Talk:Pi

Section "Definition"

Most of the preceding thread is about section § Definition. This resulted in dramatically shortening this section, and removing all other definitions than ⁠$C/d$⁠. I expanded the section by explaining why this definition is not fully satifactory in modern mathematics and providing elementary non-geometric definitions.

As further improvements are certainly possible, I open this thread for discussing them. D.Lazard (talk) 16:18, 2 October 2025 (UTC)

In case it wasn't clear above, when I said that the version from 2012 was probably overall better for the article than the version from recently, I didn't mean that we had to revert to that, but only that I think we should prioritize keeping things somewhat streamlined and as accessible as possible in the top few sections, instead of trying to immediately trying to impress readers by how sophisticated we can be. I don't have a problem with mentioning alternate definitions early on or with unpacking trickier topics further into the article. I think we can still make this section more concise though. Perhaps along the lines of:

π is commonly defined as the ratio of any circle's circumference C to its diameter d: $$\pi ={\frac {C}{d}}$$ The ratio of circumference to diameter is the same number for any circle in Euclidean space, whatever its size. For example, if a circle has twice the diameter of another circle, it will also have twice the circumference.

In modern mathematics, this definition is not fully satisfactory, because it relies on measuring the length of a curved line, which is difficult to define rigorously, requiring at least the concept of a limit, and typically based on the calculus concepts of derivatives and integrals. π also routinely appears in non-geometrical branches of mathematics, and the modern trend in mathematics is to establish geometry in terms of numerical coordinates rather than treating it independently.

Analytically, π can be defined as the smallest positive zero of the sine function; that is, ⁠$\sin \pi =0$⁠ and π is the smallest positive number with this property. The sine function can, itself, be defined by an integral or by its Taylor series, or as a solution of the differential equation ⁠$f''+f=0$⁠ which characterizes simple harmonic motion. π is half the fundamental period of any solution of this differential equation.

–jacobolus (t) 17:09, 2 October 2025 (UTC)

This is fine to me: the important ideas are kept, and the phrasing is much better than mine. D.Lazard (talk) 09:20, 3 October 2025 (UTC)

I'm a little wary of speaking about the "modern trend in mathematics". And maybe differential equations are a bit too much to drop on young/inexperienced readers this early in the article. But overall this looks like a good approach. Stepwise Continuous Dysfunction (talk) 00:06, 4 October 2025 (UTC)

Feel free to rephrase, rewrite, etc. I was trying to keep the thrust of D.Lazard's addition while making it a bit more concise and direct, hopefully easier to follow for a lay reader who wants to try and also easier to skim past, but without leaving the mathematicians feeling left out. –jacobolus (t) 00:28, 4 October 2025 (UTC)

How about this: The sine function can, itself, be defined without referring to circles or triangles, for example by using its Taylor series. And then talking about differential equations in the next sentence. Stepwise Continuous Dysfunction (talk) 17:20, 4 October 2025 (UTC)

I'm not liking the current Definition section. First, for most people π = C/d is a great! definition. But the section admits that it isn't true except in ...what?...(2d+) Euclidean Space? So, I'd suggest qualifying the constancy claim. Second, "unsatisfactory"? Says who??, citation please. (Satisfaction is an arbitrary mental state, in this context claims about it are unfalsifiable.) The sentence about limits and (derivatives, etc.) is misplaced and I think in error. For instance, the sine function uses limits. You shouldn't suggest use of limits is unsatisfactory and then use them pretending to circumvent them (no pun intended?). So, the use of the differential equation just hides the use of limits, doesn't dispense with it. I DO like the mention that in some non-Euclidean metric spaces (if I've got that right?) C/d is not constant! That's good and a valid reason to want a better more universal definition. So, I'd suggest saying that for simple geometric uses C/d is fine, but a more rigorous mathematical definition, yes, using limits, is preferred for ...whatever...modern analysis? and blah blah blah. (I might also mention that assuming the trig functions aren't geometrically based seems flawed to me. I don't argue that they're best formally defined algebraically but that many readers were taught that they're the ratios of sides of right-triangles. I mean that sine(0) is 0, so "first positive zero" while true requires understanding that zero isn't positive...is it? and $f''+f=0$? A differential equation? Really?!? If you're gonna claim that these are 'better' definitions, then you should (imho) explain why. That is, give us readers a break and tell us why these more complex definitions are better. The second bullet point that it's the smallest possible difference is completely ridiculous - imho. I'm no mathematician, but I've got a pretty good understanding of general math and I have no idea what this means. The smallest "possible" ?? Whaaat?? The difference between THREE zeros??? (I don't understand why the tangent function is included here. Tangent = sine/cosine (just check the 1st illustration in the Trigonometric function article) and when cosine = 0 how exactly do you define the tangent? I'd dump it as without any pedagogical merit here. ~2026-92463-4 (talk) 05:53, 16 March 2026 (UTC)

