Talk:Proof that π is irrational/Archive 1

A nice proof
Thanks to Michael Hardy for writing a very nice article. I did notice one typo, which I've fixed:



(F'\cdot\sin + F\cdot\cos)' = F''\cdot\sin + F'\cdot\cos + F'\cdot\cos - F\cdot\sin = (F'' - F)\cdot\sin + 2F'\cdot\cos \neq f\cdot\sin $$

but (corrected version)



(F'\cdot\sin - F\cdot\cos)' = F''\cdot\sin + F'\cdot\cos - F'\cdot\cos + F\cdot\sin = (F'' + F)\cdot\sin = f\cdot\sin. $$

Notice that this also makes the integral work out as a positive integer, since cos&thinsp;&pi; = &minus;1.

I also think the argument about f(j)(0) and f(j)(&pi;) should be phrased in terms of f(2j)(0) and f(2j)(&pi;), in light of the way the function F is defined, but I thought I'd bring that one up on the talk page before I stick it in the article. DavidCBryant 12:16, 17 August 2007 (UTC)


 * OK, I see that Michael has been here and gone, without doing much to the article. So I'm putting the change about the (2j)th derivative of f into the article, plus a bit more explanatory language about why F(0) and F(&pi;) are always positive integers – actually, F(0) = F(&pi;), if you think about it. DavidCBryant 15:47, 17 August 2007 (UTC)


 * The change about the (2j)th derivative of f was wrong. By using Binomial theorem you can see that f is a polynomial like $$a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_nx^n$$. So for 0 &le; j < n f(j)(&pi;)=0; but it is not correct that for 0 &le; j < n f(2j)(&pi;)=0 (example: Let n be 2. Then $$ f(x)={x^2(a-bx)^2 \over 2}={a^2x^2-2abx^3+b^2x^4\over 2}$$ and $$ f(0)=a^2$$ (in this example j=1). So I corrected this and the following sentence. Furthermore, I added why f''(j)(0) is an integer (by using Binomial theorem). You will see that the proof is correct now. If you see anything that can be simplified, then let me know. Egndgf 15:28, 23 August 2007 (UTC)

What about Lambert's proof?
Lambert was the first to prove the irrationality of π. Should we not include his proof on this page? Nschoem 01:18, 8 September 2008 (UTC)


 * Probably. If I knew the details I might have added it myself. Michael Hardy (talk) 18:27, 8 September 2008 (UTC)

A little help here
Sorry if this is wrong place to ask, but nobody could tell me. Why you can't just say that if pi is rational, then a circle has a finite number of sides? 24.218.46.235 17:59, 4 November 2007 (UTC)


 * Why could you say that? A rational number is an integer over another integer, e.g. 355/113.a  If $$\scriptstyle\pi$$ were exactly 355/113, how would that imply that a circle has a finite number of sides? Michael Hardy 00:13, 5 November 2007 (UTC)


 * Is the anon user perhaps thinking that the perimeter of any polygon (finite number of sides) is rational? That's not true (it is algebraic, however); but even if it were, you still couldn't use it to prove that pi is rational. - dcljr (talk) 03:06, 2 August 2009 (UTC)

Proofs before 20th century
Given the title of the article, some reference to proofs from before the 20th century would seem to be called for. (In particular, who developed them and what were they based on?) - dcljr (talk) 03:17, 2 August 2009 (UTC)

Lambert's proof not correctly described?
Description gives implication, but assumes equivalence?

"If (x!=0) is rational => right side is irrational."

Should (probably) be:

If, and only if, (x!=0) is rational then right side is irrational.

