Talk:Approximations of π

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:

• History of π - this currently says more about the use of the letter π than about calculations (which make me correct some links) - but maybe this was not intended?
• History of approximations of π
• Numerical approximations of π - this moves it somehow out of category:history of mathematics, where I wanted to settle it in order to delimit precisely the scope. Maybe under this title we could put the technical / mathematical aspects only, and put a reference to the "history" page.

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)

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)

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)

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 $\pi$ 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)

The graph's x-axis

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

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)

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)

Is anybody else a little bewildered by this claim? I think such a strange claim requires at least a reference so I will delete the passage in the article unless a reference can be provided JHobbs103 (talk) 20:07, 17 June 2009 (UTC)

Agreed. What Hebrew numerology has to do with calculating a ratio from physical measurements is obscure at best and certainly confusing to the reader. I didn't know what to make of the text when I first saw it, either. Unless the spelling of the Hebrew words actually changes their numeric meaning (from "three" to "three and a half", for instance), I don't see the relevance. — Loadmaster (talk) 23:24, 18 June 2009 (UTC)

I don't see the problem. The section in the article itself is well referenced. It says: "Here is an explanation which has been given." It is the case that that explanation has been given and it is worth the two lines that it gets to mention it. There are pros and cons to that explanation and discussions of it provenance, but I don't think that belongs in the article. The word for line is 'kav' which would be spelled (as in the article) 'qoph vav'. I am not sure what the idea of this 9 line May 2007 comment with resh-qoph is but that is not relevant for the shorter section in the article. --Gentlemath (talk) 02:03, 19 June 2009 (UTC)

The problem is that it states that the numeric value is supposedly calculated from "the ratio of the numerical values of the Hebrew spellings". What has that got to do with the actual physically measured dimensions of the temple bronze bowl, or with the real value of pi? As I said before, unless the alternate spellings change the meaning of the Hebrew words (to something other than "thirty" or "ten"), it does not seem relevant. — Loadmaster (talk) 22:40, 1 July 2009 (UTC)

"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)

After 18 months that is still a Good Question. My take on it is that proven mathematical facts, no matter who proved them, are a form of Absolute Truth that is not subject to opinion, faith or ongoing research. The concerns of WP:OR will surely be satisfied if a mathematical fact can be readily verified by any appropriately skilled user. WP:N applies also of course.

In the case of the stated aproximation of π above, I am not able to readily verify it, mainly because neither on my PC nor in my drawer do I have a calculator that can handle such numbers. Failing a WP:RS, if someone provided a checkable s/w code that can be run to prove the fact then I would judge it includable nevertheless in the article.Cuddlyable3 (talk) 20:04, 19 February 2009 (UTC)

It is true that in mathematics there is seldom dispute about whether a proof is valid or a computation is correct. But deciding whether a proposition is interesting is still a judgment call to be made by flesh-and-blood mathematicians. Of course we want the propositions in the encyclopedia to be true, but we also want them to be worth the reader's time. (It would be easy to make a computer generate an infinites stream of provable theorems if we did not care about whether they are interesting or important). In this respect "no original research" works as well in mathematics as in any other field as an easily-administered initial test for notability -- which is all it ever is anywhere in Wikipedia. In a field where noteworthy claims tend to be published in respectable sources (this includes mathematics), it is a practical working principle to assume that claims that are not so published are not noteworthy enough that we should include them in the encyclopedia.

By the way, it is not difficult to test the proposition here with standard tools; you can reduce the risk of rounding trouble by taking the log of everything before you begin. –Henning Makholm (talk) 20:46, 19 February 2009 (UTC)

Makholm you explained the WP:N guideline cogently. The WP:N guideline and WP:NOR policy should not be interchangeable.

If my standard tools included a pushbutton log and antilog calculator accurate to at least 18 figures then as you say, it would not be difficult to test the proposition. Where do you get one of those? Cuddlyable3 (talk) 21:24, 19 February 2009 (UTC)

The proposal was to allow original research in mathematics articles. My counterargument is that mathematics that has not found publication in respectable sources is unlikely to be notable, so creating an exemption from WP:OR would just lead us to reject the same material for notability reasons. There would be no benefit to the encyclopedia to offset the increase in complexity and confusion that would result from creating exceptions to a core rule.

