Talk:Proof that 22/7 exceeds π

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This article was previously considered for deletion. The consensus was keep. An archived record of the discussion can be found here.

Required calculus

There are plenty of pages that involve math on Wikipedia. Why is this one getting so much attention? There is no rule that all encyclopedia articles must be accessible to all people.

"The following argument will be readily understood by persons with no knowledge of mathematics beyond first-year calculus. "

Well, what about those of us who don't even have that? It's all just funny looking symbols to me :) Adam Bishop 03:25, 29 Nov 2003 (UTC)

Well, "simple" is a subjective term. I don't think its possible to give a simpler proof than this, because even defining π involves calculus (the length of a curve is the limit of a sum, an integral). -- Arvindn 03:52, 29 Nov 2003 (UTC)
The only part I had trouble with was that... I'm not sure what its called... curly thing with the 1 and 0 Nat2 23:04, 2 February 2006 (UTC)
So, in other words, you didn't understand the proof at all!

I suspect that any reasonable proof of the same proposition that avoids knowledge of calculus would be more complicated. As for calculus being involved in defining π, if someone who knows no calculus asks me what π is, I would not hesitate to answer that it is the ratio of a circle's circumference to its diameter; multiply the diameter by π to get the circumference. Michael Hardy 21:26, 29 Nov 2003 (UTC)

Do you know a (formal) definition of circumference that doesn't involve integration? If not, what I said still holds, doesn't it? -- Arvindn 03:27, 30 Nov 2003 (UTC)
It can be formally defined in terms of the limit of circumferences of polyhedra. This does need the notion of a limit, but not the fully machinery of integral calculus. However, since the formalization of the limit was done precisely to facilitate the development of calculus, the distinction is perhaps not a strong one in the history of formalism. However, in the history of Pi the earliest estimates of Pi make informal use of limits, but not of integration. (BTW, thanks for the name change from A very elementary...) ~ Jeff 18:33, 1 Dec 2003 (UTC)

Many characterizations of π do not mention integration. Which among them should be considered definitions is perhaps a subtler question. The notion of limit is not really needed, since one can say simply that it's the least upper bound of the set of all perimeters of inscribed polygons. Michael Hardy 21:40, 1 Dec 2003 (UTC)

If we use the continued fraction as definition, the proof is even simplier!!!wshun 21:49, 1 Dec 2003 (UTC)

How can the continued fraction be used as a definition? Somehow, you would have to say which continued fraction you're talking about without relying on some prior characterization of π. Michael Hardy 21:55, 1 Dec 2003 (UTC)

To elaborate a bit further on the point above: It is not as simple to understand how it is known that π = 3.1415926535... or that π = a certain continued fraction, as it is to learn calculus and then read this Wikipedia article. Michael Hardy 01:12, 2 Dec 2003 (UTC)

Arctan integral

So, where does one go to brush up on the fact that integrating 1/(1 + x2) yields arctan? I had added a link to a page that included the formula, but it was removed. ~ Jeff 03:23, 4 Dec 2003 (UTC)

I suspect there may be a page that covers it, but I will add it to trigonometric substitution. Some of the calculus pages on Wikipedia have lots of problems. Michael Hardy 20:52, 4 Dec 2003 (UTC)
Thanks. Jake

Which polygon is the crossover?

So how many edges would a regular polygon (centre O, vertex V, edge bisector P) need before the ratio of the perimeter to OP was less than 22/7 ? That proof would not require caluclus, nor even limits, but only trigonometry. mike40033 03:01, 2 Apr 2004 (UTC)

oops. I meant, the ratio of the perimeter to 2*OP should be less than 22/7. Anyway, the answer is n=91. Using n = 96 and only trigonometry should yield a proof (with no calculus) that pi is less than 22/7. But I don't think the proof would count as "simple"

The title is misleading

The point of this whole page is the funny integral. That is no "simple" proof for the fact that 22/7 exceeds pi, but rather a mathematical joke; the page is relevant and informative, but the caption is misleading at best; this is a page about a putnam competition problem and some subsequent analysis on a particular definite integral. Funny, but miscaptioned. This is not THE proof for that thing; THE proof is indeed naively using the calculator so readily available to anyone who has gotten this far... --sigs

The article is of interest quite independently of the Putnam Competition. Of course, the brute-force calculation yields no understanding of this simple argument, and the brute-force calculation is far more complicated than this argument. Michael Hardy 02:50, 20 February 2007 (UTC)


the title of the page says π , not π , somebody should fix that!

I've change the name to "A simple proof that 22/7 exceeds Pi" hopefully I changed all the links correctly ;-) Paul August 21:14, Jul 23, 2004 (UTC)
And now I've moved it to "a simple proof that 22/7 exceeds pi" (with a lower-case "p" in "pi") as a compromise. Until a couple of days ago, the π in the title looked like the lower-case Greek letter. I don't know why that changed. I'll look into it. But probably not today. Michael Hardy 23:53, 23 Jul 2004 (UTC)
I disagree with the title - its meant to be ironic. It should be obvious that ironic titles are not NPOV.-SV
I don't think the title is meant to be ironic. Why do you say this? Paul August 11:18, Jul 25, 2004 (UTC)
:) -SV
I suspect that he is thinking of the word "simple", because the proof isn't simple if you don't know any mathematics (see above). Pcb21| Pete 12:18, 26 Jul 2004 (UTC)
I suck reely bad at numbers and stuff, so I think your page should be named more simpler, no offense, but it makes the reader feel stupider than I think I is. -SV

I am the person who named the article originally. I did not intend irony! It's just a prosaic descriptive title. Michael Hardy 01:52, 1 Aug 2004 (UTC)

