Talk:Numerical approximations of π

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[edit] Creation

I think the section about "Numerical approximations" on the π is too long, it is about 1/2 of the whole π page, maybe even more.

So I made an attempt to compile some material here at History of numerical approximations of π. I think this still needs some work, and probably merging with History of pi would be a good idea.

In fact, the cited section "Numerical approximations" on the [[π] should be divided up in two parts: the historical aspect, and the technical (mathematical) aspect. I think there is largely enough material to make up a correct page for both of them. Once it is OK and covers all of the cited section, I suggest to reduce the latter to a brief description of the most important events / facts only.

Maybe the name I chose is not so well chosen w.r.t. the content. Other possibilities would include:

Help from everybody is appreciated. — MFH:Talk 23:23, 16 March 2006 (UTC)

Why not just put all of this information into a "numerical approximations" section on History of pi? There no reason to split our pi information up like this. Night Gyr 23:43, 16 March 2006 (UTC)
I agree. In fact, 2 points "prevented" me from doing this initially :
  1. The current content of History of pi concerns only the name of π, but finally I think this was not a major intention, but the reason is just that completion of that page had been interrupted.
  2. There are 2 possibilities to "file" this information:
    1. History of π with subsections : "history of the name" (not much to say), "history of numerical approximations", and maybe other subjects of "history of π" (history of formulae involving pi and/or number theoretical issues about π) etc
    2. Numerical approximations of π with subsections "history", "formulae used for calculations", "uses of numerical approximations" etc.
In view of that ambiguity, and the fact that each of the 2 aspects (historical + technical) have enough material to fill up a honest page (which might rapidly grow into the order of magnitude considered as "limit of pagesize" for editing and readability reasons), I was tempted to make both, separately. (Finally, the mathematical aspect is not really closely related to the historical aspect.) — MFH:Talk 13:29, 17 March 2006 (UTC)
what do you mean they're not related? The entire history of pi is mathematical. Night Gyr 18:19, 17 March 2006 (UTC)
read technical instead of mathematical if that helps. - I mean: details on convergence of different formulae, algorithms, ... — MFH:Talk 21:52, 17 March 2006 (UTC)

In its present form, history of pi is essentially a stub. I moved the table from that page, because the page created an erroneous impression that history of numerical computation of pi is the whole of the subject. Before this long page gets merged into that stub, the latter should be expanded beyond the stub stage. Michael Hardy 00:01, 26 March 2006 (UTC)

[edit] please note

I will now severely cut down the section on "numerical approximations" on the main pi page. So please don't delete material here, without previously cross-checking if it no more there.— MFH:Talk 14:38, 21 March 2006 (UTC)

[edit] Biblical value

I have a web page describing the supposed Biblical measurement discrepancy (here). I'm not going the add the link, as this would probably be construed as shameless self-promotion (or possibly even original research), but obviously someone else can if they deem it worthy of mention in Wiki. — Loadmaster 15:58, 4 October 2006 (UTC)

I'll bite - it's a good write-up. - DavidWBrooks 23:14, 4 October 2006 (UTC)

[edit] Intro graf

we're getting into a back-and-forth edit here, so how about some discussion. The current edit by User: Henning Makholm is, IMHO, poorly written (starting out with "That pi is ... " is atrocious), clumsy, and excessively wordy. (We'll ignore the typos) How does "no practical system for calculating with numbers is able to express pi exactly", differ from the shorter, tigher, less redundant "but no method of calculation was available until fairly recently"? - DavidWBrooks 20:02, 16 October 2006 (UTC)

The last couple of edits both seem like nonsense. Anyone who thinks π is the ratio of circumference to radius rather than circumference to diameter should wake up before editing this article.

Now what in hell does this mean:

“ Unfortunately no practical system for calculating with numbers is able to express π exactly. Though this fact was only proved rigorously in recent time, it has been suspected since the earliest times, ”

???

