Talk:Approximations of π/Archive 1

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.&mdash; MFH:Talk 14:38, 21 March 2006 (UTC)

Creation
I think the section about "Numerical approximations" on the &pi; is too long, it is about 1/2 of the whole &pi; 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 [[&pi;] 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 &pi; 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. &mdash; 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 :
 * The current content of History of pi concerns only the name of &pi;, but finally I think this was not a major intention, but the reason is just that completion of that page had been interrupted.
 * There are 2 possibilities to "file" this information:
 * History of &pi; 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
 * 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.) &mdash; 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, ... &mdash; 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)

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 &pi; 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 &pi; 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 &pi;'. &pi; 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, &pi; 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 &pi;; 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 &pi;'. Thus, both exact calculations with &pi;, and methods of calculating &pi; 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 &pi; is less than 3 + 1/7, and some time later he proved that &pi; 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 &pi;
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 &pi; 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)

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)

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. — Preceding unsigned comment added by Pmanderson (talk • contribs) 16:57, 10 March 2009 (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)

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)

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://upload.wikimedia.org/math/5/c/2/5c23b9c8b622a015a3da437bbbc85426.png is just absurd... it contains more digits itself than correct digits of pi! —Preceding unsigned comment added by Zamadatix (talk • contribs) 02:49, 17 March 2010 (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)

Still a very interesting question, all this is totally obsolete, if you don't know: why oh why? I don't care for the 1000000th place of pi, but as it seems, some people do - WHY?! 141.20.192.254 (talk) 11:30, 27 May 2010 (UTC)

Simple example needed
This article has plenty of in-depth information about Pi. That is fine. What is also needed is a simple example program in a readily available language (Python or C) which calculates and prints the value of pi in base 2 or 10. This way people could calculate their own values without having to use someone else's idea of how many digits are needed. This would be invaluable for students.
 * Agreed! This article is only useful to people with advanced math backgrounds who are also interested in history.  It does nothing for people with an ordinary math background who want to know how we come up with the long strings of numbers that we're told represent pi.  "Summing a circle's area" is by far the most accessible section, but it's buried in the middle of the article, and comes with no code. Lunkwill (talk) 09:11, 12 June 2010 (UTC)

What were the actual assertions of early calculations?
As we all ought to know, pi is wrong as a choice of "fundamental constant" in preference to 2pi. I was hoping to find here some information on what the actual assertions were which imply these values of pi, so as to learn when pi itself (as opposed to, say, 2pi) began to take hold as the ratio one should focus on. (It can't have only as late as Jones that that convention was fixed, can it?) So does anyone have any of the texts of these original sources to hand? 4pq1injbok (talk) 01:49, 2 July 2010 (UTC)


 * Just a guess: It might potentially have something to do with the fact that with relatively crude instruments it's somewhat easier to find the diameter of a circle than its radius.Ptorquemada (talk) 17:07, 14 March 2011 (UTC)

Pyramids and pi
I removed the citation of Edwards, The Pyramids of Egypt p.269. This book says "The normal angle of incline was about 52° - a slope which in the Pyramid of Meidum and in the Great Pyramid would have resulted if the height had been made to correspond with the radius of a circle the circumference of which was equal to the perimeter of the pyramid at ground level." This does not support the assertion that these pyramids were involved in an early computation of pi. Kenilworth Terrace (talk) 17:16, 10 September 2010 (UTC)


 * Here is a recent book by an Egyptologist who says that the AE did not know Pi and that even problem 48 in the Rhind papyrus is not a computation of ? at all. They are approximating the circle by an octagon and we can extract an approximation to ? from that, but they would not have thought to look for it. Dougweller (talk) 18:19, 10 September 2010 (UTC)