It is a tricky one because the page plainly isn't simple if you are not mathematican (so don't feel bad Steve :)), however the reason for the pages existence is that it proves something about pi without relying on "heavy machinery" (at this point you kinda have to take it on trust that despite appearances all the machinery is fairly lightweight. And without some suitable adjective (we had elementary before, but that evokes the same probs as simple) in the title this raison d'etre gets lost. I vote that we merge into pi then the problem goes away. Pcb21| Pete 17:34, 26 Jul 2004 (UTC)

The proof uses only high school maths, so I think it deserves to be called "simple". Gdr 17:46, 2004 Jul 26 (UTC)
Yeah, and others disagree. How are we going to compromise? Pcb21| Pete 17:48, 26 Jul 2004 (UTC)
By teaching these others some simple calculus? Or perhaps by expanding the Pi#Numerical approximations to pi section to a new article with discussions of the techniques used to establish approximations to π such as Archimedes' inscribed/circumscribed polygon method. This page could then be merged and redirected. Gdr 19:35, 2004 Jul 26 (UTC)
Something like that could be ok, as long as this proof is kept too - it is kinda neat. Pcb21| Pete 22:39, 26 Jul 2004 (UTC)

Simple is a comparative term. As compared to all other mathematical proofs this one is extremely simple. Even in the context of proofs in a high school calculus class it is quite simple. It is simple because:

  • It is short
  • It is straight forward (no twists and turns)
  • It uses only basic mathematical tools (i. e. high school level advanced high school level (see below Paul August 23:49, Sep 12, 2004 (UTC)))

However because it is a "simple proof", does not mean it is necessarily "simple to understand". It is simple to understand if you know calculus. If not then not only is it not simple (to understand) it is impossible. In which case, calling it a "hard proof" (to understand) would also be misrepresentative. Paul August 19:59, Jul 26, 2004 (UTC)

I don't think anyone has suggested that calling it a hard proof would help. I repeat, I understand why mathematicians call this proof simple/elementary. Those words (simple/elemnentary) convey a special meaning to all mathematicians from first year college level up. However outsiders do not have that mathematical experience or "culture" if you like. This is not mathepedia. Calling it a simple proof can all too easily, in the eyes of the general reader, create an impression of arrogrance and aloofness. An impression we are not doing too well in dispelling on this talk page! Pcb21| Pete 22:39, 26 Jul 2004 (UTC)
Yes, you are right Pete, no one has suggested calling it a "hard proof", that wasn't my point. My point was, that the word simple, in this case, is being used to describe an objective property about this proof, namely it's brevity, directness and (as you put it very well above) it's lack of "heavy machinery" - rather than a subjective property concerning it's ease of understanding. I agree that this distinction might not be clear to every reader, and therefor the title might be misunderstood. This might point out the need for a better title, but I can't think of one. One could try to be more explicit, by " A short, direct proof that 22/7 exceeds pi, using only high school calculus" but this strikes me as inelegant. Paul August 16:58, Jul 27, 2004 (UTC)
(An aside, you say this proof is about a simple a proof as you can get - I would say that a geometric proof of Pythagaras would be simpler still. In its way Euclid's proof of infinity of primes is simpler too). Just some examples. Pcb21| Pete 22:39, 26 Jul 2004 (UTC)
I agree with you Pete, there are many proofs even simpler than this one ;-) Paul August 16:58, Jul 27, 2004 (UTC)

There are many proofs that are simpler than this one, but are the any proofs of this proposition that are simpler than this one? Michael Hardy 01:56, 1 Aug 2004 (UTC)

That's an irrelevant question, since "simplest" does not imply "simple" (think about asking the same question about the proof of the 4 color theorem). -- Jibal 09:58, 15 March 2007 (UTC)

High School?

I would be very curious to know where most of you went to high school. In the U.S., even simple calculus is not commonly thought of as a high school level math. It is taught in high schools, but the majority of students don't take it, ie: it is usually offered as an advanced elective course, and often under a name like pre-calculus or as a part of elementary functions, etc. To the average U.S. citizen, calculus is a college-level math. Indeed, the very small percentage of, students who take calculus in high school are commonly said to be doing college level work. func(talk) 01:41, 12 Sep 2004 (UTC)

That is true, but for the most part, people reading this article probably did take advanced courses in math, or are naturally curious (in that case, more power to 'em). Just wondering, does anyone know when this integral was a Putnam problem? Jonpin 01:03, Oct 2, 2004 (UTC)
I think it was at least 25 years ago. Michael Hardy 22:02, 2 Oct 2004 (UTC)
According to John Scholes' archive [1], it was 1968. Mindspillage 02:38, 24 Dec 2004 (UTC)

There is a tradition by which differential and integral calculus is the first college math course. But in recent decades, it has frequently been available to high-school students. Most members of that "vast majority" who don't take calculus in high school never go into a field where they need mathematics anyway, so they're not really relevant to this discussion. Perhaps most college students who take calculus don't take it as their first college math course, but most of those are also not people who go into fields where they think about math daily, so they're not really relevant to this discussion either. Michael Hardy 21:18, 12 Sep 2004 (UTC)

Michael said: "Most members of that "vast majority" who don't take calculus in high school never go into a field where they need mathematics anyway"

That may be true at MIT or Berkeley or other elite schools, but for the rest of us, it's not uncommon at all to see many math and science majors taking calculus for the first time in college, so your own experiences may be clouding your conclusion, here. Revolver 02:59, 9 September 2005 (UTC)

Ok, not "high school level" but "advanced high school level", I stand corrected ;-) Paul August 23:50, Sep 12, 2004 (UTC)

0 < integral of [x^4(1-x)^4/(1+x^2)] from 0 to 1 = 22/7-pi

/ \


That's not very simple!