83.172.125.164 (talk) 22:33, 14 March 2010 (UTC)Themerion

Nah it's not necessary, the proof just uses the contrapositive of "If (x!=0) is rational => right side is irrational" If the right side is not irrational (rational) => (x!=0) is not rational (irrational) 67.49.243.3 (talk) 05:22, 30 August 2010 (UTC)

A minor notation issue
In the present version, there is a slight problem with the use of derivatives: in the sentence (–1)kf&thinsp;(k)(π – x) = f&thinsp;(k)(x), there should be a better distinction between (f(π – x))(k), i.e., the kth derivative of the function g(x) := f(π – x), and f(k)(π – x), i.e., the kth derivative of f applied to π – x. —Preceding unsigned comment added by 94.245.127.12 (talk) 12:05, 23 September 2010 (UTC)

Zhou's article On Discovering and Proving Pi is Irrational
Unpublished papers should not be put in Wikipedia. I have read Zhou's article and it seems very different than published math articles. In fact I know of no published article whose mission is to criticize a published article. Also, it uses a fictional story to make its points and this seems to be a strange way to present mathematics. It seems to be at the level of an opinion piece. I point out that it is now March 2011 (6 months after Jones' article was published) and the Monthly has not published an end note correcting Jones' article. Jones' article is published and it has been removed from all references in Wikipedia's article per one reader's recommendations (999er). MathSciNet does have the author's summary of the article under the heading review; they had an opportunity to say more and did not.

It seems strange to allow an unpublished article about an article to be referenced without allowing the original published article to be on the site. It is likely that Zhou himself (999er) is the author who has placed this reference into Wikipedia's site -- this seems to violate the suggested guidelines for the encyclopedia: don't have a personal interest in the topic. —Preceding unsigned comment added by 76.101.178.158 (talk) 13:41, 1 March 2011 (UTC)

Explanation of move from "$\pi$" to "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, "pi" is a far more common usage than "π". On Google books, pi irrational gets vastly more hits than π irrational. "Pi" is the usage of CNN, BBC, the New York Times, Merriam-Webster, and Oxford. The "pi" entry in Britannica is a good model. They clearly have no policy or technical issue that prevents them from publishing a "π", yet they use it only for equations. Van Nostrand's Scientific Encyclopedia does the same (p. 4105 -- sorry no link). If the symbol "π" represented established specialist use, one would certainly expect Van Nostrand`s to use it that way. Math journals can go either way. Check here and here for journal articles with "pi" in their titles. Kauffner (talk) 02:26, 19 April 2011 (UTC)


 * CRGreathouse reverted this move, citing a "strong consensus" -- I'm not quite sure where this came from, since it doesn't appear ever to have been discussed here. --SarekOfVulcan (talk) 11:10, 21 April 2011 (UTC)


 * Probably the math project talk page. Personally, I prefer the title "Proof that π is irrational" as well. The article on Dirac's &delta; function could be moved correspondingly; there's no real reason for spelling out letters like "pi" and "delta". Or perhaps we should move M & M's to Emm & Emm's? &mdash; Carl (CBM · talk)


 * The wpm discussion is here plus there are similar ones scattered across a bunch of article talk pages. Kauffner has done quite a lot of these moves, and made a bunch of edits to actual article text changing &pi; to pi.  This seems to me to be disruptive.  I think they should all be undone per the WT:WPM discussion.  The in-text changes are especially bad; we should do what reliable math sources do, and I've never seen one that spells "pi" in the text.  CNN, NYT, etc are not math sources so they don't count.  Van Nostrand's encyclopedia for that matter was 1) first published in 1938, so may have stuck with an older style; and 2) sounds like it's not really a math source either (science&ne;mathematics).  I don't understand why Kauffner is on such a tear about this anyway.  I do know that chapter 16 of a well known calculus book (ISBN 0521867444) is titled "&pi; is irrational", the topic of this article. 69.111.194.167 (talk) 12:37, 21 April 2011 (UTC)
 * Ah, thanks for the link to the discussion. That seems like a pretty clear consensus. --SarekOfVulcan (talk) 13:30, 21 April 2011 (UTC)
 * I agree that the in-text changes are more important than the title changes. I made a proposal at WT:WPM. &mdash; Carl (CBM · talk) 12:45, 21 April 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: Not moved. Jafeluv (talk) 07:09, 5 May 2011 (UTC)

