Talk:Pi/Archive 7

A-class review
I've nominated this article for A-class review through WikiProject Mathematics. I'm transcluding it below. Please help me address/respond to their concerns. I'd eventually like to nominate this article for FA. —Disavian (talk/contribs) 19:25, 12 December 2007 (UTC)

Baffling and unnessisary links in the External Links section
Such as the pi is wrong! link - amusing and irreverent as this may be, it's utterly unnessisary and reads like the type of dross that me and my friends used to write at the back of Math class when we were 14. It doesn't really deserve a place on this page because it adds nothing to the discussion of what Pi is, does or is useful for. —Preceding unsigned comment added by 87.194.49.123 (talk) 19:00, 4 May 2008 (UTC)

I agree, the links need to be sorted, some just repeat themselves (Bonzai273 (talk) 04:39, 24 May 2008 (UTC))

The "π Is Wrong!" article makes a serious point, and the link to it should be kept. —Preceding unsigned comment added by 86.130.60.154 (talk) 15:55, 6 June 2008 (UTC)

Redirect to Pi
3.1415926535897932384626515 redirects to Pi. Although this is not the actual value of Pi, it may seem so at the beginning, but 515 is to be replaced with 832 (3.1415926535897932384626832795) Androo123 (talk) 00:41, 23 May 2008 (UTC) I eat PI

Well I decided to see if I could figure out exactly what pi was and figured everyone is doing it wrong... it shouldn't be explained in a decimal try explaining a rational number... rationally... but sadly google's calculator ran out of compatibility and then my calculator did the same but the closest I was able to get it was somewhere in between (352/(squareroot(12554.09993))) and (352/(squareroot (12554.09995))) if anyone needs explanation squareroot - means to take the square root of the following character or number in a shiny set of 's / - means divided by or also making a fraction of the characters or numbers before and after it —Preceding unsigned comment added by 206.74.75.177 (talk) 02:37, 13 January 2009 (UTC)

If I didn't remembered it wrongly, Androo123 missed out 433.(3.141592653589793238462643383279...) Visit me at Ftbhrygvn (T alk |C ontribs |L og |U serboxes ) 07:05, 29 May 2009 (UTC)

Hex
Hexadecimal 3.243F6A8885A308D31319, as stated in article does appear as Hex. I have seen Hex. when programming. 0, I believe is a null, the lowest value; not zero as in a number line; and should not appear in a decimal series. Coding of Hex., e.g, on OS MVS/XA uses a letter and number on the bit-map, no series of numbers. Please check. Your table might only apply to ASCII?

ASCII:

http://www.asciitable.com/

EBCDIC:

http://www.legacyj.com/cobol/ebcdic.html

Notice, in EBCDIC there are two for the bit, usually expressed using two lines, one line over the other line. To confirme check with a mainframe system programmer.

Thanks. —Preceding unsigned comment added by 58.110.206.219 (talk • contribs) 11:11, 28 May 2008 ; moved from Talk:Pi/to do by —Disavian (talk/contribs) 02:09, 29 May 2008 (UTC)
 * 0 exists in all bases. The way it's expressed in COBOL is irrelevant to the purposes of this article. —Disavian (talk/contribs) 02:11, 29 May 2008 (UTC)

Unsourced addition
I reverted the following addition to the article:


 * However, by computing this series in a somewhat more clever way by taking the midpoints of partial sums, it can be made to converge much faster. Let
 * $$\pi_{0,1} = \frac{4}{1}, \pi_{0,2} =\frac{4}{1}-\frac{4}{3}, \pi_{0,3} =\frac{4}{1}-\frac{4}{3}+\frac{4}{5}, \pi_{0,4} =\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}, \cdots\! $$
 * and then define
 * $$\pi_{i,j} = \frac{\pi_{i-1,j}+\pi_{i-1,j+1}}{2}$$ for all $$i,j\ge 1$$
 * then computing $$\pi_{10,10}$$ will take similar computation time to computing 150 terms of the original series in a brute force manner, and $$\pi_{10,10}=3.141592653\cdots$$, correct to 9 decimal places.
 * $$\pi_{i,j} = \frac{\pi_{i-1,j}+\pi_{i-1,j+1}}{2}$$ for all $$i,j\ge 1$$
 * then computing $$\pi_{10,10}$$ will take similar computation time to computing 150 terms of the original series in a brute force manner, and $$\pi_{10,10}=3.141592653\cdots$$, correct to 9 decimal places.
 * then computing $$\pi_{10,10}$$ will take similar computation time to computing 150 terms of the original series in a brute force manner, and $$\pi_{10,10}=3.141592653\cdots$$, correct to 9 decimal places.

As no reference is given, this appears to be original research. If someone can find a source, please feel free to re-instate with reference. Gandalf61 (talk) 19:48, 31 May 2008 (UTC)

It is somewhat OR in that I'm not sure whether it has ever been published. I posted an article giving this result on Usenet many years ago--you can find it by going to Google Groups and searching on "ash@sumex-aim.stanford.edu sci.math". I sent a proof via private email to Noam Elkies; we agreed that the result is valid and I'm trying to ping Noam now to see if he remembers the correspondence or can point us to a published reference.--Dash77 (talk) 20:45, 31 May 2008 (UTC)

Per a note from Noam, it appears that this is not OR but is an example of the Van Wijngaarden transformation. I am about to reinstate the deleted text with a link to that Wikipedia article, which in turn contains a link to another article that is well referenced.--Dash77 (talk) 01:05, 1 June 2008 (UTC)

Avoid peacock terms
As my simple edit ( http://en.wikipedia.org/w/index.php?title=Pi&diff=216691447&oldid=216456829 ) was reverted by someone who 'disagreed'. Allow me to explain. I didn't state than I don't think Pi is 'important', just that claiming 'importantance' is not, by itself, encyclopedic.

Please refer to Avoid_peacock_terms

I leave it to someone else to now be bold.

Dr. Zed (talk) 17:58, 6 June 2008 (UTC)


 * In this case, the term is justified because it has been often said; so I copied the source from e for it. Dicklyon (talk) 00:10, 7 June 2008 (UTC)

Question about not being immediately obvious
"While that series is easy to write and calculate, it is not immediately obvious why it yields π." - in the discussion of 4/1 - 4/3 + 4/5 - 4/7

Why does it say this? The result is easily derived from the expansion of arctanx. I will add this in unless someone objects. Helenginn (talk) 16:32, 7 June 2008 (UTC)
 * I think the point is that 4/1-4/3+... is a conditionally convergent series, which means you have to be careful when manipulating it (for example it can become divergent, or converge to a different limit, if you reorder its terms). The easy derivation of the series for tan&minus;1(x) is valid for |x|&lt;1, the region where the series is absolutely convergent, and it doesn't follow instantly that it works for x=1, even though the series does converge (conditionally) at x=1. --Trovatore (talk) 09:15, 12 October 2008 (UTC)

SVG
For anyone who can be bothered, it would be good to replace the gif at the top with an SVG. — [  semicolons  ] — 21:18, 14 June 2008 (UTC)
 * Semicolons, I disagree. SVG with animation has incomplete support in browsers while GIF animations are universal and recommended.Cuddlyable3 (talk) 06:54, 25 June 2008 (UTC)
 * Wikipedia doesn't directly serve SVG images to the browser, they are rendered to PNG using rsvg. Animation is thus out of question, as it is supported neither by rsvg nor by PNG. — Emil J. (formerly EJ) 15:08, 25 June 2008 (UTC)

Pi calced at 3 when figuring the circ/dia ratio of a thin walled vessel.
 [Re: Biblical value of Pi] 

Wrong. That would make Pi exceed 3.14. As wall thickness increases, the ratio goes up, not down. —Preceding unsigned comment added by 69.122.62.231 (talk) 15:59, 18 June 2008 (UTC)


 * No. If the circumference C is measured on the inside edge of the brim and the diameter D is measured from the outside edge, then the ratio $C/D$ is exactly π if the brim had zero thickness. But as the brim thickness increases, assuming that the outer diameter D stays the same, the inner circumference C decreases, so the ratio decreases as well. If instead we say that the inner circumference C remains the same and the outer diameter D increases, the ratio still decreases. So when measured this way, a non-zero brim thickness produces a ratio less than π. Therefore at some thickness (where the brim is about 4 inches wide), the ratio is exactly 3.0. | Loadmaster (talk) 16:49, 16 January 2009 (UTC)

Pi cropcircle
As the article is locked, I'll add it here: Crop circle repesenting pi 86.150.97.50 (talk) 17:13, 20 June 2008 (UTC)


 * I see the Telegraph has gone downhill.... No, it shouldn't be added to the article.  &mdash; Arthur Rubin  (talk) 17:55, 20 June 2008 (UTC)

Cosmological constant
Does anybody know where the formula for the cosmological constant came from? In particular, the c2 denominator is not in the listed source (and not in any of the other sources I checked). Also, is ρ supposed to be the vacuum energy density? Thank you.&mdash;RJH (talk) 18:12, 22 July 2008 (UTC)

Who discovered it?
Who discovered\invented Pi?--64.79.177.254 (talk) 18:08, 22 July 2008 (UTC)
 * See Pi. The concept was known at least about 4000 years ago. No single known personal inventor can be nominated. The use of the symbol π was introduced by William Jones (mathematician) A.D. 1706,  Rgds / Mkch (talk) 00:24, 24 July 2008 (UTC)

Suggestion to the registered users
Since the page is protected, I cant changes it myself. There is another identity for calculating Pi which I did not find (almost) anywhere else - not even on Mathworld. I have talked about it on my blog: http://blog.hardeep.name/math/20080725/value-of-pi/. Please consider if a link to my blog entry can be posted here. —Preceding unsigned comment added by Hardeeps (talk • contribs) 08:05, 28 July 2008 (UTC)


 * This formula is called a Machin-like formula, because it is a variation on the following formula discovered by John Machin in 1706:


 * $$\frac{\pi}{4} = 4 \arctan\frac{1}{5} - \arctan\frac{1}{239}$$


 * The particular formula quoted in your blog is formula (22) at this MathWorld page. Your blog would probably not be considered a reliable source, and so linking it from a Wikipedia article would not be appropriate. You could source the formula by referencing the MathWorld page or another reliable source. However, I am not sure that this particular formula will pass the notability test, because there are so many similar formulae. I think you would have to show that this particular formula has some property that distinguishes it from other Machin-like formulae. Gandalf61 (talk) 08:39, 28 July 2008 (UTC)

History
In the History section, it says:

''Geometrical period

''That the ratio of the circumference to the diameter of a circle is the same for all circles, and that it is slightly more than 3, was known to ancient Egyptian, Babylonian, Indian and Greek geometers. The earliest known approximations date from around 1900 BC; they are 25/8 (Babylonia) and 256/81 (Egypt), both within 1% of the true value.[2] The Indian text Shatapatha Brahmana gives π as 339/108 ≈ 3.139. The Tanakh appears to suggest, in the Book of Kings, that π = 3, which is notably worse than other estimates available at the time of writing (600 BC). The interpretation of the passage is disputed,[24][25] as some believe the ratio of 3:1 is of an exterior circumference to an interior diameter of a thinly walled basin, which could indeed be an accurate ratio, depending on the thickness of the walls...''