It's a routine calculus problem. One expects secondary-school students to do such problems in a couple of minutes. Sheesh. Michael Hardy 22:46, 14 Mar 2005 (UTC)
No, one expects calculus students to do such a problem in a couple minutes, but certainly not your average secondary-school student. I think your time at MIT has warped your sense of reality. Revolver 03:03, 9 September 2005 (UTC)
I don't think so. I never said one expects this of all high-school students. But at typical high schools, there are some students who take calculus, and are expected to evaluate integrals. Some of those integrals are challenging, but for this one you just plod through the steps mechanically and you've got it. Michael Hardy 03:40, 2 April 2006 (UTC)


Although many people know this numerical value of π from school, far fewer know how to compute it.

I have no idea what this sentence is supposed to mean, or how it is supposed to relate to the rest of the introduction. Revolver 03:10, 9 September 2005 (UTC)

I don't understand what's puzzling about it. Many people know that π is about 3.14159... etc. But few know where that came from, i.e. if you're stranded on a desert island and need to know the first ten digits of π and have only paper and pencils, how would you compute them. How do you find something hard to understand in that? Michael Hardy 03:41, 2 April 2006 (UTC)


Does anyone have any idea what's going on behind this result? I will clarify what I mean, and if anyone knows an answer to my question, it might belong in the article. Those who don't know calculus might not get much out of what I have to ask - sorry about that. I certainly don't advocate making the article more confusing, so if that's the only possible result of this, let's drop it. At this stage though, it can't hurt to ask the question, which follows without further ado:

The integrand expands to a polynomial minus 4*arctan(x). That polynomial is p(x) = x6 - 4x5 + 5x4 - 4x2 + 4. Have a look at a graph of that integrand - it's rather symmetrical. That prettiness leads me to believe that p(x) is some kind of canonical approximation of the function 4*arctan(x), and not just a polynomial that happens to equal 22/7 at x = 1. It's not a Taylor approximation, or it would be closest at one point and diverge on either side. What is it, and is there a reason based on numerical approximation theory that one would be certain it's an over-approximation, and that its error should have such a pretty factorization? -GTBacchus(talk) 06:26, 28 November 2005 (UTC)

GTBacchus has clarified on my talk page that it is p(x) - 4/(1+x) that is symmetrical — in the range [0,1]. Also, p(x) could only approximate 4/(1+x2), not 4*arctan(x). -- Paddu 18:31, 18 March 2006 (UTC)

Details of Archimedes' Proof

There are, obviously, many different proofs for this result. While this page was loading up, I was fully expecting to see the Archimedean proof given here; it is obviously the most historic and arguably the most notable proof. I think the article would gain from having that proof reproduced here more fully. The proof presented is certainly notable, it a proof I have seen before (and indeed, I am sure far more people have seen the details of this proof than Archimedes') but I have only seen it given as a piece of "fun" maths, an exceptionally elegant method that is presented for instructive purposes on the topics of calculus and polynomial long division, with the interesting historical backdrop that 22/7 was a classical approximation to Pi. Archimedes' proof, on the other hand, was a serious (and relatively important) piece of mathematical research, so it seems a little unbalanced to cover it with a one sentence summary! On a slightly different note, it would be really nice to know who discovered, or at least first published, the particular integral given here; it would add a lot to the context of the article. TheGrappler 13:58, 19 March 2006 (UTC)

Move suggestion

Does anyone object if I move this article to 22/7 (number)? The article is really about the number; the proof that it exceeds pi is interesting & notable because 22/7 is an important approximation of pi. It was suggested on the AfD debate that the article be renamed, and this seems like the best suggestion to me. Mangojuice 15:48, 20 March 2006 (UTC)

Absolutely should be moved! This page is not about the proof but rather about a definite integral involved in a putnam competition; imho you should move this under Putnam competition 1968, first assignment or similar and link to it from 22/7 (Number). —The preceding unsigned comment was added by (talk) 21:01, 14 February 2007 (UTC).
It's not about a Putnam problem. This argument is of interest quite independently of its appearance in the Putnam. See the AfD discussion. Obviously the argument has great charm, and that has nothing to do with the Putnam. Michael Hardy 03:53, 15 February 2007 (UTC)
I object. The article is about the proof, not the number. I can't make heads or tails of your reason for disagreeing. We might as well argue that the Pythagorean theorem article is really about right triangles so should be moved to right triangle. --C S (Talk) 01:02, 21 March 2006 (UTC)
I think the best is to keep the main part of the article (proof part), and merge some of the introduction as a stub at 22/7 (number). Certainly a lot more could be written about 22/7 and its historical importance. But the rest of the article should be here.--C S (Talk) 01:06, 21 March 2006 (UTC)

I don't think this should be moved to that title.