Proof that π is irrational → Proof that pi is irrational – From Alpha beta transformation to Omega constant, Wikipedia spells out Greek letters in article titles with no exception for math usage. The authoritative references do the same. Although Wiki does have not have the same need to alphabetize entries as reference books do, there is no established alternative model for article titles. In fact, Greek characters have been on a title blacklist since 2008. Kauffner (talk) 17:41, 24 April 2011 (UTC)


 * Proof that 22/7 exceeds π → Proof that 22/7 exceeds pi
 * Approximations of π → Approximations of pi


 * Even non-Latin symbols that are universally understood are generally spelled out in titles, e.g. it's The Two Gentlemen of Verona, not The 2 Gentlemen of Verona. Latin script is used in the article titles of all the Greek letters: alpha, beta, gamma, etc. The less familiar the symbol, the stronger the case for spelling it out. There is Beta wavelet, Beta (finance), Dirac delta function, Delta potential, Gamma function, Gamma distribution, Zeta function, Riemann zeta function, Theta function, lambda calculus, Lambda cube, Sigma-algebra, Chi distribution, Chi-squared distribution, and last, but not least, Pi. Greek letters used in mathematics, science, and engineering includes a long list of relevant titles and the only one that uses a Greek letter is Θ_(set_theory), which is a stub.
 * "Pi" is a far more common usage than "π", as you can see from this ngram. On Google books, pi irrational gets vastly more hits than π irrational. "Pi" is the usage of CNN, BBC, the ‘’New York Times’‘, Merriam-Webster, Oxford, McGraw Hill, ‘’Britannica’‘, and Van Nostrand's Scientific Encyclopedia (p. 4105 -- sorry no link). Britannica and Van Nostrand’s use “pi” both in their articles titles and in their running text. As they both use π in equations, this is not for technical reasons. Both encyclopedias have many articles with “alpha”, “gamma", “pi", or “delta” in the title, but nothing with an actual Greek letter in it.
 * For mathematical usage specifically, there is the The Princeton Companion to Mathematics. This book uses "π" exclusively, never “pi”, even in the index where it looks a bit out-of-place. IMO, this is more a gimmick than a usage model, since the book consistently spells out other Greek letters. Springer-Verlag's Online Encyclopaedia of Mathematics splits the difference with "Pi (number π)". In the titles of journal articles since 1980, it is 1 plus 13 for “pi", compared to 5 plus 21 for "π".
 * Other moves and titles do not establish precedents. But nonetheless editors do ask, “Why don't we adopt the pi symbol for math articles across the board?” There was a recent proposal of this kind here, which was voted down. My own view is the best model is usage by the encyclopedias with more developed scientific coverage and that a usage which is at most peculiar to mathematics is not desirable. Kauffner (talk) 17:41, 24 April 2011 (UTC)

The title blacklist is mostly a spam filter; I just now created User:CBM/π. We also have the article μ-recursive function. There is no general prohibition on Greek letters in titles; all that matters is that there should be a redirect from the Latinized title to make it easier to type into the Go box. I don't see any strong reason to rename articles to avoid using Greek letters. If we do anything, we should go through and use DISPLAYTITLE to make sure that the Greek letters appear in the titles when they should, and otherwise leave the titles alone. &mdash; Carl (CBM · talk) 18:29, 24 April 2011 (UTC)


 * Whatever the origin of the blacklist, it is used to disallow "mixed-script titles", which includes all three of the above. Perhaps there is no "strong reason" for any activity on Wiki, since the world would survive without it. But I think I have shown that there is a well-established convention not to put special characters in titles, so putting them it when there no particular need for them looks unprofessional. If you think that this convention is a by-product of obsolete technology, look at titles on Amazon. Their policy is take the special characters out of titles, even in the case of the movie π, which made a big fuss about using the pi symbol as its title proper. Kauffner (talk) 07:01, 25 April 2011 (UTC)
 * The blacklist has nothing whatsoever to do with well-established conventions or accessibility. It is there solely to slow down a certain page-move vandal. —David Eppstein (talk) 07:08, 25 April 2011 (UTC)