I believe the last sentence should have the positions of the adjectives "exterior" and "interior" reversed. I. e., the last sentence should read:

The interpretation of the passage is disputed,[24][25] as some believe the ratio of 3:1 is of an interior circumference to an exterior diameter of a thinly walled basin, which could indeed be an accurate ratio, depending on the thickness of the walls. --AjitDongre (talk) 01:09, 7 August 2008 (UTC)

Why did you not mention the name of Aryabhat from India? He is very famous for his contribution to 'pi' much before William Jones and Lambert. He was the first to realize that Pi (π) is irrational. —Preceding unsigned comment added by Dsg512 (talk • contribs) 08:20, 6 March 2009 (UTC)

Pi vs. π
This article waffles on using "pi" and using "π". This should be remedied. —Preceding unsigned comment added by Tastemyhouse (talk • contribs) 15:49, 25 August 2008 (UTC)


 * I am in agreement. Though this may not be the right place to say this, however the Greek pronunciation of "pi" needs to be updated.  Greek does not have an "i" (or "eye") in it.  iota, eeta, upsilon are all the "ee" sound making the proper pronunciation "pee".  —Preceding unsigned comment added by 199.253.174.9 (talk) 19:10, 4 December 2008 (UTC)


 * The pronunciation is given in the standard International Phonetic Alphabet, which denotes by "i" the sound you write above as "ee". — Emil J. 15:07, 15 June 2009 (UTC)

Pi
Hello, sir. I happen to know the first 32 digits of Pi, if it helps the article. Any comments, feel free to leave messages on my talk page. Chris Wattson (talk) 18:25, 11 September 2008 (UTC)
 * We have all of the digits we need, thanks. —Disavian (talk/contribs) 20:21, 11 September 2008 (UTC)

Italicization
The Greek letter pi is italicized in some parts of the article, and roman in others. That needs to be fixed, but which is correct?

67.171.43.170 (talk) 02:37, 17 September 2008 (UTC)
 * Pi should be italicized; π should not be. Septentrionalis PMAnderson 16:58, 20 September 2008 (UTC)


 * Not exactly consensus, but I agree and boldly made the change throughout (in celebration of this glorious day), but e still appears italicized. Note that, as a constant, this is different from the several variables that appear italicized (d, r, etc.), although I'm not 100% sure what's proper there, either. Out of curiosity, would $$\pi$$ be an option for the article body? Or $$e$$? /Ninly (talk) 16:04, 14 March 2009 (UTC)

'''WRONG. $π$ should always be italic''', for the following reasons.. The tradition of mathematical typography is to italicize $π$. There is discussion of whether this is 'correct' but the lower-case italic form $$\pi$$ is the most widespread in the literature. ($π$ is italic when it denotes the standard meaning of pi in trigonometry, calculus, physics, etc -- but deviant statisticians use π and Π to symbolize other stuff .  MathML, LaTeX, and Wikipedia tags

or all italicize $$\pi$$.  From Edward Tufte's website: These traditions are hundreds of years old. Italics help distinguish between variables and other things like numbers and operators. For instance cosine of x is written "cos($x\,\!$)". (This issue comes up in other symbolic systems; programmers need that distinction too, but today almost universally use color coding instead.) But like any centuries-old tradition, there are a lot of quirks, for example vector variables are sometimes bold, non-italic. Also which letters you use for which variables is important. A lot of today's traditions were introduced by Descartes, like using the end of the alphabet for unknowns and the beginning for known quantities. At one point people used vowels vs. consonants, I think, but no longer. According to Stephen Wolfram: One of the things that Euler did that is quite famous is to popularize the letter $\pi$ for pi--a notation that had originally been suggested by a character called William Jones, who thought of it as a shorthand for the word perimeter. Most fonts are non-mathematical and render &amp;pi; as π (non-italic). But mathematical typesetting packages render \pi as "$$\pi$$". I believe this article should follow mathematical convention. So I'm thinking about going back and italicizing the pi's. An objections? Ropata (talk) 05:38, 17 June 2009 (UTC)


 * The Wikipedia Manual of Style clearly asks not to italicize π (or any other Greek letters for that matter, except possibly when they are variables). The discussion really belongs to WT:MSM, not here. — Emil J. 11:54, 17 June 2009 (UTC)


 * The discussion is already here, not at WT:MSM. The Manual of Style does not enforce rules, it merely states a few conventions. Wikipedia ought to follow mathematical convention not reinvent it. Ropata (talk) 11:00, 20 June 2009 (UTC)


 * That the discussion already started here does not make the choice any more appropriate. People who are interested in conventions for formatting Greek math symbols are supposed to watch the MoS page, they are not supposed to watch every article which uses the symbols such as this one. — Emil J. 14:25, 23 June 2009 (UTC)


 * OK, if anyone's interested, I've copied the above to Wikipedia talk:Manual of Style (mathematics)
 * cheers, Ropata (talk) 03:59, 24 June 2009 (UTC)


 * The correct answer to this question is neither. The mathematical symbol used for pi is not merely the Greek later pi italicized, but a specific drawing of that letter included with mathematical fonts. Since web browsers have poor support for mathematical fonts, we settle for a rough approximation. One approximation is not clearly "better" than another. Dcoetzee 02:23, 2 December 2009 (UTC)

The rationality of Pi sequence
I have removed this material again: it is probable WP:OR, WP:PEACOCK terms, unsourced and contentious (pi is not rational). I have asked the author to discuss it here before trying to add it in again. Richard Pinch (talk) 11:36, 20 September 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. —Preceding unsigned comment added by Xelnx (talk • contribs) 07:36, 24 September 2008 (UTC)

Also, at several places = needs to be replaced by \approx (or \approxeq). —Preceding unsigned comment added by Xelnx (talk • contribs) 07:43, 24 September 2008 (UTC)


 * Yes, "approximately equal" should always be \approx, not \approxeq. I have replaced \approxeq with \approx in the Geometrical period section of the article. Gandalf61 (talk) 10:34, 24 September 2008 (UTC)

Misleading numeric value of 50 digits
It is misleading to show Pi to only 50 decimal places. Note the red digits, below, show how the zero (the last digit shown in black) would actually be a 1 were such a 50-decimal-place expression of Pi actually be properly written down. For instructional purposes such as here on Wikipedia, truncated expressions of Pi should ideally end on a digit where the following (hidden) digit would fall in the range of 0 to 4. This issue is resolved by expressing Pi here on Wikipedia another three digits because the digit after the 2 (shown) is a 1 (hidden).

Specifically,

Here is the value to 53 decimal places:

3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 10 5 82

Greg L (talk) 20:55, 7 October 2008 (UTC)


 * Noted. — Arthur Rubin  (talk) 22:06, 7 October 2008 (UTC)


 * If the quark-splitting research at CERN is based on Wikipedia, this could be of vital importance!! ;) Ropata (talk) 08:10, 16 January 2009 (UTC)


 * Ropata please explain your statement. Cuddlyable3 (talk) 13:45, 16 January 2009 (UTC)


 * Sorry, just being sarcastic. Actually on second thought Greg_L has a point, but I think it is "tidier" to truncate at 50 digits, than to round the last digit or extend to 53. There are many many sources on the web for longer expansions of $$\pi$$. -- Ropata (talk) 10:40, 26 January 2009 (UTC)


 * Ropata, perhaps in your effort to regale us with your quarkish humour you have missed Greg_L's point. It is simply wrong to truncate at any number of places where if correctly rounded the last digit would be different. Correctness trumps over your idea of "tidiness". Neither Greg_L nor anyone else is proposing to extend the value in the article to 53 digits. Cuddlyable3 (talk) 11:35, 27 January 2009 (UTC)


 * Cuddlyable3, Greg_L really is proposing exactly that: "This issue is resolved by expressing Pi here on Wikipedia another three digits". Which I consider quite pointless, hence the sarcasm. Who cares about a theoretical rounding of the last digit.. -- Ropata (talk) 01:59, 31 January 2009 (UTC)


 * I read Greg's words "is resolved..here" to mean on this Talk page, but Greg may correct me if I have misread his post. Ropata, maybe you care more about justifying your sarcasm than whether the digit is correct or wrong but Greg and I certainly agree that it must be correct. Our purpose is improving the main article so why are you posting at all if you think that is pointless?
 * Truncation is admissible with trailing ellipsis so this is correct: pi = 3.1415... Cuddlyable3 (talk) 10:55, 31 January 2009 (UTC)

I'm having a really hard time wondering why there is all this discussion. If the 50th digit is a zero, then so be it. As long as we don't say "rounded to 50 places", it should be no problem at all. "The first 50 digits of pi are:" or "the value of pi truncated at 50 decimal places is:" would be perfectly fine. Anybody who doesn't understand the difference between "rounded" and "truncated" isn't going to wonder about that zero; and anybody who does know the difference already knows that they're not going to rely on the value shown in this article anyway - most especially not if they have any reason to believe they even need 50 places. I would also note that this was re-opened several months after no discussion on the topic. Frank |  talk  00:37, 1 February 2009 (UTC)


 * The perfectly fine statements above are indeed correct. However stating a numerical value of pi carries the assumption and obligation that the digits printed are as accurate as they can be. That by default i.e. with no qualification such as "truncated" nor use of ellipses means correctly rounded at the last place. The subject of how many digits to show of an irrational number is moot since the consensus is what we have. Wikipedia has lots of information that someone may already know or never need to know, so no surprises here. See WP:TIND Cuddlyable3 (talk) 14:47, 5 February 2009 (UTC)


 * The 50th decimal digit of Pi is 5 not 6. You are advocating for an incorrect digit. Ropata (talk) 10:55, 20 June 2009 (UTC)


 * It seems to me that it is most important that the article is correct in that it does what it says it does. It says 'truncated' and it is truncated, so, no problem. Martin Hogbin (talk) 14:07, 21 June 2009 (UTC)

delimiting values every five digits
This technique of delimiting values every five digits is amateurish. The international standard (according to BIMP: 5.3.4 Formatting numbers, and the decimal marker, and NIST More on Printing and Using Symbols and Numbers in Scientific and Technical Documents: 10.5.3, Grouping digits is that digits should be grouped in threes. Greg L (talk) 21:05, 7 October 2008 (UTC)
 * It may not meet with professional standards, but it's much more sensible than threes. For what it's worth, I have never seen a number of more than 15 digits in threes.  We should be dealing here with mathematical standards, rather than engineering or physics.  — Arthur Rubin  (talk) 21:58, 7 October 2008 (UTC)


 * It is the international standard and is extraordinarily common. Greg L (talk) 22:04, 7 October 2008 (UTC)
 * P.S. If you don’t believe me that physicists delimit numeric equivalencies every three digits, check out this at the NIST: Energy Equivalents Calculator . Try any other conversion; they’re all delimited the same way. And, these guys know physics. Greg L (talk) 22:08, 7 October 2008 (UTC)


 * This is a mathematics article. What care we for physics standards?  And the number of physics articles which have more than 12 decimal places for a number which is not a defined constant is probably fewer than the number of decimal places we have in this article.  — Arthur Rubin  (talk) 22:11, 7 October 2008 (UTC)


 * For all purposes— especially mathematics—the international standard has always been to delimit every three digits. The only thing that varies is what sort of delimiter one uses and what the decimal marker is. Some countries use spaces before and after the decimal marker and the decimal marker is a comma. Regardless of whether it is commas or spaces for delimiting before the decimal marker, or spaces after the decimal marker, the delimiting—both before and after the decimal marker—is always every three digits. See BIMP: 5.3.4  Formatting numbers, and the decimal marker. Delimiting every five digits is thoroughly non-standard. Pi needs to follow the way numbers are used throughout Wikipedia. See also Natural logarithm.  Greg L (talk) 22:25, 7 October 2008 (UTC)


 * Fixed, now, in natural logarithm. See e (mathematical constant).  — Arthur Rubin  (talk) 22:33, 7 October 2008 (UTC)


 * Please don’t disrupt Wikipedia to make a point WP:POINT. What you are doing amounts of vandalism. The BIPM, the ISO, and the NIST all require three digits. You are wrong. Greg L (talk) 22:42, 7 October 2008 (UTC)

There are hundreds of books that list pi split into 5-digit groups. Also hundreds in 3-digit groups. I'd go with the majority. Dicklyon (talk) 22:47, 7 October 2008 (UTC)


 * Yes, Greg, please stop this. We follow usage in the literature, not what some standards body has arbitrarily promulgated. --Trovatore (talk) 22:50, 7 October 2008 (UTC)

Same applies to e in 5-digit vs. e in 3-digit groups. Dicklyon (talk) 22:51, 7 October 2008 (UTC)