I also don't think Archimedes' proof should appear here. Perhaps a separate article on Archimedes' computation of π should be created if it does not already exist. The reason why it should not be in this article is that this article is really about a particular method rather than about the bottom-line result. If the title doesn't make that clear, perhaps another could be found. Michael Hardy 21:06, 20 March 2006 (UTC)

I am in favour of this move. In the AfD discussion this renaming was suggested by AySz88 and me, deemed acceptable by Fg2 and GTBacchus, and opposed by Oleg Alexandrov. I still think this is a better name for the article, even though (currently) the proof takes up the better part. The way the article is written, it is already "about" 22/7; the only thing that needs to change is the title. The proof is only "notable" because 22/7 is a widely used approximation of π, so it derives its importance from 22/7. An equally straightforward proof that 85/27 exceeds pi would not be interesting, would it? LambiamTalk 00:39, 21 March 2006 (UTC)

No, a proof that 85/27 would probably not be as interesting. 85/27 is actually further from π than 22/7 despite the fact that the denominator, 27, is bigger than the other denominator, 7. You can't come closer to π than 22/7 with any rational number until the denominator is actually more than 100. I think the fact that 22/7 is a convergent in the continued fraction is relevant here. Michael Hardy 04:02, 30 March 2006 (UTC)
I would have to see the proof that 85/27 exceeds pi in order to deem it interesting or not. Much of the interest of this proof is because of certain mathematical aspects of the proof. --C S (Talk) 01:02, 21 March 2006 (UTC)
OK. We start with the observation that  :) . LambiamTalk 22:00, 22 March 2006 (UTC)
-- Paddu 08:10, 1 April 2006 (UTC)

Don't move to "22/7"

This article is not primarily about that number. It's about the fact that this proof is though-provoking and causes one to suspect some deeper and enlightening fact is a large iceberg whose tip is this argument. Michael Hardy 04:07, 30 March 2006 (UTC)

Am I "hostile"?

Does it constitute "hostility" or "vitriol" or some of those other characterizations of some of my responses on the AfD page if I say clearly that people who know nothing about a subject should not pontificate about it as if they are authorities? If one "assumes good faith" initially, must one continue to assume good faith when people do that? Is that sort of behavior consistent with "good faith"? Michael Hardy 21:06, 20 March 2006 (UTC)

I have no context regarding the comment you make, as I'm new to this. However, keep in mind that an expertise in math does not equate to an expertise on the editing of encyclopedia articles, even ones on mathematics (no doubt they will be authoritative regarding matters of factual accuracies in an article, but an encyclopedia article's quality is more than just that). Also keep in mind that an encyclopedia is different from a paper you might published in a math journal or a textbook you might write; different goals and standards apply.
You are free to make you own judgment about "good faith" and "bad faith" as you see fit. Of course, if this judgment is somehow involved in a dispute or discussion, obvious some may agree and others disagree. 00:51, 15 June 2006 (UTC)

Discussion on recent edits on June 2006

I forgot to add this in my edit comments, and since your recent revert was so broad I don't really know which part you are actually trying to revert:

Based on your edit summary for the revert, I would like to point out that the "citation needed" template is tagging this specific sentence:

"It is easier than most Putnam Competition problems, but the competition often features seemingly obscure problems that turn out to refer to something very familiar."

I don't believe the first link in the external link section (which you mentioned in the edit summary) makes any claim about "seeming obscure problems that turn out to refer to something very familiar". Without a citation this sentence is subjective and cannot be included in an encyclopedia, no matter how accurate the case might be. 23:51, 14 June 2006 (UTC)

Thank you; this is why such matters should be discussed on the talk page to begin with. You are correct that this statement could use a source; you exaggerate to claim that it is unencyclopedic without one. I'm sure a search of AMM would come up with a statement that this is the intent of the PE, as well as the experience of those who have taken it. Septentrionalis 00:04, 15 June 2006 (UTC)
First-hand accounts are generally not considered suitable sources, so "as well as the experience of those who have taken it" doesn't matter, no matter how true. If you are sure that the search of AMM will yield a suitable reference, then please do the search and let us know where you find it.
There's also the separate fact that this opinion, again no matter how true, doesn't really add much to the article. The statement seems better placed in the article on PE. So in addition to the missing citation, there's also the issue of whether the setence in issue should even be kept.
Finally, your reverting ignores the fact that I made multiple edits for different purposes. Regarding the edit where I took out the paragraph in the intro, that's because it is POV. Not to mention it is wordy and redundant: if the proof is so simple and beautiful, why not just let the reader see the proof and decide for themselves? 00:22, 15 June 2006 (UTC)
No, in fact, it doesn't. I have considered all the changes; one is now explained, but the others seem valueless. Septentrionalis 00:46, 15 June 2006 (UTC)
Ok, see below for the discussion on the paragraph in contention. 01:12, 15 June 2006 (UTC)

There's also the fact that the template I put on top of the page is talking about not the lack of sources, but the way the citation is shown. For example in one of the sections, the citation is done by having the text "(see references below)". That doesn't seem like a good style to use for citations. I have leave the tag off for now in case you're talking about something else, but do expect me to either re-add the tag or possibly just make the edits I'm aiming for. 23:59, 14 June 2006 (UTC)

Either fix it, or leave it alone; the purpose of references is to be clear about sourcing, and as long as they fulfill that purpose, all else is Christmas tree decoration. If your calling on Wikipedia is to paint the lily, do so by all means. Septentrionalis 00:04, 15 June 2006 (UTC)
Ok, but just because I see a problem doesn't mean I have the time or interest to invest to actually fix it; that's what cleanup tags are for. 00:22, 15 June 2006 (UTC)

if the proof is so simple and beautiful, why not just let the reader see the proof and decide for themselves?