 * I wrote several paragraphs above about the conventions for using special characters in titles in various sources. You might try reading them. Kauffner (talk) 07:34, 25 April 2011 (UTC)
 * Why is this comment formatted as a response to David Eppstein's 07:08 comment? Hans Adler 08:13, 25 April 2011 (UTC)


 * Oppose. There are technical problems that make it advisable not to use anything other than ASCII symbols and the most basic accents in titles of certain kinds of documents. These do not apply to Wikipedia. Wikipedia should do what almost all sources do that don't have such problems: Write π, not pi. Hans Adler 18:56, 24 April 2011 (UTC)
 * On Google books, pi irrational gets vastly more hits than π irrational: Bad search choice. If an article mentions π (the number), there is usually no need to include the word "irrational": people know already that π is irrational. Anthony Appleyard (talk) 21:50, 24 April 2011 (UTC)
 * If you to compare "pi" to "π" directly, the margin in favor of "pi" is even more overwhelming as you can see on the ngram I gave above. But of course the results are more relevant if they relate to same general subject as the article. If you think I am cherry picking, you can compare pi approximations to π approximations. Kauffner (talk) 03:57, 26 April 2011 (UTC)


 * Strong oppose. (1) As long as the corresponding article with "pi" redirects here, for those who are unable to enter π directly, there's no harm.  (2).  The "greek" blacklist was entirely intended as a spam filter to block a certain move vandal; I remember when it was created.  (3)  Google search for π is problematic, at best.  (4) In my experience, "pi" is only used the first time the concept appears, even in non-mathematical books.  — Arthur Rubin  (talk) 22:19, 24 April 2011 (UTC)
 * Oppose moving this article and strongly oppose moving the others en masse. The number is referred to as π, not "pi", by mathematicians. CRGreathouse (t | c) 23:00, 24 April 2011 (UTC)
 * Oppose; redirects make it all OK. I agreed with the main article being called Pi, but that's no reason to move all these others.  Dicklyon (talk) 00:02, 25 April 2011 (UTC)
 * Oppose. Next we'll be referring to Wikipedia as "that big web site with all those facts in it" because the real name is too long and complicated to spell correctly. Only π isn't even long and complicated. —David Eppstein (talk) 04:22, 25 April 2011 (UTC)
 * π is "complicated" in the sense that it is a character that doesn't appear on a standard keyboard. Kauffner (talk) 03:57, 26 April 2011 (UTC)
 * False. "π" is exactly as easy to type as P and therefore slightly easier to type than Pi on my standard OS X keyboard. And what you actually typed, π, is not a character and is typeable on your own keyboard. And anyway your response doesn't exactly address my concern: which is, more bluntly, that I think that attempting to dumb down an article like this one is wrongheaded. —David Eppstein (talk) 04:35, 26 April 2011 (UTC)y
 * I thought the same way at one point. But if look at the reference books, everyone is using "pi", at least for titles. Britannica and Van Nostrand's Scientific Encyclopedia also use it in running text. Kauffner (talk) 10:00, 28 April 2011 (UTC)