Subsidiary data accuracy
The digital value of Pi at 5 digits is 3.1416 and is pretty accurate (ratio = 1.000002+ on my calculator). And it's interesting to note that some other values related to Pi are pretty close to 4 digit numbers. Like Pi/4 =0.785398+ (= .7854 x 0.9999976), and Pi/6 = .523598+ (=.5236 x .9999976). So its pretty obvious that extended decimal digit numbers of the Pi value are not going to add much to the accuracy that would be accomplished by their usage. However, I am intrigued by what an extended binomial value of Pi would show in the array of 1's and 0's involved. That's because a binomial numbering system is better at showing the value of extended numbering than a decimal numbering system because each extension number is an indication of whether it is more or less than 1/2 of what is left. And a study of that might more significantly show the accuracy accomplishing value of the calculation than just inspecting the decimal digit notation.WFPM (talk) 23:23, 17 August 2009 (UTC) I'll bet that if we used a binomial numbering system where instead of writing down all the 1's and 0's, we just counted them each time they occurred and wrote that number down we would soon arrive at a more accurate determination of the Pi value (with fewer numbers) than we could using a standard decimal number method of notation.WFPM (talk) 23:46, 17 August 2009 (UTC)

39 digits is incorrect
Where the article currently says as follows:

…this is obviously in error regarding 39 digits. The supposed citations—if they really say as much—are in error. There is all sorts of bunk out there. Wikipedia’s own Universe article states as follows:

Any child can do the math: Circumference of universe $93 ly$ × Pi = $292 ly$ = $2.764 m$ divided by $1.15$ = 2.4 Å (the diameter of hydrogen atom). Greg L (talk) 21:24, 7 October 2008 (UTC)


 * Given that the number you used is a lower bound on the size of the universe, 37 digits is almost certainly not enough. But 39 is almost certainly enough.  No need to put your WP:OR into the article. Dicklyon (talk) 22:24, 7 October 2008 (UTC)


 * Fallacious argument. The value for the size of the universe comes right out of our own Universe article. It’s pretty much the same value as in the Observable universe article. The Universe article makes it clear that sizes even twice this big are urban legends. If you can cite a much value that withstands scrutiny, cite it. Otherwise, accept that the measurements of the universe are certainly not in error by two orders of magnitude. Greg L (talk) 22:29, 7 October 2008 (UTC)


 * It's not our job to try to justify a tighter bound than the 39 digits found in a reliable source, even if we can. Dicklyon (talk) 22:35, 7 October 2008 (UTC)


 * Your citation is demonstrably false and is beyond worthless. It would require a universe be 100 times larger than the values accepted for use in two of Wikipedia’s articles. Greg L (talk) 22:37, 7 October 2008 (UTC)


 * You haven't showed that it's false, only that you have found a tighter bound that you prefer. But that's WP:OR. Dicklyon (talk) 22:52, 7 October 2008 (UTC)

I have no opinion on 39 versus 37, but to change it to 37 while keeping the source which says 39 is contrary to the verifiability policy. What we have in the article should agree with the source we give. And, as Dicklyon says, if 37 works, then 39 works as well &mdash; leaving a comfortable margin for error. siℓℓy rabbit (  talk  ) 22:57, 7 October 2008 (UTC)


 * Honestly I'd question whether it makes sense to include this silly calculation at all. The calculation really doesn't make much sense even on its own terms.  Space may be asymptotically flat, but it isn't flat to 30 significant digits.  So if you wanted to know the circumference of the observable universe to 30 significant figures, or even fifteen, you wouldn't do it by multiplying the diameter by &pi;; you'd have to do something more complicated. --Trovatore (talk) 21:08, 10 October 2008 (UTC)
 * The main point of this argument is of course that there is no practical geometry-motivated application for computing millions of digits of pi. Perhaps we should just say this instead. Dcoetzee 00:09, 11 October 2008 (UTC)


 * That's the underlying point of the info from the cited source, yes, but is hardly the point of this argument. This argument is about WP:V vs. WP:RS. Dicklyon (talk) 04:55, 11 October 2008 (UTC)


 * By "this argument" I meant the argument described by the text under discussion. The idea was to suggest a compromise, since it's entirely irrelevant exactly how many digits of pi are needed for this particular application. Dcoetzee 09:31, 11 October 2008 (UTC)


 * Someone who has the math should just extrapolate how many digits it takes to measure the circumference of the observable universe to the accuracy of a Planck length, and that's how many digits should be in the article. Icestryke (talk) 08:16, 24 October 2009 (UTC)

10000 digits of Pi
I know 10,000 digits of pi, if it helps. Disliker of humanities 08:53, 12 October 2008 (UTC)
 * Not for me to say whether it might help something, but as far as this article goes, I'm afraid not. If we wanted to put in 10,000 digits, or even a million, we could easily look them up, or generate them. The consensus is that this would not be an improvement to an encyclopedia article. Impressive memory feat though! --Trovatore (talk) 09:00, 12 October 2008 (UTC)

Vandalism
This article is getting a depressing amount of vandalism from IPs. I thought it was semi-protected? Richard Pinch (talk) 21:55, 14 October 2008 (UTC)
 * It was, but the protection expired. I've renewed it. Thanks for the note. --Ckatz chat spy  22:23, 14 October 2008 (UTC)
 * Thanks for that. I was confused, as it still had the little padlock icon.  Richard Pinch (talk) 05:45, 15 October 2008 (UTC)

Pi day 2009 has long passed. Can we unprotect it now? 146.115.34.7 (talk) 22:31, 31 March 2009 (UTC)


 * The place to ask for unprotection is at WP:RPP  A new name 2008 (talk) 22:39, 31 March 2009 (UTC)

Higher analysis
Is it worthwhile to mention the following derivation?
 * $$e^{i \pi} + 1 = 0$$
 * $$e^{i \pi} = -1$$
 * $$i \pi = log(-1)$$
 * $$\pi = \frac{1}{i}\, log(-1)$$
 * $$\pi = -i\, log(-1)$$

for the principal value of log(–1). | Loadmaster (talk) 16:20, 9 December 2008 (UTC)


 * Do you have a reliable source that puts it that way? If not, it's just your own trivial rearrangement, not really useful. Dicklyon (talk) 18:03, 9 December 2008 (UTC)


 * Yes, that's correct, it is my own trivial rearrangement, showing that π is equal to a complex log. Perhaps something like this more rightfully belongs at List of formulae involving π. | Loadmaster (talk) 21:27, 9 December 2008 (UTC)


 * This is not a valid derivation of the principal value of log. Try starting with $$e^{3 i \pi} + 1 = 0$$, do the same manipulations, and you'll end up with the same RHS and a different LHS. Fredrik Johansson 21:54, 9 December 2008 (UTC)

Just a thought.

Could´nt the fact that :$$e^{i \pi} = -1$$ be used to prove whether e and pi are algebraic indenpedent ? The formula clearly shows a relation between e and pi 86.52.155.97 (talk) 15:03, 6 October 2009 (UTC)

Not again!
A page constructed by total nerds for an encyclopeadic article for someone looking for the quick explanation, not 10 complex mathematical theories and formulas! Please, replace it. It is incredibly annoying for someone who could not understand the mathematical formula who is in Year 9 in the 2nd top group with one of the highest levels in the class. It is totally pointless and gawky. Koshoes (talk) 19:01, 25 January 2009 (UTC)
 * This page may be more use to you. Cuddlyable3 (talk) 21:59, 25 January 2009 (UTC)

Seconded. There is absolutely no need for you to attempt to justify your outrage at the "nerdiness" of this article with your dubious academic qualifications. I am sure there is a stylistic guideline somewhere against shameless self-aggrandizement. 70.233.173.118 (talk) 05:57, 29 January 2009 (UTC)

70.233.173.118 is a nerd. —Preceding unsigned comment added by 68.202.97.110 (talk) 00:48, 13 April 2009 (UTC)

Carl Sagan
Why not including some reference to the presence of pi in Sagan's book "Contact"??? It's pretty important to the story! —Preceding unsigned comment added by 143.107.178.145 (talk) 18:03, 30 January 2009 (UTC)


 * You could add that under the heading Pi in popular culture, but I suggest only as a reference. WP:UNDUE. Cuddlyable3 (talk) 11:08, 31 January 2009 (UTC)

Error in statement of physics
The article states that Heisenberg "shows that the uncertainty in the measurement of a particle's position (Δx) and momentum (Δp) can not both be arbitrarily small at the same time". This is not true and is not derivable in quantum mechanics. The derivable result is that the product of Δx bar and Δp bar cannot both be arbitrarily small at the same time. This is true not just for QM but for any wave mechanics - this applies to all 'wavicles'.

Δx bar is the average of a set of measurements of position, either of the same particle in subsequent measurements or of a set of particles at the same time. It is easy to see that this cannot hold for a single particle. Prepare a beam of electrons with precise momentum. The beam is necessarily spread out because of the corresponding uncertainty of position. Then measure any one electron's position to arbitrary accuracy. At that time you have a precise momentum and position.

The article should state that Heisenberg showed that a set of measurements of position and momentum will have an accuracy not better than h / 4 pi. 212.167.5.6 (talk) 16:32, 4 February 2009 (UTC)


 * When you say "prepare a beam" and "measure...position to arbitrary accuracy" are you describing a physical demonstration or a thought experiment? If it's a physical demonstration please tell how you achieve the steps described. If yours is a thought experiment, have you offered anything more than a conjecture about how an uncertainty in position might arise? The article Uncertainty principle refers explicitly to a single particle thus:

When the position of a particle is measured, the particle's wavefunction collapses and the momentum does not have a definite value. The particle's momentum is left uncertain by an amount inversely proportional to the accuracy of the position measurement.
 * That supports the statement to which you object. Cuddlyable3 (talk) 15:55, 5 February 2009 (UTC)

In popular culture
In the In Popular Culture section, the following should be added: "On November 7, 2005, Kate Bush released the album, Aerial.  The album contains the song "π" whose lyrics consist solely of Ms. Bush singing the digits of π to music, beginning with "3.14. . ." —Preceding unsigned comment added by NE Voter (talk • contribs) 18:09, 15 February 2009 (UTC)


 * A few years ago, the article contained a very long list of such random "in popular culture" sightings. There are so many that a trying to create a list of them all is essentially useless -- each individual example does not contribute relevant encyclopedic information about pi. Rather, to avoid the slippery slope starting again, I think the two last sentences in the current "in popular culture" section should be culled. –Henning Makholm 21:22, 15 February 2009 (UTC)


 * As an inclusionist I respectfully disagree and say hang the slippery slope! Notable appearances of pi in popular culture are, by definition, not random but related in a specific way. If the list gets too long the solution is to fork it to its own article, like the many we have at . Cuddlyable3 (talk) 21:36, 15 February 2009 (UTC)


 * Who said we're dealing with "notable appearances of pi"? Most of the appearances that people typically want to add strike me as definitely not notable in their relation to the mathematical constant. The album referenced here may or may not be notable in its own right, but its connection to pi still isn't. Connections between two notable topics need not be equally notable from both ends. For example, it is a notable fact about John Lennon that he was killed in New York City, but it is not a notable fact about NYC that it is where John Lennon was killed. –Henning Makholm (talk) 16:12, 14 March 2009 (UTC)


 * I agree with Henning. — Emil J. 13:23, 16 February 2009 (UTC)


 * I'm all too aware of the tendency for a "___ in popular culture" section to degenerate into a list of every passing mention of a phenomenon in every trivial television episode, etc., but I do think that there should be a Popular Culture section, particularly for Pi which is, as the article rightly notes, just about unique in its hold on mathematicians and non-mathematicians alike (maybe the numbers 3 and 7 compare, but that's about it). A scholarly, encyclopedia-worthy discussion of the phenomenon of π and the non-mathematician world is well called for.  We've just got to be vigilant about the "my reference, too" approach and remove the fluff.


