Talk:Pi/Archive 1

From Wikipedia, the free encyclopedia

Archive This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.
Archive 1
| Archive 2 →

Contents

Bailey-Borwein-Plouffe

The article says:

   It was shown that the existence of the above mentioned Bailey-Borwein-Plouffe 
   formula and similar formulas imply that the normality in base 2 of <eth> and various 
   other constants can be reduced to a plausible conjecture of chaos theory.

What is the conjecture? Who showed this reduction? Can you give a reference for the papers involved? Or their websites? Even a few keywords suitable for web searching would be very much appreciated!

Bailey and Crandal

I see you've added "Bailey and Crandal in 2000". That's great. It would be good to add a link to their paper (or it's abstract) if it exists somewhere on the web. I've tried searching for it, and haven't had any luck.

But the article does give an URL, and you can find the paper there, in both PDF and PostScript. (The PDF is very badly done, however.) --Zundark, 2001-08-21

You're right. My mistake. I'd missed the URL in the earlier section.

not OK to link to the image

I deleted the image at http://www.nersc.gov/~dhbailey/dhb-form.gif from the article. It might be OK to swipe the image and put it on the Wikipedia server, but it's not OK to link to the image itself on the government server. We don't own their bandwidth... --LMS

See bandwidth theft. --Theaterfreak64 23:28, Apr 24, 2005 (UTC)

I removed ...

I removed the following badly garbled sentence:

The PiHex Project have 1000000000000000, 1^15 bits of pi. (1^30 digits of PI) PiHex,PiHex Site

—Herbee 2004-02-13

particularly long value of pi

I've got no problem with showing a particularly long value of pi, assuming it's accurate as far as it goes. It may be curious and totally useless, but it's not vandalism. Eclecticology

  • I called it vandalism because it's merely part of a days long pattern from this character. Rgamble

Why approximate pi?

Is there any point in computing many digits of pi?

Is this true?

pi to 40 places is sufficient to measure the circumference of the known universe with an error less than the width of a hydrogen atom.

If so, what is the purpose of calculating pi to large numbers of digits?Ortolan88

What is the purpose of climbing the Everest? It is there. Same with pi.--AN

But is the statement true? If so, it, or some variant, should be in the article to give some proportion to this quest. I would also like to know what is so hard about calculating that number of digits. Is it anything more than long, long, long division? In what way does this advance the art of mathematics? These are serious questions by an uninformed member of the encyclopedia-reading public. Ortolan88

IIRC, this BBC programme mentions facts relating to this. Mr. Jones 14:23, 8 Mar 2004 (UTC)

Computing many digits of pi is trivial and not mathematically interesting. It is often done to test (super-)computer hardware. I don't know if the statement about the universe is true, but something close to it is probably correct. You don't need more than a couple dozen digits in real word applications. AxelBoldt 21:07 Nov 23, 2002 (UTC)

The specific knowledge of KNOWING the digits of pi is trivial, but the mathematics used to GET more digits is mathematically interesting and not trivial. For instance, many of the advances made in calculating more digits of pi in a shorter amount of time came not from faster computers, but from better identities for pi, i.e. either sums, products, or continued fraction expansions with faster rates of convergence. Coming up with these identities with faster rates of convergence is NOT easy; nor is it a trivial and uninteresting math problem. In fact, PROVING that many of these algorithms for getting more digits of pi are true, will give more (correct) digits, and will give them faster (i.e. proving the rate of convergence), all of these are very interesting questions, and this is not just to get more digits of pi. Often, the ideas behinds these proofs of the identities are related to other problems in math, i.e. an IDEA used to prove an identity used to get more digits of pi might be used to prove another result, which does have other "real-world" applications, or at the very least, other applications within other areas of pure math. Revolver

According to the article on pi, "The most pressing open question about π is whether it is normal, i.e. whether any digit block occurs in the expansion of π just as often as one would statistically expect if the digits had been produced completely randomly." This can only be answered by a mathematical study of the digit sequence. Obviously one wouldn't study it by looking it up in an encyclopaedia, but even so, people at home might want to do a bit of a check for themselves, to see if it sounds plausible. :) -- Oliver Pereira 00:20 Nov 24, 2002 (UTC)

Good luck. Even looking at the first 10,000 or 100,000 digits isn't enough to give some kind of statistical assurance. The short-term random luck factor is just too high. Anyone who has played poker for any length of time will understand what I mean by this. Revolver

Evidence for or against the normality of pi can obviously never be produced by looking at a finite initial segment of the digit sequence. That's why I said above that computing digits of pi is mathematically uninteresting.