 * Support per discussion about Pi, where screenreaders do not read &pi; instead they read p, so this is an accessibility concern. All article titles should be accessible, as should at least the introductory paragraph. 65.94.45.160 (talk) 05:26, 25 April 2011 (UTC)
 * Have you tried looking at the content of the article? I think a π in the title is likely to be the least of the problems with screen readers. —David Eppstein (talk) 05:59, 25 April 2011 (UTC)
 * So, because the page can't be read, the title shouldn't be able to be read either? Keep it less accessible, because it's hardly accessible at all? 65.94.45.160 (talk) 10:19, 25 April 2011 (UTC)
 * Modern Greek language speakers also read &pi; as "p", and they are able to do math too. So if screen readers pronounce &pi; in the Greek way it's not wrong, and certainly not inaccessible. You seem to be saying that the title can be read but you wish it was read some other way. &mdash; Carl (CBM · talk) 10:22, 25 April 2011 (UTC)
 * Just to verify that the title blacklist doesn't affect these things, I created and deleted an article named "&Pi; article" just now . It worked fine. The title blacklist is just there to prevent vandals from making titles like "th&iota;s is &delta;umb!!". It does not affect established users. We have been using Unicode in non-vandalism titles for years. &mdash; Carl (CBM · talk) 10:25, 25 April 2011 (UTC)
 * Apparently, I'm not an "established user." I tried creating a blacklisted title and it's a no-go. I certainly can't create it myself. Even after doing an RM and getting a consensus, it was still rejected. Kauffner (talk) 14:12, 25 April 2011 (UTC)
 * It may need administrator rights to create the page. This is just because the particular vandal in question (known as "Grawp") would create accounts, let them sit dormant for a long time, and then pull them out to do page move vandalism. The restriction was not intended to prevent the use of non-latin characters in titles more generally. &mdash; Carl (CBM · talk) 14:18, 25 April 2011 (UTC)
 * Oic, you are an administrator and you can override the black list. So you don't even have experience with it. Kauffner (talk) 01:28, 26 April 2011 (UTC)
 * Whether CBM himself is prevented from moving or not has no relevance. The only thing that matters is this: The fact that all titles that mix Latin and non-Latin script are blacklisted has nothing whatever to do with deciding legitimate disputes about such titles. It is merely one of several technical measures necessary to keep this site working when it is under attack from Grawp. Any legitimate page move prevented by this filter, and every new title that is not of the type "ҜÂμƒʄŋἒг ₫°€$₦′₮ łἷ∫ҭёи" is legitimate if it's not part of a move war, will automatically be done by an admin if you ask one, and if it's a more general problem and you ask nicely, the edit filter will probably be adjusted. Hans Adler 07:25, 26 April 2011 (UTC)


 * Personally, I find that a day without being insulted in bizarre script is like a day without sunshine. I tried to add diacritics to the article title for Truong Tan Sang, the current leader of Vietnam. Such diacritics are blacklisted, so I did an RM, which was rejected with the notation, "The case of reliable English language sources is the strongest." I suppose that's fair enough, at least in the sense that the international press does not use diacritics when referring to him. But the general news media does not use special characters of any kind, including the pi symbol. So there is no getting around the blacklist under this standard. But just as you can find the pi symbol in various math sources, you can find Vietnamese diacritics used in various Vietnam-specific English-language sources. Kauffner (talk) 11:18, 26 April 2011 (UTC)


 * Strangely, Vietnamese doesn't have an entry at Romanization, although Chinese, Korean, and Japanese do. There are endless arguments over the right way to do article titles for foreign names, so I'm not surprised if there is still some inconsistency. But this is all unrelated to the intention behind the blacklist; apparently the blacklist has an unintended side effect of making it harder for non-admins to rename articles with non-latin titles, but I think that overall the blacklist is a red herring here. &mdash; Carl (CBM · talk) 14:22, 26 April 2011 (UTC)


 * The Vietnamese alphabet is a Latin alphabet with diacritics based on Portuguese. The languages on the list you linked to are all written in non-Latin script. Kauffner (talk) 14:39, 26 April 2011 (UTC)


 * Actually, there is also Naming conventions (Vietnamese), I just missed it. Unfortunately there is not Naming conventions (Mathematics). &mdash; Carl (CBM · talk) 15:11, 26 April 2011 (UTC)


 * I am not sure it's unfortunate. AFAICT so far we had the best sort of consensus: Things just were that way and everybody agreed it was the right thing. If a single editor who got a strange idea in his head is enough to upset everything and cause huge disruption that requires new guidelines to be solved, then we have a serious problem. We have enough policy bloat as it is. Hans Adler 20:32, 26 April 2011 (UTC)


 * Oppose, for reasons already stated at . Some of the highlights of that discussion are: (1) reliance on Google searches and library catalog searches to determine symbol prominence are misleading (for various reasons), (2) the solution to the screen reader problem is to fix the screen readers, not to change stylistic elements of all of our &pi; based articles just because third-party screen readers are broken.   Sławomir Biały  (talk) 07:21, 27 April 2011 (UTC)


 * support Accessibility and the fact that the mixed text comes in to play. Chaosdruid (talk) 20:13, 29 April 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.