 * And that "Pi Plate" is wonderful. I want one. --Minturn (talk) 17:11, 18 February 2009 (UTC)


 * What I object to is just the my-reference-too fluff. I agree that the general fact that many references to pi occur in popular culture is of encyclopedic interest, and there should be a section for it as soon as we can find reliable secondary sources for a discussion of it. It is just the particular examples we should avoid (except insofar as they in fact support a general point to be made). –Henning Makholm (talk) 16:13, 14 March 2009 (UTC)


 * Here's a start for a secondary source: . Maybe Scientific American has something... --Piledhigheranddeeper (talk) 20:44, 30 March 2009 (UTC)


 * I agree with Cuddlyable3, why else have a popular culture section? Martin Hogbin (talk) 22:50, 27 May 2009 (UTC)

history
there is this tale i've heard, several hundred years ago someone tempted to calculate pi with more decimals than have had been calculated before him (the number of which averaged 17). so he has drawn circles on the ground for like twenty years, and one day a soldier stepped on his work, so he burst out toward the ignorance of the soldier and was killed on the spot.

i think he had calculated around 600 decimals, but he had done an error after the 200th or so, so the decimals were all right except the ones coming after that last-good one.

p.s. sorry for the lack of references everyone, but i think it's better to start a discussion about it and claim such an history exists than to not give anything just because one hasn't the time to do it. Twipley (talk) 18:59, 7 March 2009 (UTC)


 * This is a garbled version of story of Archimedes who according to Plutarch was killed in similar circumstances during the Siege of Syracuse (214–212 BC). According to tradition, his last words are supposed to have been, Do not disturb my circles. Archimedes was one of the first to do systematic work on approximations on pi, but his best-known result is that pi lies between 223/71 and 22/7, which gives only 2 decimals. He definitely was not concerned with computing decimals, since decimal fractions were only invented long after his time. The one who spent decades of his life computing 600 decimals (of which 527 were right) was William Shanks, as described in the Numerical approximations of pi article. Shanks lived 2000 years later than Archimedes and died of old age in 1882 in England, not close to any military action. –Henning Makholm (talk) 22:47, 7 March 2009 (UTC)


 * Acttally, as his article notes, Shanks is famous for his calculation of π to 707 places, accomplished in the year 1873, which, however, was only correct up to the first 527 places. This error was highlighted in 1944 by D. F. Ferguson (using a mechanical desk calculator). Glenn L (talk) 05:33, 25 March 2009 (UTC)


 * See also Article "A piece of Pi" by Isaac Asimov in "Asimov on numbers" Pocket books, 1978 —Preceding unsigned comment added by WFPM (talk • contribs) 22:45, 17 August 2009 (UTC)

Error in article
Digits of pi redirects to Numerical ... and does in fact NOT show the 10,000 first digits of pi. —Preceding unsigned comment added by 62.97.226.2 (talk) 12:38, 14 March 2009 (UTC)
 * ✅, it used to be the first 10,000 digits of pi, but it was redirected to an actual article. A new name 2008 (talk) 12:47, 14 March 2009 (UTC)

"The Cosine algorithm"
There is no mentioning of the cosine algorithm (or whatever it is called) for calculating pi. The one where p_0 = 1.5, p_(n+1) = p_n + cos(p_n), pi=2*p_inf. I know hardly anything about it (more than how it works), so I can't write about it, but it trippels the number of decimals each step, so I think it should be mentioned. /Petter —Preceding unsigned comment added by 79.136.98.249 (talk) 14:04, 28 March 2009 (UTC)


 * I think I understand why it's not mentioned. Each time you have to re-evaluate the cosine of a number that converges to pi/2. And re-evaluate it infinite number of times. I'd personally prefer either


 * $$\pi = 16 \arctan\frac{1}{5} - 4 \arctan\frac{1}{239}$$

or



\pi=3+\frac{1^2}{6+}\frac{3^2}{6+}\frac{5^2}{6+}\frac{7^2}{6+}\frac{9^2}{6+}\frac{11^2}{6+}\frac{13^2}{6+}\frac{15^2}{6+}\cdots \ =\frac{2^2}{1+}\frac{1^2}{3+}\frac{2^2}{5+}\frac{3^2}{7+}\frac{4^2}{9+}\frac{5^2}{11+}\frac{6^2}{13+}\frac{7^2}{15+}\cdots $$ (Glenn L (talk) 00:27, 29 March 2009 (UTC))


 * The article does briefly mention the analytical definition of pi as twice the number between 1 and 2 whose cosine is zero. The algorithm 79.136.98.249 describes directly implements that definition by finding that zero using (a simplification of) Newton's method. This gives it a certain instructive value, though on the balance I don't think we need it in this article. If we can source a textbook that uses it, it might be a good addition to Computing pi, though. –Henning Makholm (talk) 16:19, 29 March 2009 (UTC)

Why is C/d constant?
The article says:

The ratio C/d is constant, regardless of a circle's size. [...] This fact is a consequence of the similarity of all circles.

Since pi is such a fundamental number, I mean its constancy deserves a more thorough explanation (and proof) than just stating that it follows from the similarity of all circles. Will somebody with mathematical insight please consider taking a look at this? —Preceding unsigned comment added by 62.107.206.120 (talk) 20:54, 30 March 2009 (UTC)


 * Fair point. Actually it is somewhat untidy from a mathematical point of view to have to depend on a lot of Euclidean geometry (such as similarity) to define pi. For this reason working mathematicians generally prefer to consider pi defined analytically rather than geometrically. That means depending on a lot of analysis instead of a lot of geometry, but the analysis is more likely to be useful for other reasons in whatever one is doing, and is usually developed with more rigor than the geometry. I've tried to add some of this to the definitions section. –Henning Makholm (talk) 00:04, 31 March 2009 (UTC)

Pi.e trancendental
What does this sentence explain?: "However it is known that at least one of πe and π + e is transcendental (see Lindemann–Weierstrass theorem)."

Both this and the e article claim the numbers are trancendental. It it not therefore obvious that the sum or multiple of both would be too? If not, can someone explain it in the article and also explain why this is of importance to Pi? The Lindemann theorum seems to prove Pi is trancendental by using e, but as both have better independant proofs, i don't think it really says anything about Pi.Yob<b style="color:#008000;">Mod</b> 11:28, 2 April 2009 (UTC)


 * The sum of two transcendental numbers does not have to be transcendental: e.g., both e and −e are transcendental, yet their sum, 0, is not transcendental. So there is no a priori reason why π + e should be transcendental, and the same goes for πe. — Emil J. 11:56, 2 April 2009 (UTC)


 * Aha, maybe I misunderstand the objection: yes, it is obvious that the sum OR multiple of e and π is transcendental, since both e and π are roots of the polynomial x2 − (e + π)x + eπ. This follows immediately from the transcendentality of either e or π. The question is, whether this observation is interesting enough to warrant inclusion in the article. — Emil J. 12:06, 2 April 2009 (UTC)


 * I don't think it should be in the article. The context and "however" leads the user to think that the observation represents some kind of progress into the algebraic (in)dependence of e and pi specifically, but as your argument shows it is true for any pair of trancendental numbers. –Henning Makholm (talk) 00:57, 10 April 2009 (UTC)

I had a question. Did not Newton prove (implicitly) in Lemma 28 of Principia the fact about transcendentality of pi? I felt that the proof is a stone throw away from this Lemma and whats best is that its very elementary (and elegeant of course!), Someone please confirm. I cannot post the content of Lemma 28 right now. —Preceding unsigned comment added by Akashssp (talk • contribs) 03:51, 10 October 2009 (UTC)

The sentence on the "FFT which allows computers to perform arithmetic on extremely large numbers quickly" is just plain wrong
The FFT has nothing to do with performing arithmetic on extremely large numbers - either in terms of magnitude, such as 3 x 1037 or in terms of the number of digits. FFT computes the FFT of a series of data points, which can have large magnitudes, but that does not determine FFT performance, the number of data points does.

RSzoc (talk) 02:06, 22 April 2009 (UTC)


 * The point is that there are fast multiplication algorithms that are based on the FFT. See for example Schönhage–Strassen algorithm. --Trovatore (talk) 02:10, 22 April 2009 (UTC)


 * Right; it's not wrong. Multiplication of big numbers involves the convolution of their digit sequences, and the FFT is an efficient way to do convolutions. Dicklyon (talk) 02:20, 22 April 2009 (UTC)


 * It's sort of wrong. FFT is certainly not the only way to do fast large-scale arithmetic - in fact, implementations of Schönhage-Strassen are usually based on the number theoretic transform, rather than the FFT. It also oversimplifies the issue (there's a hell of a lot more to Schönhage-Strassen than just "use FFT/NTT.") Dcoetzee 22:49, 22 April 2009 (UTC)


 * It's not the only way, but it's the most well-known fast transform that can be used to do the convolution. Dicklyon (talk) 02:41, 23 April 2009 (UTC)


 * Right or wrong, it's commonly called "FFT multiplication" in the literature. — Emil J. 10:03, 23 April 2009 (UTC)


 * I would distinguish between the transform (which is independent of the way it is computed) and the algorithm. The NTT can be done without use of an FFT algorithm, although this is useless; it is the use of an FFT algorithm to compute the NTT that makes Schönhage-Strassen multiplication fast. Arguably, FFT is not a specific algorithm but a class of algorithms. As the FFT article puts it, "Many FFT algorithms only depend on the fact that $$e^{-{2\pi i \over N}}$$ is an $$N$$th primitive root of unity, and thus can be applied to analogous transforms over any finite field, such as number-theoretic transforms." The OP's comment that "FFT computes the FFT of a series..." is like saying "quicksort computes the quicksort of a list". Fredrik Johansson 13:37, 23 April 2009 (UTC)
 * Well, it depends. FFT can mean fast Fourier transform, but it can also mean finite (i.e. discrete) Fourier transform.  So if you unravel it as "the fast Fourier transform algorithm computes the finite Fourier transform", that makes a certain amount of sense. --Trovatore (talk) 20:08, 23 April 2009 (UTC)

Open Questions: What conjecture?
Open Questions states that a plausible conjecture of chaos theory would imply that pi is normal base 2. However, the citation is to a subscription-only site, could someone state what conjecture within the article and link to it? —Preceding unsigned comment added by 141.209.170.158 (talk) 17:28, 12 May 2009 (UTC)

It was a sidebar in the article:

Proof of the normality of pi may come from a link between number theory and chaotic dynamics. While this link is too complicated to explain briefly for pi, the example of log 2 (the logarithm of 2 to base e, where e is the fundamental con- stant 2.718281828 .... ) illustrates the mathematicians' new approach. Log 2 can be obtained to any desired number of decimal places from the ex- pression given below: log 2 = 1/2 + 1/8 + 1/24 + 1/64 + where each term has the form 1/k2 k starting at k = 1 This value works out to 0.6931471805599453 .... Bailey and Crandall have proposed that the normality of log 2 to base 2 is linked to a particular iterative process, or dynamical map, that gen- erates a sequence of numbers be- tween 0 and 1. Here's the mathemati- cal form for this dynamical map: Xn = (2xn - 1 + I/n)mod 1 Starting with xo = 0, each iteration, n, uses the previous result, xn - 1, as the in- put for calculating the next number, xn. The term "mod 1" is an instruction to use only the fractional remainder of each iteration's result as input for the next iteration. In other words, no input is ever larger than 1. The process generates the following sequence: xo = 0, xl = 0, x2 =1/2, x3 =1/3, x4 = 11/12, x5 = 1/30, x6 = 7/30, x7 = 64/105, x8 = 289/840 .... If it could be proved that the erratically fluctuating numbers xn are evenly distributed between 0 and 1, log 2 would be deemed normal to base 2. Establishing the same equidistribution property for a different, more complicat- ed dynamical map would lead to a proof that pi is normal to base 16 (or, equiva- lently, to base 2). That would be a signifi- cant step toward the long-sought goal of proving pi's absolute normality. —Preceding unsigned comment added by 141.209.34.49 (talk) 17:54, 12 May 2009 (UTC)

Base conversion.
How were the values of Pi in hexadecimal and binary calculated? Have these values been verified as being correct? 203.211.74.185 (talk) 02:48, 19 May 2009 (UTC)

Rationality of Pi.
The last digit of pi was discovered. Here is the link. —Preceding unsigned comment added by 69.225.243.9 (talk) 01:39, 23 May 2009 (UTC)


 * Although the Japanese team has claimed to obtain the same result three times, that does not by itself prove that pi (π) is rational. There may be a problem with the computer used to evaluate it that caused the result. Only if another digit-hunting team is able to duplicate this result should we really decide to re-evaluate the proofs that pi ls both irrational and transcedental. Glenn L (talk) 03:11, 23 May 2009 (UTC)
 * Did you happen to read the line just below the story? The one that says The Top Ten Satire News Stories of 2008 - as chosen by award-winning journalists and writers from NPR, the BBC, Fox News, and Boston University!?  I'm guessing not :-) —Preceding unsigned comment added by Trovatore (talk • contribs)

Yeah, and a space alien filed a paternity suit against Hilary Clinton.