That's exactly what I did when I wrote this article originally. Then someone nominated it for deletion, citing the fact that there was no authoritative source attesting to its elegance (as if readers could not see that for themselves). Michael Hardy 00:41, 15 June 2006 (UTC)
Ok, but since there is already a citation in the reference section to the source you are citing, you paragraph would seem to be duplicating the information.
Rather, the great elegance and simplicity of the argument may serve to make the reader suspect that this is the tip of a deeper iceberg of understanding of the theory.
Unfortunately, the article provides little context to this "theory" being mentioned. If you are talking about diophantine approximations, the article should better explain the connection of this proof to the theory. Also, this sentence, depend on interpretation, is either making a speculation on how a reader may react, or is trying to make the readers react in a certain way. That would definitely has POV issues (no matter how justified the view) which shouldn't belong in the article. The sentence might warrant being mentioned in the talk pages here as it has more relevance to the debate about the article's importance etc.
Overall, I would say that it suffice to just keeping these sentences
Lucas (cited below) calls this proposition "One of the more beautiful results related to approximating π". Havil ends a discussion of continued fraction approximations of π with the result, describing it as "impossible to resist mentioning" in that context.
The rest seems mainly useful in regards to the debate people have about the article's merit and such, and are therefore better off being put in the talk pages here instead. 01:12, 15 June 2006 (UTC)

Here's a proposed rewrite of the disputed parts of the intro which I think will serve the intent of those who wrote them, but more in line with Wikipedia standards:

What follows is a different mathematical proof that 22/7 > π, requiring only an elementary understanding of calculus. It is notable for its connections to the theory of diophantine approximations. Lucas (cited below) calls this proposition "One of the more beautiful results related to approximating π". Havil ends a discussion of continued fraction approximations of π with the result, describing it as "impossible to resist mentioning" in that context.

I don't think we need to say anything in this article about how short and straightforward the proof is. Even amongst mathematicians, "short", "straightforward" and "elegant" are informal terms generally derived from intuition and defying precise definition. That it is "elegant" is probably better expressed by noting (and hopefully better explaining) its connection to diophantine approximations. That it is "short" and "straightforward" is meaningless to those who have little experience with calculus or mathematical proofs anyway, and readily apparent to those who do, so explicitly saying them doesn't really help anyone in that regards. 01:21, 15 June 2006 (UTC)

I want to note that I intend to give this discussion about 3 days for initial feedback. If there are no comments at all after that, I will resume editing by substituting the proposed replacement above into the article. Just a hats-up. 01:25, 15 June 2006 (UTC)

I believe this requires some editing, but I will put a version in to see how it looks. Septentrionalis 14:15, 16 June 2006 (UTC)
This is pretty much my first edit outside sandbox so ignore me but wouldn't all the "short" "elegant" stuff come under rainbow words which should be removed? Also as a 16 year old with an ok grasp of maths taking it in an english college i found this proof easy to understand but it would be nice to understand where it comes from Extended phenotype 21:45, 19 February 2007 (UTC)

Diophantine approximation

It would be useful if Diophantine approximation explained why this gadget works, which it doesn't now; in fact, it's rather stubby. (That is the title, the present link from the article is a redirect.) Septentrionalis 15:36, 16 June 2006 (UTC)

Equal signs

In the 'details' section, shouldn't those equal signs be less than signs? --Carifio24 21:48, 22 July 2006 (UTC)

Certainly not. Why would you say that? Michael Hardy 22:32, 22 July 2006 (UTC)
Don't mind me, I didn't really read the whole thing right...wasn't totally awake. Now I've made myself like a fool, great --Carifio24 00:33, 23 July 2006 (UTC)

Thanks, people

You all know what brought me here, but that isn't what I want to talk about. What I want to say is simply "thanks". Because even though I had seen the proof before, I found it nowhere as succinctly and yet as clearly stated as here. Shinobu (talk) 16:01, 16 December 2007 (UTC)

Possible rename?

I hate to nitpick, but I'm so busy these days I don't have time to make substantial contributions. But I was thinking, would this not more properly be titled "Proof that 22 sevenths exceeds π"? Maybe I'm splitting hairs. --Cheeser1 (talk) 05:12, 17 December 2007 (UTC)

I disagree. I haven't recently looked at Wikipedia:Manual of Style (dates and numbers), but there is a difference between "every number except one" and "every number except 1". When writing about the number 1, rather than using it in such a sentence as "There is one excecption", I think it should be written as a digit, and this seems to be a somewhat similar case. Michael Hardy (talk) 05:20, 17 December 2007 (UTC)
I completely understand your point about "all but one" vs "all but 1" - I don't know that the same could be said here (how could one confuse "sevenths" with say "7ths" or "over 7" - not to mention "over seven"). It's just always been my experience that unless ambiguity is possible, one says "nths" whenever n is a reasonably small integer (as opposed to "over n"). It's not so much whether or not we spell it out, but whether we say "over 7." I am aware that I'm certainly being a bit picky, but I'm just curious about it. I presume a / is not possible in a title, is it? --Cheeser1 (talk) 20:14, 19 December 2007 (UTC)
I don't know if a slash is a good idea, but it is certainly possible per WP:Subpages#Slashes in article titles (main namespace). I don't have any strong opinions on the title, but I would prefer twenty-two sevenths or 22/7 or 22 over 7 rather than 22 sevenths. I find "22 sevenths" hard to read, but I cannot reliably distinguish 2+b from 2+5 from s+5, so feel free to ignore. JackSchmidt (talk) 20:30, 19 December 2007 (UTC)

From the MOS on numbers:

Fractions are normally spelled out; use the fraction form if they occur in a percentage or with an abbreviated unit (⅛ mm, but never an eighth of a mm) or they are mixed with whole numerals.

Looks to me like it should be spelled out. However, it may also make sense to write 22/7. I am uncertain of what we'd want, although my preference would be 22/7 now that I've realized that a / is acceptable in articlespace titles. --Cheeser1 (talk) 20:59, 19 December 2007 (UTC)

Another consideration is that how fractions are written when writing about mathematics is different from how they are written in other contexts. It makes sense to say that in order to override the president's veto, at least two-thirds of the senators must vote for the bill, with "two-thirds" spelled out like that. But if one says the fraction 2/3 is in lowest terms, then one should write "2/3", not "two-thirds".