Tools: Standard IEEE 754 double precision values (as provided by any programming language you can lay your hand on, or the Windows calculator, or bc or apcalc on Unix systems) have about 19 digits of precision. That should be plenty to check the first 10 digits of the eventual result. –Henning Makholm (talk) 18:23, 26 February 2009 (UTC)

As to whether this formula is interesting: this is a different question from WP:N but my opinion is that it is not. It is interesting when one puts a small number of digits of information into a formula and gets a much larger number of digits of precision back out. Here the number of digits in and the number of digits out are roughly the same, making this likely to be just a numerical coincidence rather than anything with a deeper meaning. —David Eppstein (talk) 22:19, 19 February 2009 (UTC)

All rational approximations to the irrational pi are numerical coincidences, starting with the 22/7 that I was given at school. I agree with Henning Makholm who I believe is saying that "of interest to flesh-and-blood mathematicians" is a workable test of WP:N. The number of digits 101 of the numerical (not textual) expression of a googol is considerably more than the number of correct digits of pi here. Cuddlyable3 (talk) 13:32, 21 February 2009 (UTC)

Charitably construed, the googol might count for only the 5 digits in 10¹⁰⁰. But the amount of information on the left-hand-side would still be large compared to the precision of the approximation. –Henning Makholm (talk) 18:23, 26 February 2009 (UTC)

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)

Removed. The author may well have meant any of the Victorian publications of log tables, but it is nonsense as it stands.

Graph improvement

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)

The graph should also show Archimedes' feat (in about A.D. 220 B.C.) to provide an upper and a lower limit for his estimated π. These limits are approximately 223/71 ≈ 3.14085 and 220/70 ≈ 3.14286. The interval width is then ≈ 0.002 . Rgds / Mkch (talk) 21:27, 13 April 2009 (UTC), rv Mkch (talk) 08:20, 20 June 2009 (UTC)

Citation needed: Oliver Shanks story on non-relation of Daniel Shanks and William Shanks

There's an interesting but unsourced story in the current article, attributed to Daniel Shanks's son Oliver Shanks, that they are not connected to William Shanks. I have not been able to locate this story elsewhere. This story was provided by 205.188.116.5, currently a blocked address. The diff is here. In the same edit he states incorrectly that Daniel Shanks calculated pi to 1,000,000 decimals (the true figure is 100,265), so I lack confidence in his Oliver Shanks story. I have marked it as needing a citation. Thanks for any information on this story. --Uncia (talk) 21:25, 27 June 2008 (UTC)

Spaces

There are many instances in this article where there are no spaces between a statement and a link. I was too tired to fix them. 68.200.239.84 (talk) 17:57, 19 July 2008 (UTC)

Use of the symbols \approx and \approxeq

The symbols \approx and \approxeq are used apparently with the same meaning. Also, \approxeq does not appear in the list of mathematical symbols. I could not find an explanation for \approxeq, while \approx is explained in the list of mathematical symbols. I suggest to use only one symbol: \approx. An alternative is to include an explanation for \approxeq in the list of mathematical symbols. Xelnx (talk) 08:26, 24 September 2008 (UTC)

Idle question

The article mentions

• accurate to 30 decimal places:

${\frac {\log(640320^{3}+744)}{\sqrt {163}}}$

This is a consequence of the closeness of the Ramanujan constant to the integer 640320³+744. This does not admit obvious generalizations, because there are only finitely many Heegner numbers and 163 is the largest one. One possible generalization, though (also a consequence of value of the j-invariant on a lattice with complex multiplication), is the following, perhaps not as impressive, but accurate to 52 decimal places:

${\frac {\log(5280^{3}(206371+60606(1+{\sqrt {61}})/2)^{3}+744)}{\sqrt {427}}}$

Related to the first fact is

• accurate to 17 decimal places:

${\frac {\log(5280^{3}+744)}{\sqrt {67}}}$

I would expect that EITHER the 52 decimal place fact is a fluke (and not super worthy of inclusion) OR there is a related fact with X>>52 digits involving 640320 at least equally worthy. I certainly support the inclusion of the 30 digit fact and maybe the 17 digit fact. I take no position with regard to the 52 digit or X digit fact. --Gentlemath (talk) 18:50, 9 March 2009 (UTC)

Please explain what you mean by "a fluke". Have you tested the expression? Cuddlyable3 (talk) 13:46, 25 March 2009 (UTC)