Just in case any innocent person is reading this page: see Proof that π is irrational.

And, BTW, the terminology in the joke article is clumsily and stupidly incorrect. Michael Hardy (talk) 05:41, 26 May 2009 (UTC)

Three steps to Pi
When a clear and simple understanding of Pi is presented, a 10 year old can probably grasp it with minimal guidance. This is such an explanation of Pi. The Main page is unnecessarily complex.

There were 3 steps to Archimedes determination of Value (3:14..) for Pi

1. Determine the angle at which Arc length equals Radius length. This angle was determined to be 57.2958.. and it was given the term “Radian” 2. Divide Radian value (57.2958..) into Circle value (360) to produce a Radian / Circle ratio. As Radian Arc length equals Radius length, the ratio produced (6.28..)is also a Radius/Circumference ratio. 3. Correct the Radius/Circumference ratio to a Diameter/Circumference ratio. As 2 Radius equal 1 Diameter, the value 6.28.. is divided by 2 to produce value 3.14..

To determine Arc length for 1 degree of Arc, divide Radian value (57.2958..) into Radius length value. Then multiply that determined value by the degrees of Arc under consideration, to determine the length of the Arc.

It should be noted that Pi 3.14.. is also a ratio for Radius / Semi-Circle.

The image of Radian can be found by searching Wiki, Radian. If I have erred in Posting this as a new section, I offer my apology. --Layman1 (talk) 17:51, 8 July 2009 (UTC)


 * I don't see that as reasonably accurate, although, if corrected, it might be usable as a new simplified section (not in the lead). — Arthur Rubin  (talk) 19:25, 8 July 2009 (UTC)


 * Layman1 has demonstrated that the values of (Circumference/Diameter) and (Semicircle angle / Radian angle) are both equal to Pi. That is derivable from the definitions of circumference, diameter, semicircle and radian. It is easy to find an approximate value in degrees of a radian in a reference book but that has nothing to do with actually calculating a value for Pi. The sourced method used by Archimedes does calculate a value for Pi to any desired accuracy. The evidence is that Archimedes had no use for the alleged 3 steps. It is irresponsible to pretend that they were his determination of the value of Pi, not least if one is guiding a child. Cuddlyable3 (talk) 22:32, 8 July 2009 (UTC)


 * The original poster does not allege, I believe, that Archimedes used this method. Rather, it's just a simple method for computing pi to limited precision. The problem with this method is that it assumes you know the angle at which arc length equals radius length - determining this arc length itself is typically done using the value of pi (arc length = 2&pi;r&theta;). If you're talking about measuring a physical circle, it's probably far simpler to just measure a piece of string the length of the radius and then see how many times it goes around the circle. Dcoetzee 02:46, 9 July 2009 (UTC)
 * Woops, looks like they actually did make that claim. Regardless it doesn't seem particularly useful, as I don't see any way of computing the angle in question without the use of pi itself. Dcoetzee 02:49, 9 July 2009 (UTC)
 * Dcoetzee please be more careful. From your page I see that you are no newbie but in fact an admin with respect worthy academic degrees so it is reasonable to expect diligence from you. How could you have ignored the OP's statement There were 3 steps to Archimedes determination.. before your first response? How can you imagine that a piece of string the length of the radius can go around a circle many times when it can't do that even once? You have posted obvious mistakes that should not be here. Cuddlyable3 (talk) 13:06, 9 July 2009 (UTC)
 * The answer to 'how many times' is '1/2pi', which happens to be less that 1 - I don't think that's an unreasonable use of words. As for the other bit, everyone makes mistakes. There's no need to chastise him for it after he's corrected himself. Olaf Davis (talk) 14:00, 9 July 2009 (UTC)
 * That's fine if you think you can see that fractional number of a time. I wouldn't give much for my accuracy in doing so, and Archimedes certainly would not. This is supposed to be how one presents Pi to a 10-year old. The fact that everyone can make mistakes is a good reason to be careful. Cuddlyable3 (talk) 11:30, 10 July 2009 (UTC)
 * What I meant was to successively move the string so that its tail lies where its head was previously, as you would measure a person with a ruler. It's not that amazing of a concept. I realise this is not a sufficiently precise procedure to determine pi to more than a couple digits. Dcoetzee 06:33, 20 October 2009 (UTC)

Reports japanese scientists have met the end of pi ... recurrs
Hoax? —Preceding unsigned comment added by 86.132.229.81 (talk) 14:23, 12 July 2009 (UTC)


 * Do you mean this, discussed above? Olaf Davis (talk) 14:27, 12 July 2009 (UTC)

i am mark kevin tom —Preceding unsigned comment added by 203.87.176.2 (talk) 09:00, 13 July 2009 (UTC)

Please correct this disinformation concerning the Gregory--Leibniz series
Under the subsection "Calculating pi" it currently says

In addition, this series [Gregory--Leibniz series] converges so slowly that 300 terms are not sufficient to calculate π correctly to 2 decimal places.[23]

Nevertheless 300 terms is certainly sufficient. It seems to me that 294 terms is enough to be precise. Also the reference to Lampret's article (ref no 23) is a bit weird. There is nothing in the article that supports the claim about 300 terms.

Could someone with an account correct these things, please. -- 91.156.39.161 (talk) 10:17, 15 July 2009 (UTC)

The "300 terms" citation appears to be nearly a word-for-word quote from Petr Beckmann's A History of Pi (1971-75 with numerous reprints) on page 140. Beckmann wrote about the Gregory--Leibniz series, "its convergence was so slow that 300 terms were insufficient to obtain even two decimal places." Unfortunately, it was not pointed out to him the exact progression, otherwise he would have corrected this before his death in 1993:

Term Adjustment Pi Estimate 291 +4/581 3.1450291 292 -4/583 3.1381680 293 +4/585 3.1450061 294 -4/587 3.1381913 295 +4/589 3.1449825

Clearly the sum of the first 293 terms rounds to 3.15, while the sum of the first 294 or more terms always rounds to 3.14, as pointed out above.Glenn L (talk) 16:03, 15 July 2009 (UTC)

Accurate to 62 places
$$	\frac{80\sqrt{3}g^2((9(5+g)^3+(g-10)^3)(5+g))^\frac{3}{2}}{9((36(5+g)^3+4(g-10)^3)(827g^2(5+g)-\sqrt{89}))-5\sqrt{89}(10-g)}=\pi$$

$$g=\sqrt[3]{500+53\sqrt{89}}$$

source: iamned.com math page

Format problem
These lines appear in the article:



I tried to indent this by putting a colon in front of it. That doesn't work. The failure to indent violates WP:MOSMATH. Is there some simple way to fix this? Michael Hardy (talk) 20:22, 23 August 2009 (UTC)
 * Fixed. Ropata (talk) 13:32, 27 August 2009 (UTC)

Edit protect
Can someone fix citation [17]? Here is a suggested formatted reference which renders as

Thanks. 12.164.217.162 (talk) 12:45, 1 September 2009 (UTC)

Open Questions section
I don't really have a working knowledge of editing wikipedia, but I came across this article recently and thought that it addresses at least partially the question of digit distribution and occurence in pi. Maybe someone could edit this in?

The first digit frequencies of primes and Riemann zeta zeros tend to uniformity following a size-dependent generalized Benford's law by Bartolo Luque, Lucas Lacasa —Preceding unsigned comment added by 142.157.215.139 (talk) 04:21, 4 September 2009 (UTC)

Pi in the Bible
It is an interesting historical fact that there is a reference to the ratio of the circumference of a circle to its diameter being 30/10 = 3 when the bible was written.

''1 Kings 7:23 He [Solomon] made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it. ''

There are a number of ways people react to this: (1) How could an infallible God have an inaccurate approximation? (2) If one interprets the Hebrew word for circumference as a number (using the natural way of doing so) and the Hebrew word for diameter as a number then the ratio, times 3 is an extraordinarily accurate approximation for Pi (given the date that the bible was written.

I am not sure how much of this adds rather than distracts. Personally, I think at least some of this is interesting. Wikinewbie123 (talk) 15:15, 9 September 2009 (UTC)


 * It's covered sufficiently in the article, and even further in History of numerical approximations of π and Rabbi Nehemiah. The very next Bible verse mentions that the wall of the bowl was "one handbreadth" thick, which could have made a difference in the way it was measured . Also, it's not a question of the infallability of of the Judean God, but rather of the inaccuracy of the Hebrew scribes. — Loadmaster (talk) 17:18, 9 September 2009 (UTC)

Good and interesting points all around Wikinewbie123 (talk) 03:02, 10 September 2009 (UTC)

Alternative Geometrical Period Formula
I have created an alternative formula for calculating Pi (which I would like to put on this wiki). It is based on Calculating Pi via Polygons and comes under the sub-section of Calculating π. Would it be okay with everyone to put it on the wiki; and do you all think I have chosen the right sub-section to put it in? The formula is as follows (also on my user page):


 * $$\pi = \lim_{n \to \infty}\frac{1}{2}n\cos\left(90-\frac{180(n-2)}{n}\right)$$

I have also create a reciprocal formula (which may also be posted - undecided as of yet):


 * $$\pi = \lim_{n \to 0}\frac{1}{2n}\cos\left(90-\frac{360n(n-2n^{2})}{n}\right)$$

Read My User Page for more information. If no one relpies within a few weeks I will assume there is no problem with it and go ahead. I just wanted to check with everyone first; and please comment with any additional ideas.