But people should definitely be encouraged to experiment with the digits. So we should tell them how to generate the digits for themselves, so that they have them in a format which allows all sorts of statistical tests. Right now, using our data to find out how often the digit sequence "33" occurs among the first 10,000 digits would require a perl program that's not much simpler than directly generating the digits from scratch. AxelBoldt 20:42 Nov 24, 2002 (UTC)

Axel is absolutely right here -- if we want to know something for an infinite number of cases, (all natural numbers, all digits of pi, etc.) just because we only have a finite # of cases, doesn't mean it's worthless. Looking at the known finite # of cases allows one to make reasonable conjectures. It doesn't prove anything, of course, but it keeps us from wasting our time going down alleys that (almost certainly) dead-ends. Revolver
Yep, I agree - except about the simplicity of writing a perl script to compute the value of pi! :) Since we are writing an encyclopaedia for general use, most people referring to it will just be ordinary people without any programming ability. Some of whom genuinely believe that pi is exactly 22/7! Such people might want to see a concrete demonstration of just how random the digits of pi are. Plotting bar charts of the frequencies of digits might be an amusing mini-project for a schoolchild who is starting to learn about statistics. Well, maybe not, but I expect there are many more plausible suggestions that people could come up with. I just mean that members of the public might be curious to see the digits of pi and play with them, even if they don't have the ability to do anything serious with them. I'm thinking of a past version of me when I was at school, for example. -- Oliver Pereira 21:38 Nov 24, 2002 (UTC)
The problem with plotting frequency of digits to test normality is it's not just single digits, it's all finite sequences of digits, and the distribution of these, the more digits you get, becomes much more susceptible to short-term luck. Revolver

width of a hydrogen atom

I removed this:

It is said that pi to 40 places is sufficient to measure the circumference of the known universe with an error less than the width of a hydrogen atom.

Who says that? If somebody has done the calculation, we can simply omit the qualifying "it is said that" which essentially renders the whole paragraph pointless. AxelBoldt 20:00 Nov 25, 2002 (UTC)

I'm confused about it myself. Surely the accuracy of the result is dependent on the accuracy of the figure we have for the diameter of the universe, as well as the accuracy of pi. -- Tarquin 20:07 Nov 25, 2002 (UTC)
It isn't pointless. It gives a sense of proportion to the whole thing. 22/7 is sufficiently accurate for pi to get the circumference of a can of corn. With a circle of one mile diameter 3.14 yields 16572.2 feet circumferences, 3.142 gives 16589.76, 3.1416 gives 16587.648, and 3.14159 yields 16587.595, which is where my calculator poops out, but, obviously, the more digits of pi are adding accuracy. If one of you cosmo brains would do the same arithmetic for some galactic measure or other (instead of silently deleting my interesting, if unproven statement about hydrogen molecules) using, say 40 places for pi, then maybe we could put this million digits of pi business into some kind of proportion. What is confusing or pointless about that? This is an encyclopedia. My whole intent is to give readers some idea of the value of the additional digits of pi.Ortolan88
Okay, i'll have a go... for a universe 12 billion light-years across, that's about 1.135e26 m. An atom of hydrogen is about 1e-10 m across, meaning you'd need about 36 decimal places in pi to get error levels below the diameter of hydrogen. So 40 does it admirably. Graft
The observable universe currently has a radius of about 50 billion lightyears, because of the past expansion, but your estimates still work. Of course, if you want the volume of the observable universe precise to the volume of a helium atom, you need about 270 digits of pi. AxelBoldt 23:17 Nov 30, 2002 (UTC)
Cool, so I assume you'll add this to the article? Ortolan88 05:57 Nov 29, 2002 (UTC)
PS - OK, no takers, some time this weekend I'll add it to the article. It isn't just whim, folks, it is something the average encyclopedia reader deserves and will only get from the Wikipedia. I may move one of the more interesting paragraphs up a bit too. And, mathematicians, I read in a rival encyclopedia [1], that Euler came up with a connection between pi and natural logarithms that isn't mentioned in our article. Ortolan88 15:40 Dec 7, 2002 (UTC)
Sure it is -- take Euler's identity, take the log of both sides, you get pi = (-i)(log (-1)), where we take principal branch of log function. At least, I assume that's the relation they had in mind. Revolver
This just in, pi to 1.24 trillion digits. [2] How about an article listing them all? Ortolan88 22:12 Dec 7, 2002 (UTC)