What "fact" that was recently prooved is referred to? And the edit before that said that no method of calculating π was known until recently. What about the word of Archimedes in the 3rd century BC? Michael Hardy 20:15, 16 October 2006 (UTC)

"Radius" - ha! didn't even notice that; I was fixating on the typos and English ... - DavidWBrooks 21:23, 16 October 2006 (UTC)

I think your idea of taking a time out for a discussion is rather good, DavidWBrooks. However, between the three of you, you've lost a couple of other edits. Perhaps worth to salvage them, anyhow. JoergenB

Now, for the main issues: Both the version of DavidWBrooks and the one of Henning Makholm could be a little confusing as regards what we reasonably could mean with 'calculating exactly with a number', or 'methods of calculation of π'. π was recognised as an exact entity - a 'proportion' - by matematicians in antiquity - probably already by Eudoxos, and definitely by Archimedes. Archimedes employed Eudoxos's method of defining general proportions by means of 'commensurable proportions'. In modern terms, π is defined exactly by means of an infinite number of approximations by rational numbers. (This is slightly misleading; the ancients formalised their ideas in geometric terms more than in terms of numbers as we recognise them.) Archimedes did indeed use this in order to 'calculate exactly' with π; one of the most important results is that exactly the same proportion holds between the circumference and the diameter of a circle on the one hand, between the areas of a circle and of a square on the radius of the circle on the other, and between the areas of a sphere and of a square of its diameter 'on the third hand'. He also gave concrete methods of constructing approximations of this proportion with arbitrary precision; and this is what most of us today mean with 'a method of calculating π'. Thus, both exact calculations with π, and methods of calculating π with arbitrarily good approximations are known from the days of Archimedes and on.

As Henning Makholm very aptly noted, a nicely written introduction is not to be preferred, if it is factually wrong. Therefore, please let the old introduction (as reinstated by Michael Hardy) stand for a while, until you've discussed new ideas on this talk page.

Browsing through the article, I did notice some other errors, which I think could be corrected in the meantime (assuming the edits do not get lost in revert wars). I'm especially thinking of the sentence starting

In the third century BC, Archimedes showed that 3 + 10/71 < π < 3 + 1/7, and later formed a proof that 22 over 7 exceeds π...

Now, it is a funny matter for all of us to laugh at together, that no one of the editors for some time noticed this contradiction. Probably, now that you look at it, you see that it is a little queer to state that first Archimedes showed that π is less than 3 + 1/7, and some time later he proved that π is less than 22/7.

However, this is the kinds of oversights we all make. I wouldn't dream of implying that all who have edited the article without noticing this thereby proved that they didn't know that 3+\frac{1}{7} = \frac{22}{7}, and therefore 'should wake up before editing this article'. I make this kind of laughable oversights all the time myself, so it would be rather stupid of me anyhow. JoergenB 22:27, 17 October 2006 (UTC)

Aw gee, what a killjoy you are: some editors won't want to play if they can't call people idiots! But as the guy who wrote "radius" I can't have the fun of being on a high horse - sigh. It would be nice if this article started with a layman's description of the situation, so the casual reader who doesn't go beyond the first couple of grafs has a general understanding and might be interested enough to pursue it further. A dry sentence like "This page is about the history of numerical approximations of the mathematical constant." isn't going to enlighten many folks. - DavidWBrooks 22:41, 17 October 2006 (UTC)

I don't know that there's anyone who won't participate unless they can call people idiots. I don't think I'm alone in preferring those who make astute contributions. It's not easy to be patient with someone who claims a certain proposition was recently proved while being unable, even after some attempts at explanation on various talk pages, to say precisely what it was that was proved, and just leaves us guessing. Michael Hardy 23:52, 17 October 2006 (UTC)

Since the very first time I heard of or from you was an expletive-laden, over-punctuated, self-righteous comment thrown on my talk page and copied in other places, it wasn't unreasonable for me to assume you were - say, about 15? The fact that your arguments were correct didn't lessen the immaturity; lots of 15-year-olds are smart. Perhaps that's something you can work on, just as I need to work on not making sloppy edits. - DavidWBrooks 10:19, 18 October 2006 (UTC)

"Expletive laden"? I said what you wrote is "bullshit"; maybe that's an expletive; there were no others. What you wrote was irresponsible and dishonest. Do you think writing an introduction with a nice format in complete disregard of its truthfulness constitutes a good-faith attempt to improve Wikipedia? I don't see how you can say it's mere "sloppiness" to say that no method of computing pi was known until recently, in a context making clear that "recently" means certainly no more than 500 years ago, at the top of an article that gives historical details of computing pi well over 2000 years ago. As I said, what's the matter with you? Michael Hardy 20:12, 18 October 2006 (UTC)