Jaymie 20:40, 16 September 2009 (UTC)


 * If you want to add these formulae, you should provide a reliable source for them. If you have discovered them yourself and cannot cite a third-party source then they may be counted as originbal research, which is not allowed on Wikipedia. Gandalf61 (talk) 08:08, 17 September 2009 (UTC)


 * I agree with Gandalf. Also, in your first formula you can simplify the expression on the right and apply a basic trigonometric transform to get:


 * $$\pi = \lim_{n \to \infty}\frac{1}{2}n\sin\left(\frac{360^\circ}{n}\right)$$ ,


 * which is easy to derive using polygons. I expect that evaluating this using a power-series for sin would be relatively inefficient compared to many direct series for pi, so I wouldn't be surprised to find it didn't appear much in reliable sources. Olaf Davis (talk) 10:05, 17 September 2009 (UTC)


 * Thank you Olaf for showing me the formula could be simplified. I have since further simplified my formula to:


 * $$\pi = \lim_{n \to \infty}n\sin\left(\frac{180^\circ}{n}\right)$$


 * Because of the inefficiency of it, I won't be putting it on the wiki; although it is a neat class-room tick, which was where it was developed. Jaymie (talk) 10:26, 19 September 2009 (UTC)

Aproximating pi with a rational p/q with p,q relatively prime
I have read that |pi - p/q| is never smaller than c*{q^(-42)}, where c is a fixed real number. Perhaps this should be put in the article, but by a more knowledgeable person than me. For all I know, the exponent may have since been improved to something greater than (- 42), that is, even harder to approximate than an algebraic number of degree 42.Rich (talk) 02:27, 7 October 2009 (UTC)


 * If you've read it, tell us where, and we can use it. Dicklyon (talk) 03:03, 7 October 2009 (UTC)


 * Algebraic numbers of degree 42 (or any other irrational algebraic numbers for that matter) have approximation exponent 2, not 42. See Thue–Siegel–Roth theorem. It cannot get any harder than that. — Emil J. 10:22, 7 October 2009 (UTC)
 * oh. Rich (talk) 22:19, 8 October 2009 (UTC)

Every circle contains an equilateral triangle of 60 degree
If a circle has a circumference of 30 then pi=3 circumference divided by squareroot of 12, a constant, equals to the lenghts of triangle. the length of the triangle * 3 is the sum of the perimeter then divided by square root of 12 equals to the heights of the triangle.30 divided by 12=2.5  2.5*4=10 the diametere of the circle.Can't be done without calculator. the circumference of the circle divided by the perimeter of the triangle=1/sin60. three divided by square root of 12 equals to sin60. Twentythreethousand (talk) 16:53, 11 October 2009 (UTC)


 * Is this confusing assertion connected with this sketch ? Cuddlyable3 (talk) 15:26, 12 October 2009 (UTC)


 * circumference of circle of 3 divided by radical 12 equals to sin60
 * sin60 times 3 divided by radical 12 equals to the height of triangle
 * 0.75 divided by three equals 0.25
 * 0.25 times 4 equals to diameter 1
 * 3 divided by 1 equals three

pi —Preceding unsigned comment added by 74.198.10.74 (talk) 22:23, 18 October 2009 (UTC) mesfin mengistu gemechu(that's me) Twentythreethousand (talk) 14:09, 27 October 2009 (UTC)

Good luck I'm a Prophet,I have seen the Lord from psalm 18(which they call him the dragon from the book of job chapter 41,everything under heaven is his)The king of the north and the King of the south.Greeks verses Persia.
 * I have no idea what you are trying to prove. that pi=3? that all circles contain a sort of triangle? what is your assertion and postulate.-- Procrastinating@ talk2me 17:17, 27 October 2009 (UTC)

To say pi equals three is changing all the textbooks in the classroom,and it can't be done whithout a calculator machine.the eye and the pupil of the human eye contains an equilateral triangle and two circles for a proof that pi =3 —Preceding unsigned comment added by Twentythreethousand (talk • contribs) 19:48, 30 November 2009 (UTC)
 * base, sides and heights

:60 degree, Sides of polygons 3 sides=30/radical 12 *3 :30 degree, Sides of polygons 6 (30/radical 12 *sin(30))Square + 2.5 square=5, *6=30 :15 degree; polygons 12 sides 2.5823769308862773429589687286191*12=30.988523170635328115507624743429 :7.5 degree polygons 24 sides :3.75 degree polygons 48 sides :conclusion: 5 the radius times ((sin60,sin30,sin15,sin7.5,sin3.75,sin1.875...)...3*22.... times sides of polygons reaches to the ratio of pi .Twentythreethousand (talk) 17:33, 3 January 2010 (UTC)
 * Lenght of polygons 5*sin60/2x*3*2xfor a radius of 5.

Twentythreethousand (talk) 22:47, 25 December 2009 (UTC) Twentythreethousand (talk) 16:47, 3 January 2010 (UTC)

Removal of infobox
Based upon a discussion at Wikipedia talk:WikiProject Mathematics, I've removed the infobox from the article. If anyone disagrees, could you please join the discussion there. Thanks, Paul August &#9742; 12:36, 18 October 2009 (UTC)


 * I disagree with the removal of the infobox. The information is relevant. Drakcap (talk) 02:26, 19 October 2009 (UTC)


 * So do I. This was done with no consensus of the editors of this article, indeed behind our backs by closing a quick discussion at the Math Project talk page. I restored the infobox. Finell (Talk) 04:14, 19 October 2009 (UTC)


 * Keep the infobox. This talk page is the appropriate place for such a discussion, wikiproject or no. The infobox is a good feature of many articles. __Ropata (talk) 11:29, 19 October 2009 (UTC)

Just a comment: It really doesn't do much good to discuss the issue here. There is an entire community of editors over at Wikipedia talk:WikiProject Mathematics who contribute to the upkeep and improvement of all of the mathematics articles. Since this is an issue that affects seven or eight different articles, it makes more sense to discuss it on the wikiproject page.

In addition to this philosophical argument, it is also simply a reality that the WikiProject Mathematics community does have the power to change this article as it sees fit. For example, the project includes several administrators, who will presumably respect any consensus reached on the project page.

Right now, there is not a consensus on the project page, and if you continue to participate in the discussion there I suspect that you will manage to forge a compromise that everyone can live with. What you need to do right now is read over the objections that have been raised to the infoboxes and propose an alternate form that they could take that addresses some of these concerns. The strategy of trying to move the discussion back here is not going to work out for you. Jim (talk) 16:40, 19 October 2009 (UTC)


 * Jim: Could you please point to a Wikipedia policy or guideline that says that a project can override the consensus of an article's editors? I have not heard of this before. This is a question, not a confrontation. Thank you. Finell (Talk) 03:19, 20 October 2009 (UTC)


 * The thing is, there is no mechanism in Wikipedia for recognizing who constitutes "the editors" of a given article. Anyone can show up to this article at any time and demand changes, and the policy is that an effort must be made to achieve a consensus.  If ten people show up at this article all demanding changes, then collectively they will end up getting their way, unless they can be talked out of it.


 * I sympathize with you that you suddenly have to deal with an entire wikiproject landing on your doorstep demanding changes, and it must be particularly frustrating that they want to discuss the matter on their talk page. However, ignoring the wikiproject is simply not going to work, and the best option is to try to sway the consensus developing on the project page.  (A different option would be to remain hostile, and hope that the members of the wikiproject will grow tired of the argument and move on.)  My guess is that if you were to offer a compromise proposal on the project page for modified infoboxes that eliminate a few of the more obscure constants and arcane information, you would find most of the members receptive and willing to work with you to improve the infoboxes.  If you don't try to reach consensus, it is possible that you will get your way if the wikiproject members give up, but it is also possible that you will lose your chance to participate in the consensus. Jim (talk) 07:00, 20 October 2009 (UTC)


 * At the moment, I don't see anything to compromise with. There doesn't appear to be any consensus toward anything building on the project talk page. If you want to develop a consensus navbar for articles on irrational numbers, and make a real template of it for consistency (not one proposed for the sole purpose of deleting it) to use in all the articles (in place of the 2 or 3 slightly different versions that exist), that's fine with me. If you want to do away with the infoboxes on articles like Pi and Golden ratio, I and several other editors would rather keep them as they are. If the project members decide that that the boxes aren't sufficiently objectionable to warrant the instruction creep of trying to prohibit them altogether, that's OK with me because, as I said, I prefer to keep them as they are. If Project Math takes the position that it can change articles without notice to, or genuine consensus of, the articles' editors (although I recognize that Project Math participants have as much right as anyone to participate in building consensus about articles), I am going to oppose that as a matter of community policy. But let's please be clear about facts:
 * I didn't change articles without consensus.
 * I didn't hold a 20-hour de facto RfC about changing 10 articles without any notice on the articles' talk pages.
 * I didn't replace the short navbars that were long in use to an unwieldy one just to make a point, something that no one proposed or supported on the Project Math talk page.
 * I didn't make a template for the sole purpose of nominating it for deletion to make another point, something that no one proposed or supported on the Project Math talk page (the editor who did it announced it after the fact to solicit votes).
 * I didn't express sour grapes when the template was speedy deleted.
 * I haven't done anything uncivil, disruptive, or in violation of any Wikipedia policy or guideline on the Project Math talk page or anywhere else on Wikipedia in connection with this affair.
 * Paul August had the good grace to accept responsibility for mishandling this and to apologize.
 * Aside from commenting that making a template to propose it for deletion was unproductive, have you done anything to address the behavior of Project Math participants in this matter? Do you think that might be a good idea? Finell (Talk) 20:14, 20 October 2009 (UTC)


 * I don't see that there's anything in particular that I could do at this point to address the behavior of the project math participants. I was offended that they were reverting your reverts, so I did restore the infobox on the the Square root of 2 article a couple days ago.  Though I'm a member of the wikiproject, I've been more of a lurker in this discussion, and didn't enter the fray until I perceived that you were being treated somewhat unfairly.  (In particular, I was disturbed by what I saw as an emerging dispute.)  I don't have a strong opinion on the infoboxes one way or the other, and it's hard to tell at this point whether the folks you were arguing with on the wikiproject talk page are intending to press the point any further.


 * If it helps, I would like to apologize for the behavior of the wikiproject members in this matter, particularly the zeal with which the infoboxes were initially deleted. If you hadn't complained on the project talk page, probably no one would have even noticed that you were objecting.  If the discussion does continue on the project page, I do hope that you and the other editors here will participate. Jim (talk) 03:14, 21 October 2009 (UTC)

I think some of the remarks above are a tad unfair or misleading. Here are some facts:


 * 1) I removed the infobox at e (mathematical constant):.
 * 2) Robo37 reverted with the edit summary "the same template is used in the aticle about pi and all of the other irrational numbers of interest":.
 * 3) I started a discussion at Wikipedia talk:WikiProject Mathematics saying that the infobox "doesn't seem to me to add much of use to the article (as well as the fact that the links listed seem a bit arbitrary)" and asking for other editor's opinions about that infobox as well as the one at pi:.
 * 4) Five editors (Oleg Alexandrov, Ozob, Hans Adler, RDBury and  Shreevatsa)  responded all expressing some level of opposition to the infoboxes, no editor seemed to be in favor of keeping the infoboxes:.
 * 5) Having decided that enough of a consensus existed for me to remove the infoboxes, I announced my intentions to do so:.
 * 6) I began removing infoboxes from 10 articles: E (mathematical constant), Pi, Apéry's constant, Feigenbaum constants, Euler–Mascheroni constant, Golden ratio, Silver ratio, Square root of 2, Square root of 3 and Square root of 5:
 * 7) Fimell posted an objection to the removal of the infoboxes and the way it was done and announced his intention to restore the infoboxes:.
 * 8) Algebraist posted in seeming opposition to the infoboxes:.
 * 9) Dicklyon, after being notified of the discussion by Finell,  posted in support of the restoration of the infoboxes "primarily because they were removed with insufficient consensus or even notice on the relevant articles":.
 * 10) David Eppstein began removing the infoboxes with the edit message "Please build consensus at WT:WPM rather than unilaterally restoring this useless cruft.": . (At this point support for removal on the talk page might be reasonably viewed as 8-2, although to be fair to David, since their edits were perhaps only seconds apart, I think it is probable that David had not seen Dicklyon's post prior to making this edit, and so might have reasonably thought that the only objecting editor was Finell.)
 * 11) David ultimately removed infoboxes from 6 of the 10 articles: Euler–Mascheroni constant, Golden ratio, Silver ratio, Square root of 2, Square root of 3 and Square root of 5:
 * 12) Robo37, created a new infobox with seven more constants: Champernowne constant, Prouhet–Thue–Morse constant, Laplace limit,  Catalan's constant, Mills' constant, Khinchin's constant and Reciprocal Fibonacci constant, and began adding it back to all but Silver ratio, of the original 10 articles, as well as to the 7 new articles:.
 * 13) Dicklyon began to restore the articles to their original states:
 * 14) Jim.belk restores Square root of 2:
 * 15) Finell retores Silver ratio:

I've said elsewhere that I handled this badly, and I apologized saying "It would have been better if I had waited for more editors to comment before I removed the infoboxes, and it would have been better if I had publicized the discussion I started ... more widely":. In a second post I accepted responsibily for causing this mess saying that the situation had gotten off to a bad start "for which I [was] willing to accept the blame". So if anyone feels a need to assign blame or "address ... behavior" please assign it to me or address mine. But I think it goes too far to castigate other editors in the Mathematics project, or the project as a whole. And as for Jim's comment "they were reverting your reverts" I would just point out that only one editor was reverting and as I've indicated above a reasonable argument can be made to justify those edits. I understand that reasonable people might disagree with this. In any case I think we should dispense with any more recriminations.