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)

Title Consistancy
Does it bother anyone else that the article on the mathemetical constant π is named Pi while this one still uses the Greek symbol? Brightgalrs ( /braɪtˈɡæl.ərˌɛs/ )[1] 09:21, 23 November 2011 (UTC)
 * As you can see in the discussion above, it bothers User:Kauffner (and not just in the title). It doesn't bother me because the contexts are slightly different. I believe the practice of reference works to replace non-Latin letters in this way is due to the problem of alphabetic sorting, which we don't have. However, for single Greek letters we have the problem that (1) the various article titles should be consistent with each other, and (2) they should all be recognisable as Greek letters. We cannot title the article on all Greek letters by just using the letters, because that would make distinction between A and Α (follow the links and see where they take you) tricky business. On the other hand, "proof that pi is irrational" is something that would not normally occur in well edited mathematical text.
 * There may be a consistent way of dealing with this that leads precisely to the present situation. In the title of the article about the letter "π" we mention the constant and drop the quotation marks, whereas in the present title we use it. See use-mention distinction.
 * In any case, there are a lot of mathematical articles with "π" in the title, and apart from Pi they all use this style. Any change would have to be agreed beforehand, with input of WP:WikiProject Mathematics~, to prevent disruption. Hans Adler 10:22, 23 November 2011 (UTC)
 * The problem is not with the titles of the greek letter articles. Wikipedia is not a mathematical textbook. The amount of articles that need to be changed for constancy should not be part of your argument. Furthermore wp:article_titles would strongly disagree with you on everything else. Brightgalrs ( /braɪtˈɡæl.ərˌɛs/ )[1] 20:30, 26 November 2011 (UTC)
 * I am not going to discuss with you here whether all articles with titles similar to this one should be changed, as we have had such discussions extensively after Kauffner tried to enforce his opinion through undiscussed moves. We have a consistent style for these articles, which appears to be strongly supported by the relevant WikiProject. Contrary to what you say, the section of the title policy that you are quoting is not in conflict with current practice, although there may be reasonable interpretations under which it is (and others under which the current practice is required). Hans Adler 22:56, 26 November 2011 (UTC)


 * Terser response: No, it doesn't bother me. CRGreathouse (t | c) 14:13, 23 November 2011 (UTC)
 * JUSTDONTLIKEIT?Brightgalrs ( /braɪtˈɡæl.ərˌɛs/ )[1] 20:30, 26 November 2011 (UTC)
 * Merriam Webster, Oxford, MacMillian Dictionary, Britannica, The Columbia Encyclopedia, Wolfram Mathworld, and Mathemathics all use "pi" as their entry title for this subject. The dictionaries all spell it "pi" and don't even give "π" as a variant spelling. There are thousands of books with the word "pi" in the title, maybe five to ten with "π". The Borwein's book, a hyperspecialist work about calculating the value of this constant, is entitled Pi and the AGM. Kauffner (talk) 00:43, 27 November 2011 (UTC)
 * Most book publishers (or, more important to the publishers, bookstores) are unable to put a symbol (or Greek or Cyrillic letter) in the title; we are not so limited. We may have some problems with articles which start with a symbol which is not approximately a latin letter, at least in terms of alphabetization, but we don't have any problem with this article, and it is certainly the way it would appear in text.  There's a disputed guideline at WP:TITLE which states that the title should be displayed as if it were written out in text.  That could be solved by using DISPLAYTITLE, but the article name being different would fail the principle of least surprise.
 * More to the point, there was a discussion on this a few months back, which arrived at the consensus that π should be used in article titles except for the main article. If you want to change that, you need to produce a new consensus.  — Arthur Rubin  (talk) 01:04, 27 November 2011 (UTC)