Whatever I meant, I no longer suspect so. I wasn't sure if it is considered ok practice to blank out ones own talk contributions. There are the two approximations coming from a certain theory

• accurate to 17 decimal places:

${\frac {\log(5280^{3}+744)}{\sqrt {67}}}$

• accurate to 30 decimal places:

${\frac {\log(640320^{3}+744)}{\sqrt {163}}}$

and then

• accurate to 52 decimal places:

${\frac {\log(5280^{3}(206371+60606(1+{\sqrt {61}})/2)^{3}+744)}{\sqrt {427}}}$

SO based purely on feeling I thought that there should be a related fact using 640320 (maybe 91 decimal digits since 52=3*17+1, one can dream). It could be that the two appearances of 5280 are unrelated. The numbers involved are highly composite (as are feet in a mile=8 furlongs/mi * 10 chains/furlong * 22 yards/chain * 3 feet /yard ) so it is not as amazing as one might think to see them twice or thrice. Anyway, I did not find anything else except

* accurate to 46 decimal places

${\frac {\log((640320^{3}+744)^{2}-2\cdot 196884)}{2{\sqrt {163}}}}$

I do have a question about the newest additions accurate to 18 and 25 decimal places: What should be the criterion for as worthy approximation? Something like ratio of correct decimal digits to number of decimal digits used in the expression. So 52/27=1.9.. might be marginally less worthy than 46/22=2.09... I hasten to add that I don't want to add that 46 digit fact to the article. But these new 18 and 25 digit facts have "worth" less than 1. Of course they do use the same small constants and the reference seems to have some theory behind it so who knows. --Gentlemath (talk) 16:29, 25 March 2009

Do keep in mind that the later formulas regarding the J function involve taking a log which may be considered to be 'cheating' since log is transcendental. An expression that is constructable (ie square roots, ractions, integers) is ideal.

(UTC)

Why does

Why does every pi page have to have the chudnovsky, ramanujan ,abd Borwein pi formula? it's kinda annoying and redundant. —Preceding unsigned comment added by 153.18.101.154 (talk) 02:34, 7 April 2009 (UTC)

[Numerical approximations of π] is the only page we are concerned with here. The fomulae are notable and relevant. Cuddlyable3 (talk) 12:58, 7 April 2009 (UTC)

Approximation in base 60

Is it known from where and from when does an approximation in base 60:

$(3,8,29,44)_{[60]}=3+{\frac {8}{60}}+{\frac {29}{60^{2}}}+{\frac {44}{60^{3}}}$

come from? It contains first four convergents of π:

$3;\ {\frac {22}{7}};\ {\frac {333}{106}};{\frac {355}{113}};\!\,,$

and its (periodic) numerical value is (a little bit) lower than π (correct to 6 decimals):

$3,1415{\overline {925}}\!\,.$

Ptolemy around 150 used similar (periodic) approximation (in base 60):

$\pi =(3,8,30)_{[60]}=3+{\frac {8}{60}}+{\frac {30}{60^{2}}}=[3;7,17]=\left\{3,{\frac {22}{7}},{\frac {377}{120}}\right\}=3,141{\overline {6}}\,\!,$

which is correct to 3 decimals. --xJaM (talk) 06:17, 1 May 2009 (UTC)

The reason missing

Why are all these calculations important, why do we need to know the value so darn precisely? A short explaining paragraph would be nice to have. 88.148.210.36 (talk) 07:32, 5 September 2009 (UTC)

Moore's law

The first 100,000 digits of π were published by the N.R.L.[9] :80–99 The authors outlined what would be needed to calculate π to 1,000,000 decimal places and concluded that the task was beyond that day's technology, but would be possible in 5 to 7 years. [9]:78

I guess this is OR, but it's interesting that this estimate conforms fairly well to Moore's law: 5 years is 3+1/3 times 18 months, so 3+1/3 doublings, and 2^(3+1/3) ≈ 10.08 Mr. Jones (talk) 13:17, 14 September 2009 (UTC)

"accurate to 25 digits"

I just wanted to note that the equation given for pi accurate to 25 digits http:/upwiki/math/5/c/2/5c23b9c8b622a015a3da437bbbc85426.png is just absurd... it contains more digits itself than correct digits of pi!