 * As far as I can make out the position is this:
 * Two, but not all, major pyramids have an angle of incline of about 52° and hence their height times pi is approximately equal to their perimeter
 * There is no evidence that this angle was chosen for that reason, and in fact there is no evidence as to the reason for choosing any angle, 52° or otherwise
 * There is no evidence for numerical computation of pi in Egypt before the period of the Rhind Papyrus neqrly 1000 years after the pyramid builders
 * Some Egyptologists, such as Petrie, believe that the choice was deliberate and linked to the value of pi
 * Some Egyptologists do not believe that the pyramid builders deliberately reflected the value of pi
 * Some Egyptologists believe that the pyramid builders had no accurate knowledge of the value of pi
 * So the question is how to reflect this in the article. I would propose wording along the lines of
 * It has been observed that two of the major Old Kingdom pyramids in Egypt, the Great Pyramid at Giza, built c.2550-2500 B.C and the Pyramid of Meidum c.2600 B.C. are built with an angle of incline of about 52° and hence the ratio of their base perimeter to height is approximately pi, while others have different angles.  Some Egyptologists, such as Petrie, believe that the choice of 52°  was deliberate and reflected a deliberate allusion to the value of pi; others disagree, or maintain that the pyramid builders had no exact knowledge of the value of pi.
 * Comments? Kenilworth Terrace (talk) 18:57, 10 September 2010 (UTC)


 * I agree with the comments above. The early history of the approximations of π is really somewhat murky. Experts do not seem to agree and there are conflicting opinions. I think the intro could be written to explain some of this. Below is a short paragraph. I have included the sources used for reference. --AnnekeBart (talk) 01:19, 11 September 2010 (UTC)

The early history of pi in Ancient Egypt is not entirely clear. One theory about the Great Pyramid states that the pyramid was built so that the circle whose radius is equal to the height of the pyramid has a circumference equal to the perimeter of the base. Others have argued that the Ancient Egyptians had no concept of pi and would not have thought to encode it in their monuments. The creation of the pyramid may instead be based on simple ratios of the sides of right angled triangles (the seked).

The first evidenced computation of the area of a circle comes from the Rhind Mathematical Papyrus. In problem 48 the area of a circle was computed by approximating the circle by an octagon. The value of pi is never mentioned or computed however. If the Egyptians knew of pi, then the corresponding approximation was 256/81.


 * The Egyptians probably used a measuring wheel. They decided that the base would be measured in turns of the wheel (circumference) while the height would be measured in widths of the wheel (diameter).  They would use this ratio because it approximated the angle of repose required so that the pyramids would not fall over.  In one pyramid, they overestimated this angle and had to correct it during construction.  There would be no reason for them to understand pi.  TFD (talk) 19:06, 12 September 2010 (UTC)


 * Unfortunately I can't remember the name of it, but I do recall seeing a documentary once that mentioned this relationship: if you use a wheel to lay out the base in terms of an even number of revolutions of the wheel, and make the height a multiple of the wheel's diameter, you automatically incorporate a factor of pi in the dimensions. Whether they had a "reason" to know what pi was or not, they didn't need to know it in order for the dimensions to work out, meaning the dimensions of the pyramids do not prove that the ancient Egyptians knew the precise value of pi, or even a reasonably accurate approximation thereof.Ptorquemada (talk) 17:24, 14 March 2011 (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 &pi; 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)


 * Wholeheartedly support removal. The fact that one can get ten digits of pi by a process involving an arbitrary number with ten significant figures is hardly noteworthy, even if the arbitrary number in question contains only the digits 1 and 2. At best it's coincidental piece of trivia that doesn't really illuminate any other areas of mathematics.Ptorquemada (talk) 17:42, 14 March 2011 (UTC)

Requested move

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section. 

The result of the move request was: article moved. Dabomb87 (talk) 22:59, 1 April 2011 (UTC)

Numerical approximations of π → Approximations of π — 'Numerical' seems superfluous. Ben (talk) 11:30, 16 March 2011 (UTC)


 * Support: I can't say I'd be too worried if it contained a few examples like



8\int_{0}^{\infty}\cos(2x)\prod_{n=1}^{\infty}\cos\left(\frac{x}{n}\right)dx $$


 * as well (and no it might look like $π$ but it's just a tiny tiny bit different. :) Dmcq (talk) 15:27, 16 March 2011 (UTC)


