 10 copies of the string "57" (10/1000 = 1/100), since there are 100 possible two-digit strings ("00", "01", ..." 99") and there are about 1000 length-2 substrings in a string of 1000 digits (999, to be precise). In general, for such a string of length N, we'd expect about 1/10th of the digits to be "7" and 1/100th = 1/10^2 of the substrings to be "57". If we made N very large we would also expect these estimates to get closer and closer to the truth.

You might get some strange abberations by random number generation. For instance, with astronomical bad luck you might generate 0 each time, and then your estimated fraction of "5"'s would be completely wrong. Still, the above properties are pretty good measures of how "well mixed" the digits of a number are, and they're taken (with mild generalizations) as the defining conditions of a normal number.

Specifically, for a given number x, imagine writing out its (infinitely many) digits in base b. Pick a substring of length m that you're interested in--say an encoding of Shakespeare's complete works in the original Klingon. In the first N digits, we would like to require the fraction of substrings matching our given string to be 1/b^m in analogy with the above (1/10^2 came about from b=10, m=2). That's too much to ask, so instead specify a small tolerance above and below 1/b^m. The key condition for normality is that if we look at the first N digits where N is larger than some number (which depends on the tolerances, the substring we picked, and x itself), the actual fraction of matching substrings will be within our tolerances of 1/b^m. A normal number is one where you can perform this operation in any base, with any substring, and with any tolerances.

If pi were normal, there would have to be at least one (indeed, infinitely many) occurrence of a given encoding of Shakespeare's works, since otherwise for N large enough the number of matching substrings would be near 0, and we could specify our tolerances to be between, say, 1/2 * 1/b^m and 3/2 * 1/b^m, which is strictly greater than the fraction of matches for N large enough since that fraction tends to 0, so it can't be within these bounds.

It's not too surprising that proving the normality of a number is much harder than believing it. Essentially, any number whose decimal digits appear "quite random" feels normal.

## Re:What Does This Mean? (Score:3, Informative)

## Re:What Does This Mean? (Score:4, Informative)

AFAIK, we still have no conclusive answer to the question whether Pi has finite or infinite digits.

No.

http://en.wikipedia.org/wiki/Proof_that_pi_is_irrational [wikipedia.org]

There's five different approaches. There are more, mostly closely related cousins.

http://en.wikipedia.org/wiki/Irrational_number [wikipedia.org]

rational = terminates (your "finite") or repeats (your "infinte"). Which doesn't matter because pi is irrational as per numerous different proofs and all irrational numbers are infinite in length.

If this is some sort of "holy book" "intelligent design" thing where the bible says pi is actually 3, then I can't help you there...

## Tau, not Pi! (Score:4, Informative)

That's all well and good, but what about digits of tau [tauday.com]?