However: This is an article title, and those are not as flexible in some respects as the text within the article. In particular, slashes in article names are used to indicate subpages. Michael Hardy (talk) 22:09, 19 December 2007 (UTC)

That's actually not a problem, according to this policy that User:JackSchmidt pointed us to. I remember seeing people repeat "mainspace subpages do not exist" and this appears to be the case. So I suggest renaming it to 22/7 - anyone agree/disagree? --Cheeser1 (talk) 22:28, 19 December 2007 (UTC)
I've added a clause to the "Exceptions" in the MOS on numbers, saying: "Numbers in mathematical formulas are never spelled out (3 < π < 22/7, not three < π < 22 sevenths)." (That's what I like about Wikipedia: if you think the rules as formulated get in the way, you can not only ignore but even change them.) It can be plausibly argued that in the context here 22/7 is a formula. Count this as agree.  --Lambiam 09:30, 20 December 2007 (UTC)

Move/Copy to Wikibooks

The following discussion is an archived discussion of the . 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 debate was keep, clear consensus against transwiki. --Salix alba (talk) 08:24, 17 March 2008 (UTC)

This ought to be in wikibooks:The Book of Mathematical Proofs. (Transferring to Wikibooks was proposed in a 2006 deletion debate, but that was before this particular wikibook existed.)

But we still need to decide which section it should go in. While π's most familiar use relates to geometry, that doesn't necessarily mean that Geometry is the best section. But what is? Analysis? Some whole new section?

That said, you could claim that it's still a Euclidean geometry theorem, if you rephrase it as something like

  • The circumference of a circle is less than 44/7 of its radius
  • The area of a circle is less than 22/7 of the square of its radius

but I'm not sure that that's the best course of action either. -- Smjg (talk) 14:32, 16 March 2008 (UTC)

I don't think that the article should be moved to wikibooks, i.e. copied there and deleted here. (The template at the beginning of the article suggests to copy it, which to me seems something different than "move"; I have edited the header of this section accordingly.)
If you want to have this article in wikibooks, the "Analysis" section is more appropriate than "Geometry". The number pi is important both in geometry and in analysis; but this proof that 22/7 exceeds pi does not refer to geometry at all, except for the trigonometrical fact that tan(pi/4)=1. --Aleph4 (talk) 15:14, 16 March 2008 (UTC)
I agree. It's an important, elegant, and accessible little bit of math, perfect for a wikipedia article. More depth can of course be made in wikibooks if someone wants to do that. Who is arguing to remove it, and why? Dicklyon (talk) 15:54, 16 March 2008 (UTC)
There can be little doubt that transwikiing to Wikibooks will be followed by deletion per WP:CSD#A5.  --Lambiam 02:14, 17 March 2008 (UTC)
The presence of a proof in other books (say, Proofs from THE BOOK) does not act as evidence that the proof should be deleted from Wikipedia, but rather, as evidence that it is notable enough to have an article describing it in Wikipedia. Why should a book that happens to be hosted on wikibooks be treated any differently in this respect? —David Eppstein (talk) 02:21, 17 March 2008 (UTC)

WHY is it proposed that this be moved to Wikibooks? Is there a reason? If fits beautifully where it is. It would fit far worse in Wikibooks, I would think, at least if I am right in understanding those to be somewhat like textbooks. This is not at all a textbookish sort of thing. Note that in this case, the proof, not the theorem, is what the article is about, so this doesn't belong in a list of theorems. Michael Hardy (talk) 02:47, 17 March 2008 (UTC)

  • I don't see any reason to move the page. It's possible that it should be copied (I don't know what the standards are for wikibooks), but I think it belongs on Wikipedia. CRGreathouse (t | c) 02:53, 17 March 2008 (UTC)

I don't see any reason to transwiki this. --Cheeser1 (talk)

The above discussion is preserved as an archive of the proposal. 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.
"The result of the debate was keep, clear consensus against transwiki."

A consensus to keep it, maybe, but certainly not a clear consensus against having this content on Wikibooks. I intended merely to propose transferring the content to Wikibooks, not that it should be done by one particular method or not at all, nor to hard-wire any decision on what would happen to this article should it be transferred. In any case, an open, constructive debate that considers the possible approaches and the pros and cons thereof would be more in order than what has actually happened in the discussion so far.

It's not my fault that some of the content of {{Copy to Wikibooks}} doesn't seem right, but I couldn't find a more appropriate template. Maybe it's time to write one.

Moreover, please point me to the Wikipedia policy that allows debates to be closed before the proposer has had any chance to defend his/her position, let alone for a proper consensus to be reached. This is disrespectful and ought to be changed.

My main motivation in the first place was simply that we have wikibooks:The Book of Mathematical Proofs and we ought to have this there. Even in the absence of this, it would seem to me that mathematical proofs would make good material for a Wikibook. I don't suppose there's any compelling reason to remove it from here while at it. I am sorry if the template I used to make my proposal gave a different impression. -- Smjg (talk) 16:34, 17 March 2008 (UTC)

I agree that it would be appropriate for inclusion in Wikibooks, but I was horrified by what appeared to be a proposal to delete it from Wikipedia. Michael Hardy (talk) 16:41, 17 March 2008 (UTC)
Yes, no problem with copying to Wikibooks. I suggest listing it on Wikibooks:Requests for Import to facilitate the transfer, with page history. --Salix alba (talk) 19:36, 17 March 2008 (UTC)

I don't get it.