 * Question to Ben Tillman: Can you elaborate on your statement that "numerical" seems superfluous? There are all sorts of analytic expressions that converge to &pi;, so they are "approximations to &pi;" but not "numerical" ones. Michael Hardy (talk) 15:51, 16 March 2011 (UTC)
 * Alternative proposal. This is not something I feel strongly about, but it's an idea: Make approximations of π a disambiguation page that could list both numerical approximations of π and analytic approximations of π.  Of course the latter article would need to be created. Michael Hardy (talk) 15:54, 16 March 2011 (UTC)
 * Support move proposal. No objection to analytic expressions in the article, though strictly I wouldn't consider them approximations. CRGreathouse (t | c) 16:04, 16 March 2011 (UTC)
 * ?? This has been done, right? Or is it still being proposed that the word "pi" in the title be replaced by the symbol $π$? --Kotniski (talk) 07:13, 24 March 2011 (UTC)
 * Oppose it should be called Approximations of pi 65.93.12.101 (talk) 09:30, 24 March 2011 (UTC)
 * Because ...? —Tamfang (talk) 20:09, 27 March 2011 (UTC)
 * Commonly written as "pi" on electronic works, does not require a special character not appearing on English keyboards, "pi" is a very common form found in many places where you don't have special typography. 65.93.12.101 (talk) 06:13, 28 March 2011 (UTC)
 * Those are all excellent reasons for a redirect from "pi". —Tamfang (talk) 06:38, 28 March 2011 (UTC)
 * Those are excellent reasons to keep it at "pi" 65.93.12.101 (talk) 08:21, 28 March 2011 (UTC)
 * They might have been good reasons to exclude such characters as ‹π› from article titles before Unicode fonts were commonly available, but not now. —Tamfang (talk) 16:50, 28 March 2011 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

"$\pi$" vs. "pi"

 * Very similar discussions are ongoing at Talk:Pi and Talk:A History of Pi. — Loadmaster (talk) 20:31, 20 April 2011 (UTC)

Move discussion in progress
There is a move discussion in progress on Talk:Proof that π is irrational which affects this page. Please participate on that page and not in this talk page section. Thank you. —RM bot 17:46, 24 April 2011 (UTC)

Law of Cosine's formulae
This edit (2010-12-15) added a section named "Law of cosine". A Google search does not turn up any mathematician by that name. Is this OR or spam? — Loadmaster (talk) 03:14, 16 December 2010 (UTC)


 * Hey Loadmaster,
 * I am the anonymous person who added it (sorry, I didn't have created an account yet). It is normal if my name doesn't appear on Google as a mathematician or anything because I am still a simple student. A week ago, during a (pretty boring) physics course, I found myself drawing regular polygons, and I ended up trying to find a formula to estimate Pi's value (when n→∞ (n = number of edges)). And since after doing multiple researches on internet for this formula, I did not find anything about it, I thought I'd share it on Wikipedia. As for its name, sorry for the lack of originality :D
 * I have read the article about OR, and did not know yet about that. Sorry if this should not have been added. (Feel free to remove it if it should).
 * However, I'd be glad to hear your opinion(s) about it.
 * Personally, although it is theorically correct, I think it's not a good way to determinate Pi's small decimals because of the calculation of an extremelly small cosine (all the calculators I've tried round cos(θ) to 1 when θ<10-3). But I then thought about using this formula for another purpose : calculating cosines (and sines) of very small angles, since we do know Pi's decimals.
 * => $$\cos(a) = 1-\frac{(\pi*a)^2}{64800} $$
 * (So far it's pretty useless to me, but it's cool eheh, and it allowed me to find out that my Casio pocket calculator's algorithm is worse than Matlab.)
 * Any thoughts ? Thanks! ~ Leptyx (talk) 17:04, 16 December 2010 (UTC)


 * It's original research, so it doesn't belong in the article, unless you can find a citation or book mentioning it. Anyway, it doesn't work for a = $π/3$ = 60°; I get 0.999832$+$ (for radians) or 0.451688$+$ (for degrees) instead of 0.500000. I left it in the article, but renamed the subsection. The formula really needs a citation, or an explanation of how it is derived. — Loadmaster (talk) 01:06, 20 December 2010 (UTC)