Paul August &#9742; 19:38, 21 October 2009 (UTC)

Liu
The page is semi-protected and I cannot alter it. Please improve the outcome of Liu Hiu''s formula to 3.1415894, the present outcome is too rough. Weia (talk) 23:09, 14 December 2009 (UTC)

Celebration coming up!
Soon there will be 3,141,592 Wikipedia-Articles, right? ;-) Grey Geezer 21:06, 27 December 2009 (UTC) —Preceding unsigned comment added by Grey Geezer (talk • contribs)

Pi in science fiction literature
Every now and then I run into a science fiction novel that supposes the value for pi can vary according to a gravitational constant, or a local "curvature" in the universe. Um, I don't think so, but it is getting kind of predictable that this sort of thing keeps popping up in science fiction literature.

The main article could be improved if somebody put a link in there, connecting it to another Wiki article about Pi in science fiction literature. For instance, The Infinite Man by Daniel F. Galouye is one such novel that uses this as an essential part of its plot. His story proposes that computers using standard calculations, according to a standard formula, suddenly start spitting out a different sequence of decimal digits, because the whole universe starts changing its density levels, apparently in response to a supreme being deciding to change the amount of mass in the universe.

It isn't my purpose to argue against one plot in support of another, in terms of science fiction literature employing plausible plot lines, merely that this kind of thing apparently keeps happening, and among different writers around the world. Science fiction authors keep arguing that the formulas for calculating pi, as applied, produces results that are consistent with observable data. Dexter Nextnumber (talk) 07:50, 1 January 2010 (UTC)

Perhaps in the open questions part of the article would be a reasonable place to place this type of conjectural suggestions of variant values for Pi. Even though they are not really open questions it is the place where the case might be explored. It is certainly interesting to imagine what other universes might be like if they had sufficiently different conditions that their values of Pi could be unique. What would it mean if Pi<1, or Pi=1, or Pi=Infinity? What would it mean if Pi were not a constant? What would it mean if Pi could change its sign? —Preceding unsigned comment added by 76.11.118.129 (talk) 03:41, 21 February 2010 (UTC)


 * As &pi; is a mathematical (not physical) constant, it does not depend on the physical properties of the universe, it depends on logic. It is defined in terms of idealized mathematical objects. You may be thinking of a different constant such as c or G or perhaps Λ. You can't change the value of &pi; unless you redefine circles, spheres, sines and cosines, complex numbers, etc. It sounds like the book is more fiction than science. Ropata (talk) 04:46, 20 March 2010 (UTC)


 * Pi is a mathematical constant, yes. In Euclidean space, it is the circumference-to-diameter ratio of a circle.  In curved space, this ratio, which could be interpreted as "pi", would be different (e.g. the equatorial circle in Riemannian geometry has a ratio of 2).  However, since computers don't calculate pi by measuring circles, their numbers shouldn't change.  —Preceding unsigned comment added by 174.47.110.67 (talk) 19:51, 9 April 2010 (UTC)


 * If you can find a reliable (non fiction) source for these speculations, they could possibly be added to the "open questions" section. But it looks a lot like WP:OR to me. Ropata (talk) 13:27, 23 April 2010 (UTC)


 * Supposing that in curved space and riemannian geometry the smoothness of the manifold in question is included, then yes, the ratio of circumference to radius may vary. But in the limit of very small radii, the local view of the curved space becomes—by smoothness—indistinguishable from euclidean space, so the ratio goes under this limit back to the well-known value of pi.--LutzL (talk) 14:35, 23 April 2010 (UTC)


 * I was going to comment on the introductory classification of pi as a "mathematical and physical" constant, and this talk section fits right in. I thought these were exclusive categories. As Ropata explained, pi is mathematical and not physical. If someone notable uses these words differently, I propose that a citation be requested. (Collin237) —Preceding unsigned comment added by 166.217.168.155 (talk) 13:57, 22 July 2010 (UTC)

Definition using the circle
While this is historically the correct way, and quite possibly the way the article should go, it is also dangerous, because it gives the layman the wrong impression. It is much more fruitful to consider pi an abstract constant (possibly defined through exp(i*pi)=-1), which happens to also give a certain relationship in a circle.

I would advice that, at a minimum, the introduction stresses that the "traditional" definition is a historical left-over and ultimately only a secondary characterization of pi. 188.100.196.8 (talk) 00:26, 7 January 2010 (UTC)
 * Are you sure? I think an analytic definition, while perhaps more rigorous, is still secondary to the very basic properties of circles. Ropata (talk) 04:53, 20 March 2010 (UTC)

I don't know why my edit to this page was removed. I have never seen the removal of constructive information from a talk page before. I relevantly submit that a useful definition for pi is as the only root of the sin function between 1 and 4, where sin is defined by its power series. The existence of such a root is guaranteed by the intermediate value theorem. This is the method by which pi is defined in at least two well known math volumes I'm familiar with (Principles of Mathematical Analysis by W. Rudin; Functions of One Complex Variable by Conway). —Preceding unsigned comment added by 75.140.4.134 (talk) 07:20, 17 August 2010 (UTC)


 * I wouldn't have (intentionally) removed it. However, it clearly doesn't belong in the article, and I would have said so.  — Arthur Rubin  (talk) 08:29, 17 August 2010 (UTC)

Polygons inscribed in a circle
A simplest way to calculate pi is to observe polygons (polygons from three sides, to polygons increasing by multiple of two) inscribed in a circle and watch the height and the base of the sides of the polygons and use the theorem of Pythagoras to find the degree of the angles. The more sides there are in the polygons and the less the degree of the angle of the triangles. Since the angles start with 60 degree and diminish by multiple of two and the sides increase by multiple of two we have an equation: Sin (60/2^x) * 2^x *3=pi
 * $$\pi = ( \sin (60/2^x) ( 2^x)(3)) \,$$
 * $$\ ( \sin A ) / ( \cos C)=1 \,$$
 * $$\ ( \sin 30 )/( \cos 60)=1 \,$$
 * $$\pi= (\sin (180/2^x) (2^x) \,$$

Angles of triangles of polygons.
 * 30+90+60,-60/30=2
 * 15+90+75,-75/15=5
 * 7.5+90+82.5,---82.5/7.5=11
 * 3.75+90+86.25,--86.25/3.75=23
 * 1.875+90+88.125 88.125/1.875=47
 * From 2 to 5=3
 * From 5 to 11p=6
 * From 11 to 23 =12
 * From 23 to 47 =24

……Twentythreethousand (talk) 20:57, 11 January 2010 (UTC)


 * $$\ (60/30=2)... (90/30=3)...  (90/60=3/2) \,$$
 * $$\ (75/15=5)... (90/15=6)... (90/75=6/5) \,$$
 * $$\ (82.5/7.5=11)... (90/7.5=12)... (90/82.5=12/11) \,$$
 * $$\ (86.25/3.75=23)... (90/3.75=24)...  (90/86.25=24/23) \,$$

Twentythreethousand (talk) 00:25, 15 January 2010 (UTC)

you can check what it looks like on a graph with radians and degrees and the graph is a constant on pi and on 180. you can make a graph with this equation.Twentythreethousand (talk) 00:06, 22 January 2010 (UTC)
 * $$\pi = ( \sin (6/999999999......) )*3 \,$$
 * $$\pi = ( \sin (60/x^k) ( x^k)(3)) \,$$

Twentythreethousand (talk) 19:30, 31 January 2010 (UTC)
 * $$\pi = ( \sin (6/100000000......) )*3 \,$$


 * $$\ 0.6--90--89.4 90/0.6=150---89.4/0.6=149---90/89.4=150/149  \,$$
 * $$\ 0.06-- 90 --89.94 90/0.06=1500---89.94/0.06=1499---90/89.94=1500/1499 \,$$
 * $$\ 0.006--90--89.994---90/0.006=15000---89.994/0.006=14999-- 90/89.994=15000/14999 \,$$
 * $$\ 0.0006 --90--89.9994 -- 90/0006=150000--- 89.9994/0.0006=149999--- 90/89.9994=150000/1499999 \,$$
 * $$\pi = ( \sin(0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000.......18)*10^n) \,$$
 * $$\pi = ( \sin(0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000.......6)*3)*10^n \,$$

Twentythreethousand (talk) 01:13, 2 February 2010 (UTC)
 * $$\pi= ( \sin(1/10^n*18)*10^n) \,$$

1degree,0.1degree,0.001degree,0.0001degree,$$1/10^n degree \,$$ Twentythreethousand (talk) 16:24, 12 February 2010 (UTC)


 * May I ask what this has to do with improving the article? We know
 * $$\pi = 180 \lim_{x \to 0} \frac {\sin x^{\circ}} {x} .$$
 * — Arthur Rubin (talk) 17:24, 12 February 2010 (UTC)

dear Arthur Rubin you said you know,but your equation is not even in the article and what I have stated in this discussion group is not hard trigonometry and not even hard integral calculus $$\pi.$$ If the calculator machine allowed an irrational number without any increase or a decrease of the number which is a straight line and a measure of a distance that require only two or three digits,what's the use of calculating pi to the billions of digits.Twentythreethousand (talk) 18:09, 13 February 2010 (UTC)


 * Perhaps it should be in the article, but it's a trivial consequence of:
 * $$\lim_{x \to 0} \frac {\sin x} {x}=1$$
 * and
 * 180 degrees is &pi; radians.
 * The latter probably should be somewhere in the article. (Actually, it is, in the infobox, and in a picture caption.) — Arthur Rubin (talk) 19:13, 13 February 2010 (UTC)


 * $$\lim_{x \to 0} \frac {\sin x} {x}=1$$ x can not equal to 180 in this case.It's 0.174.119.27.38 (talk) 19:55, 13 February 2010 (UTC)


 * You can believe what you want, but List of trigonometric identities has the limit I wrote, and it's unlikely you will be able convince people otherwise. — Arthur Rubin  (talk) 20:21, 13 February 2010 (UTC)
 * the derivative would be the inverse of a number.If in radians pi equals to 0 and 180 the same and we're using 1 and 0 there not much difference with degrees.You have three colors in a magnetic fields and one color has an opposite or inverted color of the other.Both proof are the same.Twentythreethousand (talk) 21:27, 13 February 2010 (UTC)

one other thing Plato the republic and criticism the major statement by Charles Kaplan about the creator and the imitator there was a movie called Coming to America that was made relating to the book.For the last generation.I am not the creator of math.We all live in a circle. —Preceding unsigned comment added by Twentythreethousand (talk • contribs) 18:39, 14 February 2010 (UTC)

New Interesting Formula
I suggest adding formulas 1.1 and 2.13 from http://iamned.com/math/infiniteseries.pdf to the main article They are interesting —Preceding unsigned comment added by 71.139.200.147 (talk) 06:14, 17 January 2010 (UTC)

$$

\pi2^{10}\sqrt{3}=\sum_{k=0}^\infty  \frac{1}{9^{k}\tbinom{8k}{4k}}	\left ( \frac{5717}{8k+1}-\frac{413}{8k+3}-\frac{45}{8k+5}+\frac{5}{8k+7} \right )

$$

$$

\pi=\frac{\sqrt{3}}{60}\sum_{k=0}^\infty \frac{(2k)!(130k+109)}{(\frac{7}{6})_{k}(\frac{11}{6})_{k}(-1296)^{k}}

$$

Extremely Accurate Approximation
$$ \pi-7.66*10^{-63} \approx \ \frac{4\sqrt{3z(5+g)}}{9} \left (\frac{-\sqrt{89}}{g^{2}(5+g)}\left ( \frac{9(10-g)^{2}}{4z}+\frac{1}{5} \right )+\frac{827}{5}