Sorry if this seems stupid, but I don't know why this article exists. Do we also need to prove that 4 is bigger than 3? That 50 is bigger than 1.2? No. I can see it now:

'Proof that 4 exceeds 3'

Proof that 4 exceeds 3 go back to antiquity, with most using the simple sum. 4-1=3. The use of the minus sign indicates taking away, therefore a positive difference. QED, 4 is bigger than 3.

Thelb4 12:17, 24 July 2008 (UTC)

We already had this discussion. At great length. Here it is. Michael Hardy (talk) 13:56, 24 July 2008 (UTC)
Ah, good point! I see now the mistake in having this page. Obviously 22/7 - π is positive. Thus 22/7 > π. QED. Wow, why didn't I think of that? --C S (talk) 15:14, 24 July 2008 (UTC)
Yes, but if: The purpose is not primarily to convince the reader that 22⁄7 is indeed bigger than π; systematic methods of computing the value of π exist., then what is the purpose? —Preceding unsigned comment added by Thelb4 (talkcontribs)
The purpose is primarily to show you something that will at once delight you and make you suspect this is the surface of an interesting iceberg. The integral is completely elementary and startlingly simple.
Consider: It is well known that an irreducible quadratic factor in the denominator of a rational function can result in a rational multiple of π showing up when the integral is found. Therefore it may not be surprising that a rational function can be found whose integral from 0 to 1 is 22/7 − π. And maybe it's not much more surprising that one can be found that is obviously positive, so that 22/7 − π must be positive. But who'd have predicted that almost the simplest rational function you could write down that could conceivably have the necessary properties (8th-degree polynomial in the numerator, the simplest of irreducible quadratic polynomials in the denominator, the numerator having such symmetry and being equal to zero at the two endpoints) would be the one that works? It's surprising, to say the least. Everyone I've shown this to who hadn't seen it before has been instantly struck by all of this. Michael Hardy (talk) 20:28, 24 July 2008 (UTC)
Why stop reading there? The lede paragraph is pretty self-explanatory. --C S (talk) 20:06, 24 July 2008 (UTC)
Just to confirm: I'd never read the article before. The lead explains the purpose of the article quite clearly and establishes notability with reliable sources. The proof itself is very nice; thanks for including it. I'll probably use it and the "upper and lower bound" section next time I teach calc II, as I usually have the students do such simple substitutions to get estimates. All the steps are covered in introductory calculus courses, such as those offered in many U.S. high schools. JackSchmidt (talk) 20:21, 24 July 2008 (UTC)

Extensions section with even n?

The section Proof that 22/7 exceeds π#Extensions is quite intriguing, particulay these subsections:

The above ideas can be generalized to get better approximations of π. More precisely, for every integer n ≥ 1,

where the middle integral evaluates to

involving π. The last sum also appears in Leibniz' formula for π. The correction term and error bound is given by


The results for n = 1 are given above. For n = 2 we get


hence 3.14159231 < π < 3.14159289, where the bold digits of the lower and upper bound are those of π. Similarly for n = 3,

with correction term and error bound

hence 3.14159265340 < π < 3.14159265387. The next step for n = 4 is


which gives 3.14159265358955 < π < 3.14159265358996.

Now if, instead of n = 1,2,3,4,..., similar formulas are developed with n=2,4,6,8 - something like:

The above ideas can be generalized to get better approximations of π. More precisely, for every even integer n ≥ 2,

where the middle integral evaluates to

involving π. The last sum also appears in Leibniz' formula for π. The correction term and error bound is given by


The results for n = 2 are given above. For n = 4 we get


hence 3.14159231 < π < 3.14159289, where the bold digits of the lower and upper bound are those of π. Similarly for n = 6,

with correction term and error bound

hence 3.14159265340 < π < 3.14159265387. The next step for n = 8 is


which gives 3.14159265358955 < π < 3.14159265358996.

then similar relationships involving n = 1,3,5,7,..., will give relationships involving the natural logarithm of 2 (ln 2 - 2/3, 38,429/55,440 - ln 2, ln 2 - 1,290,876,029/1,862,340,480, and 356,281,790,621/514,005,972,480 - ln 2) instead of pi. I shall show that info (verifyable on TI-89 and similar calculators) on this page in the near future. --Glenn L (talk) 09:32, 4 December 2009 (UTC)

Proof that natural log (2) exceeds 2/3

Is there an article that similarly proves that the value of the natural logarithm of 2 exceeds 2/3 ? It appears that the main article's proof that 22/7 > π can easily be used for this proof:

Basic idea

The basic idea behind the proof can be expressed very succinctly:

Therefore log (2) > 23 .


(expanded terms in numerator)
(performed polynomial long division)
(definite integration)
(use log(1) = 0)

-- Glenn L (talk) 18:30, 24 April 2010 (UTC)

If combined with this one without any exponent, it yields

Tags that need improvement

the section on background adn "Details of evaluation of the integral" are not sourced, makign this seem like a math tutorial instead of an encyclopaedia. likewise the "proof" and the the IIT exam stuff is WP:Trivia, likewise the Putnam part which doesnt have lasting notability based on which question appears there.(Lihaas (talk) 03:48, 18 October 2010 (UTC)).

Excuse me: The proof is trivia?? The proof is what the article is about! Without that, the reader has no way of knowing what the topic is. Like an article about George Washington that nowhere mentions George Washington. Michael Hardy (talk) 17:34, 18 October 2010 (UTC)
Obviously Lihaas has missed the entire point of the article. But perhaps that means that the point isn't expressed clearly enough? —David Eppstein (talk) 17:53, 18 October 2010 (UTC)

"Details of evaluation of the integral" uses standard techniques that are taught to first-year students. It seems worth including just so that naive readers don't think there's something mysterious about this particular integral; its evaluation is just routine stuff. If an article mentions that 17 × 47 = 799, I don't think of that as something that has to be sourced, and this seems somewhat similar. I didn't know the IIT part was there. The Putnam is a standard part of the culture, so it seems interesting.