 * Alright, I will then add an explanation of how it is derived.
 * As for your example with $π/3$, it is normal because this formula would only apply when a is very small. With a = 1°, you get an approximation of sin(a) which is at 99.99873% close of the result you get by using the sine function on a calculator. (And the smaller a is, the more precise you get) ~ 22:25, 20 December 2010 (UTC)
 * I'm sorry, but we just don't take any original research. We are not a publisher of original thought.  You need to find a mathematical journal or something and get your discovery published there, then you can cite that. Ian.thomson (talk) 22:32, 20 December 2010 (UTC)
 * Turns out it was wrong anyways. (Btw seems like "Law of Cosine" was more clear - contact me about this update if you wish)  — Preceding unsigned comment added by Leptmania (talk • contribs) 16:08, 14 September 2011 (UTC)

Confusing Sentence in Modern Algorithms
While reading the Modern Algorithms section I came across a confusing sentence.

"A former calculation record (December 2002) by Yasumasa Kanada of Tokyo University stood at 1.2 billion digits, which were computed in September 2002 on a 64-node Hitachi supercomputer with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits)."

I read this as saying that a record of 206 billion digits was acheived prior to a record of 1.2 billion digits, which would be a decrease in digits. Also, the context of the sentence suggests that there are two different dates for one record, with no date listed for the second record.

I am not editing the article because this is my first post on Wikipedia and I am taking it slow. I just wanted to bring this to the attention of others for now. Wanzeo (talk) 05:19, 24 January 2012 (UTC)
 * Good catch. I clicked on the embedded Yasumasa Kanada link and found that it was 1.24 trillion digits, so I have made the appropriate fix here. Thanks! — Glenn L (talk) 06:08, 24 January 2012 (UTC)
 * The confusion was probably caused by the two different meanings of billion: Long and short scales. Thanks for spotting the error.    D b f i r s   07:16, 24 January 2012 (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. &mdash; Loadmaster 15:58, 4 October 2006 (UTC)
 * I'll bite - it's a good write-up. - DavidWBrooks 23:14, 4 October 2006 (UTC)

Pi as just 3 is an example of Algebra or substitution. — Preceding unsigned comment added by 142.167.234.205 (talk) 06:06, 10 January 2013 (UTC)


 * What could this possibly mean, I wonder? Austinmohr (talk) 05:43, 14 January 2013 (UTC)

Best rational approximations
The last item under Miscellaneous Approximations claims to show the first few best rational approximations using the convergents of the continued fraction representation of π. The last fraction, 2383784714/758782241, is definitely wrong. It is not a best rational approximation because it has a larger error than 1068966896/340262731, for example. That last fraction also does not appear in the sequence of numerators and denominators of the convergents of π. Also, 311/99 is a best rational approximation of π, but it is not a convergent, as the text implies. I suggest providing a complete list of the first ten or twelve convergents, and noting that though all convergents are best rational approximations, the converse is not true: there exist numbers like 311/99 that are not convergents (generated by truncating the continued fraction), but are still best rational approximations (technically, they are "semi-convergents"). Sorry for not editing the article directly; this is my first contribution of any kind to Wikipedia, and it's a nontrivial fix. Edsbend64 (talk) 22:44, 11 July 2012 (UTC)


 * Thanks for spotting the error. I think the correct fraction should be 2549491779/811528438.  I'm not expert on this, so if no-one else contributes here, we'll post a request for correction on the Mathematics Reference Desk where there are lots of experts.    D b f i r s   09:36, 14 January 2013 (UTC)

"See external links below"
This sentence appeared twice:

"The first one million digits of pi and 1⁄π are available from Project Gutenberg (see external links below)".

Once in here and once in here.

However, I cannot find the "external links" below, and I cannot find the first one million digits of pi or 1/π.