\right )^{-1}

$$

$$ \begin{array}{lcl}

z=9(5+g)^{3}-(10-g)^{3} \\

g=\sqrt[3]{500+53\sqrt{89}} \end{array}

$$

—Preceding unsigned comment added by 67.161.40.148 (talk) 05:44, 19 January 2010 (UTC)


 * So, you managed to get 63 significant digits of pi by a complicated formula with 80 symbols. Well, guess what: I can do away with only 64 symbols in a much easier way. I leave it as a simple exercise to the reader. — Emil J. 11:06, 19 January 2010 (UTC)
 * For those like Finell who do not get it, the solution is "3.14159265358979323846264338327950288419716939937510582097494459". — Emil J. 11:51, 19 January 2010 (UTC)

The often quoted ramanujan approximation uses 13 digits to get 10 places: http://mathworld.wolfram.com/images/equations/PiApproximations/Inline49.gif

there's a bunch of them here: http://mathworld.wolfram.com/PiApproximations.html

The only really good one is http://mathworld.wolfram.com/images/equations/PiApproximations/Inline77.gif but it's a coincidence due to the continued fraction expansion of pi^4


 * To me the problem with the MathWorld's "Pi Approximations" page is that most of them seem to be mathematical coincidences. Even when there is a deeper mathematical reason for the approximation, it may still be a long way from a practical method for computation.--RDBury (talk) 13:57, 20 January 2010 (UTC)

@RDBury From the work I;ve done those approximations are either coincidences or in the case of Ramanujan derived using elliptic integrals. All expressions that don't involve logarithms are constructable, but obtaining the approximation probably done though trial and error via a computer without an underlying theory. As for computation, you wouldn't use a pi approximation, but a pi formula. —Preceding unsigned comment added by 67.161.40.148 (talk) 16:25, 20 January 2010 (UTC)

Einsten the first to explain river meandering?
The text currently says that Einstein was the first to suggest that rivers have a tendency towards an ever more loopy path because the slightest curve will lead to faster currents on the outer side, which in turn will result in more erosion and a sharper bend. He may well have been the first to discover the connection with pi but I cannot believe he was the first to explain the process, even though the cited source claims this. Martin Hogbin (talk) 18:44, 30 January 2010 (UTC)

Economics
Strangely enough π is also used in economics to represent profit. l santry (talk) 16:11, 4 February 2010 (UTC)

Computation in the Computer Age
The ploufe formulas need to go. You wouldn't use them to actually compute pi. They don't seem to fit in with the overall flow of the article. —Preceding unsigned comment added by 67.161.40.148 (talk) 05:38, 17 February 2010 (UTC)

Furthermore, the physicist using 39 digits to draw a circle of known universe is highly ambiguous, fully unsourced, and seems to be one of those 78% of all statistics that are made up. —Preceding unsigned comment added by 71.180.59.107 (talk) 05:15, 24 March 2010 (UTC)

New chapter to Pi
I think adding a chapter named "computing pi" addressing historical and computational aspect of this number would have a stand in this article. As it's been a difficult topic for a long time in history. Also I noticed that it's been already addressed in other articles like,

Computation of π

In one of his numerical approximations of π, he correctly computed 2π to 9 sexagesimal digits.[9] This approximation of 2π is equivalent to 16 decimal places of accuracy.[10] This was far more accurate than the estimates earlier given in Greek mathematics (3 decimal places by Archimedes), Chinese mathematics (7 decimal places by Zu Chongzhi) or Indian mathematics (11 decimal places by Madhava of Sangamagrama). The accuracy of al-Kashi's estimate was not surpassed until Ludolph van Ceulen computed 20 decimal places of π nearly 200 years later.[1] in the Kashi's article. Repsieximo (talk) 19:45, 14 March 2010 (UTC)


 * See Pi. Ropata (talk) 04:58, 20 March 2010 (UTC)

pronounciation
why not writing something about pi's pronounciation ? (i am french, we pronounce it like pea) —Preceding unsigned comment added by 77.200.68.70 (talk • contribs)  17:29, March 14, 2010
 * It is pronouncedin the United States like the singlular noun pie. —Preceding unsigned comment added by 66.30.185.201 (talk • contribs) 19:26, March 14, 2010
 * I'm fairly sure that's true in the entire English-speaking world. Even people who say "psee" and "ksee" still say "pie", because otherwise, how would you distinguish from the Latin letter P? --Trovatore (talk) 20:20, 14 March 2010 (UTC)
 * How do Italians distinguish their P from Π? They just call the last one "pi greco"! Not a big deal to say /gri:k pi/ in English... — Mikhail Ryazanov (talk) 07:21, 2 February 2011 (UTC)

It's pronounced like the Greek letter pi. Only pronunciation in the OED is /paɪ/. — kwami (talk) 16:10, 30 January 2011 (UTC)

Minor semantic error
editsemiprotected

There is a minor error in the section titled "Decimal Representation", third paragraph, first sentence. It says, "Because π is an irrational number, its decimal representation does not repeat, and therefore does not terminate." This does not make sense because repeating and terminating are mutually exclusive concepts. A number must either repeat or terminate. It must do one or the other and it cannot do both. I would suggest rewriting this to say "Because π is an irrational number, its decimal representation does not terminate, and therefore repeats indefinitely." —Preceding unsigned comment added by Fkento (talk • contribs) 14:15, 15 March 2010 (UTC)


 * I won't remove the edit request, but "terminating" decimals are often considered to be "repeating" with all 0s or all 9s at the end. — Arthur Rubin  (talk) 14:24, 15 March 2010 (UTC)


 * (e/c) The reformulation does not make sense, the fact that the decimal representation does not terminate does not imply that it repeats, it fact, the whole point of the sentence is that the decimal representation of π does not eventually repeat. The original statement is correct, since termination of the decimal representation of a number means the same thing as the decimal representation ending with repeated 0.—Emil J. 14:26, 15 March 2010 (UTC)

Yes, you are correct. I realized my mistake a few hours later but not soon enough to delete the comment. You can remove the edit request if you so desire. Thanks Fkento (talk) 18:45, 15 March 2010 (UTC)

Delations
A bunch of stuff has been deleted uncessarily from he pi discussion pages —Preceding unsigned comment added by 67.161.40.148 (talk • contribs)
 * Do you mean this junk? Or are you referring to the material archived in January? Mind  matrix  14:49, 24 March 2010 (UTC)

Edit request
{{tn| The value of PI was first calculated by an Indian mathematician Budhayan, and he explained the concept of what is known as the Pythagorean Theorem. He discovered this in the 6th century, which was long before the European mathematicians.

Amitsinghsodha (talk) 05:03, 26 March 2010 (UTC)


 * You must supply references to reliable sources for verification.  Chzz  ►  06:52, 26 March 2010 (UTC)

{{notdone}}


 * "See": Budhayan aka Baudhayana here in Wikipedia - with a number of other references from a Google search. Btw, these pages for variations of his name need to be consolidated
 * Robert Pollard (talk) 20:17, 5 May 2010 (UTC)

{{tn| Please remove repetitive information from the following consecutive sentences:


 * Both Legendre and Euler speculated that π might be transcendental, which was finally proved in 1882 by Ferdinand von Lindemann.
 * The transcendental nature of π was proved by Ferdinand von Lindemann in 1882.

Robert Pollard (talk) 17:50, 5 May 2010 (UTC)

Numerical approximations
In the section "Numerical approximations", it reads:
 * The approximation 355⁄113 (3.1415929...) is the best one that may be expressed with a three-digit or four-digit numerator and denominator; the next good approximation 103993/33102 (3.14159265301...) requires much bigger numbers, due to the large number 292 in the continued fraction expansion.

But 103933/33102 is NOT the next good approximation, because:

\begin{array}{rcl} \frac{355}{113} - \pi   &=& +2.667641890 \times 10^{-7} \\ \frac{52163}{16604} -\pi &=& -2.662132574\cdots \times 10^{-7} \\ \frac{103993}{33102}-\pi &=& -0.005778906\cdots \times 10^{-7} \\ \end{array} $$ Thus, 52163/16604 is closer to $$\pi$$ than 355/113. Since both the numerator and denominator of this are smaller than 103933/33102, the latter cannot be the next good fraction.

If what "good" means here is the number of correct decimal places, then compare:

\begin{array}{rcl} \frac{355}{113}    &=& 3.141592\,920353\cdots \\ \frac{86953}{27678} &=& 3.141592\,600621\cdots \\ \frac{103993}{33102}&=& 3.141592\,653011\cdots \\ \pi                &=& 3.141592\,653589\cdots \\ \end{array} $$ Again, there is a fraction, namely 86953/27678, which is correct to more decimal places than 355/113, but has numerator and denominator both smaller than 103993/33102. So, again, this latter fraction cannot be the next good fraction.

The explanation in the Wiki page reads:
 * ..., due to the large number 292 in the continued fraction expansion.

That's the culprit. Continued fraction is not the sole method of finding fractions close to $$\pi$$. The "better" fractions that I give above were found using Stern-Brocot tree.

219.79.176.121 (talk) 15:36, 21 April 2010 (UTC)
 * Well, continued fractions do determine all the best rational approximations, but one has to do it properly. The criterion is described at Continued fraction. In particular, the next best rational approximation to π after [3;7,15,1] = 355/113 should not be [3;7,15,1,292] = 103993/33102, but either [3;7,15,1,147] or [3;7,15,1,146]. Now, the latter indeed gives 52163/16604, in agreement with what you found.—Emil J. 16:00, 21 April 2010 (UTC)
 * And, according to the last paragraph of that section, the (continued fraction) convergents to &pi; are best approximations in an even stronger sense: n/d is a convergent for &pi; if and only if |d&pi; &minus; n| is the least relative error among all approximations m/c with c ≤ d; that is, we have |d&pi; &minus; n| < |c&pi; &minus; m| so long as c < d. — [ Unsigned comment added by Arthur Rubin (talk • contribs) 16:40, 21 April 2010 (UTC).]

On average, the number of significant places of the "best" approximation will be close to the sum of the digits of the numerator and the denominator of the approximating fraction. This can be understood as reasonable by thinking of throwing darts so they "stick" in a line 1 unit long. If there are 10000 darts then the average distance between pairs of darts is going to be 1/10000. Likewise when you think of the number of choices of possible fractions using arbitrary numerators and denominators, the estimate is simply the product of the numerator times the denominator, thus giving us a rough estimate of what to expect. This works very well except we now have the problem that 355/113 is more accurate than we would usually find for the number of digits. This is easily explained by the dart analogy because occasionally the darts will strike considerable closer to each other than what is average, and that is exactly how the 292 comes up. As the digits of Pi seem to be more or less random I'm sure a good probabilistic analysis of the likelihood of the 292 coming up would not be any earth shaking profundity.

What the article calls "Rational Approximations", are derived directly from what the article calls "continuing fraction" (I prefer to think of these as the terms of a continuing fraction, they are a series, not a fraction). In fact, given the terms of a continuing fraction series as far as it has been calculated, along with the residue left over from deriving the terms thus far, we have the complete and "lossless" description of PI because one can be exactly derived from the other in either direction (e.g. they contain exactly equivalent information).

Without proving the rational approximations derived from the continuing fraction series of Pi produce the best approximations of Pi, it is easy to show that it is reasonable that they are. In particular, any candidate fractional approximation of Pi that is not equal in value to any of the approximations of Pi derived from continuing fraction series of Pi, must itself have its own continuing fraction series. That series will be different from the continuing fraction series of Pi and if that series is extended in attempt to approximate Pi, it will fail to do so because it must converge to a different value than Pi. This is because process is reversible and if it approximates Pi it would exactly produce the unique continuing fraction series of Pi, contradicting the condition which said it was not derived from the continuing fraction series of Pi.

Stronger arguments can be made, but this one is more fun.

99.22.92.218 (talk) 10:35, 30 April 2010 (UTC)