I'm still in a mild state of shock over the seeming suggestion that we exclude from the article the thing that the article is about. Michael Hardy (talk) 18:04, 18 October 2010 (UTC)

No, of course not. the proofs are the crux of the issue. but the IIT stuff does fall into that category, no encyclopaedic notability, i think we can agree on that.
Seems, like i missed my punctuation ;)Lihaas (talk) 11:52, 19 October 2010 (UTC)

OK, if I'm understanding the "punctuation" point correctly, you meant:

"Details of evaluation of the integral" are not sourced, makign this seem like a math tutorial instead of an encyclopaedia. likewise the "proof".

and then:

and the the IIT exam stuff is WP:Trivia

My gut reaction to the part about the "proof" is that it does only things that everybody learns in infancy. Only an initial gut reaction. Michael Hardy (talk) 16:21, 20 October 2010 (UTC)

Maybe the Putnam question has a solution published somewhere that we can cite as a source for the details? I agree it's more in the nature of a routine calculation than original research. —David Eppstein (talk) 17:02, 20 October 2010 (UTC)
Yep, pretty much. Not anything against the page. In fact i think a page like this is one of the best reasons for wikipedia. certainly over the political pages ;)Lihaas (talk) 04:35, 26 October 2010 (UTC)

Values of n other than the positive integers

Has anyone looked into what

evaluates to for , or even for an arbitrary real number ? If π shows up in some of these cases that would be cool.

Thanks —Quantling (talk) 18:14, 19 October 2010 (UTC)

Actually, WolframAlpha will let you test any combination of this type.
For example, typing "4/2^0 * int (x-x^2)^0 / (1+x^2) dx, x=0 to 1" into the box results in "π ≈ 3.14159" being displayed.
It turns out that there are four patterns that oscillate in sequence:
approximates π≈3.14159265...: π, 22/7 - π, π - 47171/15015, ....
approximates (π+loge4)≈4.52788701...: (π+loge4) - 4, 11411/2520 - (π+loge4), ....
approximates loge4≈1.38629436...: loge4 - 4/3, 38429/27720 - loge4, ....
approximates (π-loge4)≈1.75529829...: 53/30 - (π-loge4), (π-loge4) - 421691/240240, ....
Glenn L (talk) 05:07, 26 October 2010 (UTC)

Missing the point

People really do know how to miss the point completely. This edit is one of the most extreme cases of stupidly missing the point that ever happened, but I'm finding out this is not an isolated case. Googling the title of this article shows a yahoo answers page about it. A huge number of people asserted that "just" (sic!) knowing that π = 3.14159265...... is enough to show that 22/7 > π. And so it is, but you have to do a lot of work in order to "just" know that. A lot more work than needs to get done in this article. It is appalling that so many people are so gullible. If all the books say that π = 3.14159265......, then they think it's the self-evident innerant word of God. Appalling.

Should we say something more than we've said in this article to correct that mentality? Michael Hardy (talk) 03:55, 24 December 2010 (UTC)

Proposed move: "Proof that 22/7 exceeds π" --> "Proof that 22/7 exceeds pi"

It is unconventional to use a math symbol in a title even when that symbol is universally understood. It's Three Blind Mice, not "3 Blind Mice", a Dirac delta function, not a δ function, and so forth. In this case, we can't avoid the use of "22/7", but that is all the more reason not to throw in additional symbols. "Pi" is a far more common usage than "π" according this ngram. CNN uses "pi", the BBC uses "pi", and the New York Times uses "pi". This entry in Merriam-Webster is entitled "pi". The "pi" entry in Britannica is a good model. They have no policy or technical issue that prevents them from producing a "π", yet they use it only for equations. Van Nostrand's Scientific Encyclopedia does the same, and that is an even better authority (p. 4105 -- sorry no link). The math journals can go either way. Check here and here for journal articles with "pi" in their titles. Although the symbol ">" is better known than "π", it is spelled out in this title as "exceeds". Kauffner (talk) 12:43, 19 April 2011 (UTC)

Oppose. Those who use pi instead of π generally do so for technical do so for technical reasons:
  • Not using anything other than ASCII in titles ensures that systems that count citations for your paper don't get anything wrong. So it's in the interest of both the journal and the author for reasons that have nothing to do with the reader or with our situation here.
  • Journalists generally don't know how to enter special characters, and even if they could, they can't risk that somewhere in the publication process something gets lost. This can happen even with TeX in mathematical journals. There is a paper by Saharon Shelah that can only be read sensibly using two versions in parallel: The final version in the journal, which has some key symbols replaced by boxes, and a very old preprint on arXiv to find out what the boxes mean.
There is no need for the move, and for esthetical reasons I oppose to it very strongly. Hans Adler 18:00, 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)

Move discussion in progress

There is a move discussion in progress on Talk:Pi which affects this page. Please participate on that page and not in this talk page section. Thank you. —RM bot 13:15, 2 June 2011 (UTC)


In a recent posting to mathoverflow, Harvard professor Noam Elkies wrote about the proof that this article is about:

Is this something one could fairly cite in this article? Would it help satisfy those who have strangely claimed they see no elegance in this proof (not that those people are active now, but they've made themselves memorable)? Michael Hardy (talk) 20:24, 13 June 2011 (UTC)

Background section

In the section currently:

Shouldn't it be like this:

or this:

What's going on in here? — Preceding unsigned comment added by Th4n3r (talkcontribs) 17:43, 14 March 2013 (UTC)