--Kc kennylau (talk) 08:37, 18 February 2013 (UTC)

Simon Plouffe’s 1996 formula for digit extraction using math in any base system
Simon Plouffe’s 1996 paper, “On the computation of the  n th decimal digit of various transcendental numbers”, is currently merely mentioned in this article’s section, “Digital extraction methods”. I think it deserves an expanded treatment with the formula written out in math notation. The original Bailey–Borwein–Plouffe formula works only in hexadecimal math and returns digits only in hexadecimal form. Also, the Fabrice Bellard formula works only in binary. What appears to be distinctive and notable about Simon Plouffe’s formula is it can calculate any digit using base 10 math (as well as in any other base).

Here is the URL for a direct-download link to Simon Plouffe’s Word-doc: www.plouffe.fr/simon/articles/articlepi.docx

I hope someone with better math skills than I will step up to the plate, read the paper, and make sure the proper form of his formula is accurately transcribed into our article. Greg L (talk) 13:20, 16 March 2013 (UTC)
 * I had to wait for a weekend to roll around before I had the brain-space, but I added it myself. I think I did it right. Greg L (talk) 13:50, 23 March 2013 (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)

Every Hebrew letter has a numerical equivalent. The only way at the time the bible was written to represent a number was either to spell it out in full, or use a letter. For example, If I had 400 sheep, I could write "four hundred" or put the numerical equivalen "ת". To represent a decimal, the only choice at that time was to use a fraction. As explained, taking the way the word is pronounced (which is not the way it is spelled) and dividing it by the way it is actually spelled, give a value of pi that is very accurate. You can put it in or not, that was not my point. My real point was why mention the bible at all if you weren't goign to mrention the fact that Jewish tradition actually had pi right to the hundreths of an inch (which could not be measured a the time)? Basically, if you are going to mention, be accurate - otherwise keep it out. One does not need a source for 1+1=2. However, here is the first google I attempted http://sumseq.blogspot.co.il/2010/03/biblical-accuracy-of-pi-hidden-in.html. FYI. — Preceding unsigned comment added by Weedmic (talk • contribs) 09:37, 9 June 2013 (UTC)

One of Ramanujan's Approximations
The article mentions that Ramanujan provided this approximation to π:
 * $$ \sqrt[4]{3^4+2^4+\frac{1}{2+(\frac{2}{3})^2}} =\sqrt[4]{\frac{2143}{22}} = 3.14159\ 2652^+$$

This seems to be commonly taught to school-children in the UK as $$\sqrt{\sqrt{\frac{1066}{11}+0.5}}$$.

Note that the use of the double square root, rather than a fourth root, makes the formula readily accessible on a simple calculator, while the fraction is extremely easy for an Englishman to remember. This presentation is probably notable as the distinctively English version of this result. Unfortunately, although almost everyone I know knows this version, there doesn't seem to be any good reference to it anywhere. Can anyone help? RomanSpa (talk) 03:29, 11 September 2014 (UTC)

Letters in generated PDF are overlapping.
If you generate PDF, 'The Gregory–Leibniz series', 'The calculation speed of Plouffe’s formula', are going outside of the page, and few other formulas are overlapping on the 2nd column ('Arcsine', 'Fabrice Bellard' formulas). — Preceding unsigned comment added by Kenorb (talk • contribs) 11:43, 3 February 2015 (UTC)

Clean up...........
We should clean up this article. Many things are redundant and misplaced. — Preceding unsigned comment added by DYKnapp (talk • contribs) 12:14, 14 July 2015 (UTC)