Thats goign to take more than 1 Terabyte of memeory!!! :-S

Regarading the rival encyclopedia article: it says "the famous formula ei 1, where i 1 ." now unless that's my browser skipping characters, they've made a complete pig's ear of Euler's identity... oh yes, by the way... we have a full article on that equation. nananana! ;-) -- Tarquin 22:22 Dec 7, 2002 (UTC)

these approximations

These approximations were once useful to the applied sciences; the more recent approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers.

I don't understand the above sentence.

  • Of what use were "these approximations"?
  • Why aren't they useful any more?
  • Are the recent approximations entirely useless, or just the extra digits?

Without answers to all the above questions, I'm inclined to delete the quoted sentence. But I hope someone who knows a lot more about math than I do (like Axelboldt) can help out here. --Ed Poor

it means this: newer approximations have more digits. They don't make the old ones obsolete. An old approximation of, say 50 digits is useful. A 10,000 digit is pointless; the extra accuracy is irrelevant because we never need it. It's hust the extra digits that are irrelevant. The 50th digit (for example) has not changed since it was first determined -- that's something for an article on approximations to explain: it is normally possible to know how good an approximation is: we don't just calculate 50 digits, we know that those 50 digits are right, and even if we went further, we'd still get them. hope that helps -- Tarquin

This entire debate is patently ridiculous and should be deleted. The children with their pocket calculators should go off and let the rest of us work in peace.

First, using pi calculations to calibrate supercomputer speeds is absolutely meaningless. Supercomputers are meant for real work, not moronic games. Calibrate speed with real work.

I don't know where you got the idea that pi calculations are used to "calibrate supercomputer speeds" nor do I know what that even means. Pi calculations (and prime number calculations) are often used to test new supercomputers. You let them compute digits of pi for a week with two different methods and then compare the results, to check for hardware bugs. AxelBoldt 05:13 Nov 26, 2002 (UTC)

Second, nobody needs to know pi to more than five digits. Period. End of discussion. Anything more than that is simply compensation. Something so delicate as to need outrageous values of pi is a toy, that's all.

Whatever. Revolver

Third, every resource spent analysing pi is a resource wasted. Spending time on pi is exactly as fruitful as spending time with the human genome.

Every resource spent writing symphonies is a resource wasted. Every resource spent pondering the meaning of existence is a resource wasted. In short, every activity YOU think is a waste of time is a resource wasted. Revolver
Some people think the human genome will someday in the future produce a useful medicine or treatment. Some people also think that pi may have a similar surprising usage. Psi 12:37 Dec 2, 2002 (UTC)
Who cares? Investigating properties of the expansion is an interesting question, even if nothing "useful" comes from it. Revolver
Every resource spent whining that every resource spent analysing pi is a resource wasted is a resource wasted. If you don't get that, don't read math articles.   ;-)
—Herbee 22:36, 2004 Mar 5 (UTC)
If mathematics were only ever pursued solely for its 'practical' purposes, much of modern thought would not exist. Mathematics is a remarkable science in that it sometimes can take centuries to find an application for a theorem. We, therefore, can make no judgement as to the usefulness of a mathematical pursuit. Calibrating usually entails using a benchmark - that's why a mathematical constant with sufficient computing time involved is used. But you don't uderstand - they don't calibrate speed. Speed is calibrated by architecture of the processor. They calibrate, or test, the logic of these computers. If an error is committed in so many trillion instructions, the processor is canned. Take into account that these supercomputers put people on the moon and coordinate defense networks, and then you might understand why you don't want to be calibrating your supercomputer with 'real work'. I'd also like to add that there has been a lot of math created as a result of efforts to calculate pi. It is not 'patently ridiculous' to attempt to calculate pi - and it takes far more than a pocket calculator.