Are you claiming that your edit summary about "dishonest idiots" contains no derogatory language? Where I come from "dishonest" implies that one is intentionally and consciously claiming falsehoods to be true. While there was greater or lesser inaccuracies in both DavidWBrooks' attempt to improve the intro paragrah, I am utterly certain that neither of us intended to write untruths. On the contrary, I spent quite some time thinking of how to express the point without making simplfications that were not technically true. As it turned out -- well after I pressed the "Save page" button -- I failed, but that does not in any meaningful way make me "dishonest". It may or may not make me an "idiot", but if you're calling anybody who sometimes make a non-perfect edit "idiots", there won't be many non-idiots left to write Wikipedia. Henning Makholm 17:17, 19 October 2006 (UTC)
You should have known that what you were writing was incorrect. DavidWBrooks' comments were obviously incorrect even to those who don't know the subject, since nearly everything else in the article contradicted it. Your comments are still unclear now. Some guessed that you meant Lindemann's transcendence proof, and tried to defend the claim that it could be read as correct. But (1) His remarks were unconvincing for reasons I've noted at Wikipedia talk:WikiProject Mathematics; (2) even if your statement could somehow be reworked into a correct statement about what was proved, it's not clear that it belongs in the introduction, rather than being just another bullet point in the long list; (3) your claim that the ancients suspected something along these lines is bizarre. Maybe some of them suspected the impossibility of squaring the circle (but even that seems like a stretch), but to go all the way from there to whatever it was you were trying to say (and I'm still unsure just what it was) is absurd. Michael Hardy 20:22, 19 October 2006 (UTC)
OK, we've all had our say - back to work! - DavidWBrooks 19:00, 19 October 2006 (UTC)

[edit] The graph's x-axis

should read "Year", not "Century". —Greg K Nicholson 21:11, 5 February 2007 (UTC)

[edit] History of continued fraction of π

The continued fraction section seems out of place in this article. If we know something about the history of the use of continued fractions for approximating or studying π that would fit. It might be worth noting that some of the classic approximations turn out to be continued fraction approximations (22 / 7, 355 / 113).

That said, I think the current treatment is a little muddled between continued fractions and best approximations. I was thinking we could somehow mark (bold or italics?) the best approximations that are continued fractions. I also think it would be worth ending with 103993 / 33102 since that is the next continued fraction approximation after 355 / 113 and the biggest jump in the early part of the continued fraction. --Jake 20:56, 30 March 2007 (UTC)

[edit] Biblical Pi

There is in fact a very simple expanation for the very bad approximation of 3, apparently used by the Bible. And it is found in the text itself...

A little textual study shows that not only does the value of pi appear to be wrong in this portion of scripture, but the spelling of the name for the measuring instrument is also incorrect. When we consider this apparent inaccuracy in terms of numerical inaccuracy (as all Hebrew letters have a numerical value), it appears to consolidate, within the limits of human vision, the very bad value of pi, it is as follows:

The word used for 'line' in the original text is spelt as follows: heh-resh-qoph. The normal spelling for this word is: resh-qoph. The initial has a numerical value of 5 + 6 + 100 = 111. The final 6 + 100 = 106. The error involved is thus: 111/106. The product of the given value of pi multiplied by the error is: 3 * 111/106 = 3.14151. Now a cubit is approximately 445 mm. So the actual length, assuming the same diameter, of the circumference of the bronze sea is: pi * 10 * 445 = 13980.08731 mm The value given in the text including the correction is: 3 * 111/106 * 10 *445 = 13979.71698 mm. Which gives a percentage error of: (13980.08731 - 13979.71698) / 13980.08731 * 100 = 0.00026%. The actual length discrepency is 0.3703 mm, which is about the limit of human vision.

Mike

220.235.172.27 05:43, 17 May 2007 (UTC)

[edit] "googolpi"

This recent item added by user:David W. Hoffman is correct:

\sqrt[193]{\frac{\mathrm{googol}}{11222.11122}} = 3.14159265364382234\dots

He stated that it differs from π only in the 10th place after the decimal point and it's only the difference between 5 and 6. He improperly signed his name to the addition (this was in the article, not on the talk page) and identified it as of his own devising. It was deleted as original research. A question is to what extent the O.R. policy should apply when it's so easily verifiable? Perhaps in some somewhat modified form this could be included. Michael Hardy 22:39, 7 August 2007 (UTC)

[edit] Shanks

Um... At one point the article mentions that Shanks' 1873 work was made possible by the recent invention of logarithms by Napier (who lived in the 17th century). Possibly the article means ' made possible by the recent publication of logarithm tables ' ? —Preceding unsigned comment added by 129.97.219.23 (talk) 16:56, 11 December 2007 (UTC)

[edit] Graph

The graph should be a semi-log plot with the y-axis being logarithmic to more accurately show the more modern progression toward the true value of pi. —Preceding unsigned comment added by 138.87.186.80 (talk) 05:03, 19 February 2008 (UTC)