Polygon approximation to a circle.
This is already in the article, so there is no need to repeat it as a numerical approximation. (Please don't keep adding it, Dubai editor 86.98.41.235.)   D b f i r s   07:53, 1 September 2015 (UTC)
 * ... later ... Marwin2015, presumably the same editor from Dubai, insists on re-adding the redundant approximation that isn't a genuine numerical approximation. I don't want to edit-war.  Is anyone else watching this article?    D b f i r s   14:52, 1 September 2015 (UTC)
 * ... later ... Is there any merit in explaining Marwin's method for calculating the polygonal approximation? It's a method rather than a formula.    D b f i r s   17:43, 2 September 2015 (UTC)
 * Doesn't look like a helpful addition to me. —David Eppstein (talk) 18:01, 2 September 2015 (UTC)
 * I didn't see any way to make this a useful addition to the article, which is why I deleted rather than corrected. Bill Cherowitzo (talk) 18:33, 2 September 2015 (UTC)
 * The paragraph on approximation by a regular polygon is very short and gives no detail of how Archimedes did his calculation. I think Marwin was trying to explain one of several possible methods.    D b f i r s   20:29, 2 September 2015 (UTC)
 * I see your point and will add something about Archimedes computation and perhaps some of its refinements. This will also permit me to get rid of the blog citation currently in that section. Bill Cherowitzo (talk) 22:30, 2 September 2015 (UTC)
 * Thanks. I can't remember which method Archimedes used.    D b f i r s   22:40, 2 September 2015 (UTC)
 * As I understand it, Archimedes looked at the areas of regular polygons inscribing and circumscribing a circle, with a number of sides equal to 3 times a power of two. If you do the same thing using a power of two without the factor of three, you get Viète's formula. —David Eppstein (talk) 22:47, 2 September 2015 (UTC)
 * Almost ... he used perimeters rather than areas. The spurious 3 (it's actually a 6) comes from the fact that he started with hexagons, and then kept doubling the number of sides. He does this all without any trigonometry and of course, no decimal notation. His approximations for the square roots that pop up in the calculations were all correctly biased (smaller when they had to be smaller and larger when they had to be larger) ... its absolutely amazing that he pulled this off. I'll start writing this up tonight. Bill Cherowitzo (talk) 23:27, 2 September 2015 (UTC)

Pi-Hex Project
The Pi-Hex project seems redundant to me in light of Simon Plouffe's 1996 formula. — Preceding unsigned comment added by Marc.gw.opie (talk • contribs) 01:52, 15 March 2016 (UTC)

This project seems redundant in light of the Simon Plouffe formula. — Preceding unsigned comment added by Marc.gw.opie (talk • contribs) 01:56, 15 March 2016 (UTC)

External links modified
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External links modified
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Futile hat note
I added the following sentence to the end of the lead, because I didn't see the same in the hat note (until afterwards) and I suspect that many readers would miss same.

(For a comprehensive account, see chronology of computation of π.)

99% of the time hat notes direct the reader to Not Quite the Same Thing rather than Entirely the Same Thing (treated differently).

My first choice would be to remove the duplication by trimming the hat note. My second choice is to leave the duplication. My third choice is returning the article to its previous state. &mdash; MaxEnt 00:43, 12 May 2017 (UTC)

Formulation and correct rendition of Liu Xin's value
My edit to the paragraph on Liu Xin's value has been [|reverted] by User:Deacon Vorbis with the comment "that's not the value that Needham gives; this is also confusingly written". That seems to me unfortunate, because that edit The value 3.1457 was copied from an old version of Liu_Xin, but I now see that it must have been a typo there, as the Talk:Liu Xin gives the reasoning behind it and the value as 3.1547, while Needham gives 3.154 (see also https://i.stack.imgur.com/jgvqb.png, as shewn in https://math.stackexchange.com/questions/2296067/how-must-have-liu-xin-calculated-pi). The formula implied is $$ \pi = \frac A { r ^2} = \frac {162} {(\sqrt 50 + 0.095) ^ 2 } \approxeq 3.154664564 $$ (though diameter and radius appear to be confused as are inches and square inches). I have already corrected that page.
 * also corrected the historical sequence,
 * also clarified that it was more accurate than previous Chinese values, to avoid giving the impression that it was more accurate than Archimedes',
 * also corrected punctuation ('.' for ','),
 * also provided the decimal values for Archimedes, to aid comparison,
 * was intended to make it easier to read! After Between the years 1 and 5 CE the Chinese astronomer Liu Xin used a now unknown method to give a geometrical figure for a Jialiang,  I changed
 * which implies a more accurate value of π at c. 3.1457.
 * to
 * which implies a value for π of c. 3.1457, more accurate than previous Chinese values.
 * .. a value for π of 3.1547 seems better English to me than a .. value of π at c. 3.1547.

I am inclined to restore my version with the following changes: Any comments? PJTraill (talk) 15:46, 6 August 2018 (UTC)
 * correct 3.1457 to 3.1547;
 * insert which is before more accurate than previous Chinese values (or make a separate sentence out of it).


 * Yes, but you'll need to give me a bit. (In the mean time, I reverted fully, which is what I had meant to do...not sure what happened) –Deacon Vorbis (carbon &bull; videos) 15:53, 6 August 2018 (UTC)


 * No problem, I do not intend to do anything more here today, nor am I in a hurry to do so tomorrow. PJTraill (talk) 17:05, 6 August 2018 (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
 * $$\frac{\log((640320^3+744)^2-2\cdot196884)}{2 \sqrt{163}}$$

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)

Some years back, too long ago for me to remember the details, someone associated with writing this article assured me that a fairly small pi representation - something like seven decimal places - was, despite its brevity, sufficient to calculate the measurements of the known universe, or the solar system, or some such enormous region. An explicit mention of this was, at that time, in the Wiki article. But now I cannot find that mention in the Wiki article and would appreciate a repetition of that helpful lesson. Sussmanbern (talk) 02:51, 18 September 2018 (UTC)

Archimedes digits
The best known approximations to π dating to before the Common Era were accurate to two decimal places; this was improved upon in Chinese mathematics in particular by the mid-first millennium, to an accuracy of seven decimal places.

Taking the rational mean of Archimedes upper and lower bounds (which is a sensible thing to do given his method of construction) gives 3123/994 = 3.141851 (averaging 1561/497 and 1562/497).

I see no good reason not to call this three decimal places, back in 300 BCE, just because Archimedes adduced rigorous bounds rather than merely futzing around with a best approximated value (implying a practical median value which I regard as entirely implicit in his result). &mdash; MaxEnt 07:49, 24 January 2019 (UTC)

Percent
To the IP changing the percent values: percents are not absolute numbers. "1.5% below $&pi;$" means $&pi; &minus; 0.015·&pi;$, not $&pi; &minus; 0.015$. –Deacon Vorbis (carbon &bull; videos) 15:43, 26 August 2019 (UTC)


 * "In mathematics, a percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, '%.'" Excerpt from en.wikipedia.org/wiki/Percentage. 1.5/100 is equivalent to 15/1000, or 1.5 percent, and 1/8 is equivalent to 125/1000. 125+15=140. 140/1000 is equivalent to 14%, or 0.14. Thus, 25/8 is 1.5% below π. Tali64^2 (talk) 14:24, 28 August 2019 (UTC)


 * Please read the above comment by Deacon Vorbis. You are not using percentages correctly. --Bill Cherowitzo (talk) 18:42, 28 August 2019 (UTC)


 * I checked, and $25/8$ is actually about 1.66% below π.--Tali64^2, always top quality. (talk) 12:52, 29 August 2019 (UTC)


 * No, it's 0.01659 ... below pi. This is an absolute value, not a percentage.  You are still misunderstanding the meaning of percentage.  You need to divide 0.01659 by pi, then turn the resulting decimal into a percentage, giving just over half of one percent.   Dbfirs  15:56, 29 August 2019 (UTC)
 * To turn a decimal into a percentage, multiply the decimal by 100 and affix the percent sign, %. I did this and got 1.659%, about 1.66%.--Tali64^2, always top quality. (talk) 16:12, 29 August 2019 (UTC)
 * Multiple people are telling you that you're mistaken. Please read and try to digest what they're saying instead of just repeating the same essential statement.  If you still don't understand, you can ask at the Math ref desk.  –Deacon Vorbis (carbon &bull; videos) 16:34, 29 August 2019 (UTC)
 * , your method for turning a decimal into a percentage is correct, but the article is using percentage of which is something completely different. For example 10% of pi is not the same as 0.1, it's actually 0.31415926535897932384... (that's as much as I remember) .   D</i><i style="color: #0cf;">b</i><i style="color: #4fc;">f</i><i style="color: #6f6;">i</i><i style="color: #4e4;">r</i><i style="color: #4a4">s</i>  19:31, 29 August 2019 (UTC)

I was looking how to find the percent error, and I found $Approximate value - Exact Value⁄Exact value$. Using this method, I found out that $25⁄8$ is actually about 0.528 percent below π.Tali64^2, always top quality. (talk) 21:47, 11 September 2019 (UTC)

Link to Stu's Pi page
The link to Stu's Pi page in the section "Software for calculating $$\pi$$" is dead. Should it be removed? --PoCc001 (talk) 11:02, 15 September 2019 (UTC)

More accurate percent values
Using the method to find percentage errors discussed in Percent, 0.5 is 1.95 percent below $25⁄8 - \pi⁄π$, while 0.53 is only 0.3 percent above it. This means that 0.53% is more accurate than 0.5% for $25⁄8$'s percent error fof π. Tali64^2, always top quality. (talk) 18:55, 21 September 2019 (UTC)


 * And 0.528% would be more accurate still, etc etc. One digit is plenty here to give a rough indication of the accuracy of the estimate; any more than that is overkill.  –Deacon Vorbis (carbon &bull; videos) 19:09, 21 September 2019 (UTC)


 * Pleased to see that you now understand percentage error, but I agree than one significant figure is sufficient here.  <i style="color: blue;">D</i><i style="color: #0cf;">b</i><i style="color: #4fc;">f</i><i style="color: #6f6;">i</i><i style="color: #4e4;">r</i><i style="color: #4a4">s</i>  07:23, 22 September 2019 (UTC)

No, I'm not saying you have to go more than two decimal digits, it's just 0.5 is about 1.95 percent below the percentage error for $25⁄8$ and π. Please accept this two-digit proposal.Tali64^2, always top quality. (talk) 22:07, 6 October 2019 (UTC)
 * No reason to. It won't help the reader.--Jasper Deng (talk) 22:55, 6 October 2019 (UTC)
 * One significant figure is sufficient here for comparison.  <i style="color: blue;">D</i><i style="color: #0cf;">b</i><i style="color: #4fc;">f</i><i style="color: #6f6;">i</i><i style="color: #4e4;">r</i><i style="color: #4a4">s</i>  06:14, 7 October 2019 (UTC)

Missing Gutenberg links
Article reads "... Project Gutenberg (see links below)" -the links to pi calculated to a million digits on Project Gutenberg have disappeared some time ago; they should be replaced and footnoted so they don't disappear again.

I added footnote references to keep them. Xenonoxid (talk) 18:38, 28 March 2022 (UTC)

Power roots
I recently discovered that $$\sqrt[3]{31} \approx 3.14138$$ is another approximation of π, better than 3.14 or 22/7 but not as good as 355/113. Looking further, I have discovered that $$\sqrt[5]{306}$$, $$\sqrt[8]{9489}$$ and $$\sqrt[9]{29809}$$ are progressively closer approximations.

A quick search of OEIS reveals and. But A080022 is just the closest integer to $$\pi^n$$, hence the integer whose $$n$$th root is closest to π, without consideration for whether taking the root gets you closer to π than any previous entry in the sequence. And A002160 isn't quite the same: not only does it count some powers twice ($$\sqrt[8]{9488}$$ is closer to π than any previous, but $$\sqrt[8]{9489}$$ is closer still), but it is based on proximity of the logarithm to an integer, which isn't the same as proximity of the power root to π even if it seems to be giving the same results!

What I'm wondering is: Have power root approximations of π been studied enough to warrant inclusion here? — Smjg (talk) 16:07, 6 June 2020 (UTC)


 * Just realised "the closest integer to $$\pi^n$$, hence the integer whose $$n$$th root is closest to π" is a flawed argument. But this doesn't affect my question. — Smjg (talk) 00:30, 8 June 2020 (UTC)