I agree that the section should be rewritten more carefully. In ordinary Euclidean geometry, pi = C/d is the standard geometric characterization and is probably the most intuitive one for most readers. But if the article wants to discuss the mathematical definition, it should explain the issue more precisely. The problem is not that C/d is "wrong", but that it is not a primitive definition, since the circumference of a circle already presupposes a notion of arc length. In classical geometry, comparing a curved line with a straight segment requires additional assumptions beyond Euclid's basic framework with which high-schoolers would be familiar. In modern analysis, arc length is instead defined by limits or integrals, and circumference is then derived from that definition. So I think the article should distinguish clearly between the elementary Euclidean characterization π = C/d and the more foundational analytic definitions. Sławomir Biały (talk) 08:19, 16 March 2026 (UTC)

Semi-protected edit request on 15 November 2025

pi is not trillions of digits it is infinite numbers after the decimal. ~2025-33517-54 (talk) 22:40, 15 November 2025 (UTC)

 Not done: Pi has been approximated/calculated out to trillions of digits so far. The statement is correct as written. —C.Fred (talk) 22:44, 15 November 2025 (UTC)

pi is finite Cenchros Roseus (talk) 20:39, 23 March 2026 (UTC)

"2π" listed at Redirects for discussion

The redirect 2π has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2025 December 14 § 2π until a consensus is reached. MEN KISSING (talk) 05:49, 14 December 2025 (UTC)

Today's Featured Article requested for pi day

I've posted a request for this article to appear on the front page of English Wikipedia on pi day, March 14, 2026. The request is at WP:Today's featured article/requests/pi. Anyone is welcome to improve the blurb text there, but please keep the character count between 925 and 1,025 characters.

Between now and March 14th, we should endeavour to keep the article in nice condition, satisfying FA criteria, and free of edit-wars or "needs improvement" tags. If someone thinks the article is needs some work, yet do not have time to remedy the article themselves, post a note here in the Talk page so others can make the improvement.

If anyone wants to make a major or contentious changes to the article, it would be nice if you could wait until after March 14.

Question: the caption below the graphic in the top-right InfoBox says "A visual proof of the value of pi". That doesn't seem accurate to me (is it a "proof"?). Does anyone else think the caption should be improved? Noleander (talk) 21:01, 4 January 2026 (UTC)

I agree that 'visual proof' is wrong. How about 'visual demonstration' or 'illustration'? Murray Langton (talk) 22:21, 4 January 2026 (UTC)

Yes, either of those sound more appropriate. Noleander (talk) 23:41, 4 January 2026 (UTC)

Claim of irrationality statement by Aryabhata

The reference says: "This approximation is accurate up to five decimal places. Aryabhata used the term āsanna to indicate that the value is not exact but rather an approximation. This is possibly the earliest reference to the irrationality of pi." Well, no, it is not a reference to irrationality at all. Being approximate is not the same as being irrational. Imaginatorium (talk) 14:09, 15 February 2026 (UTC)

What is the value of pi?

This article shows the 20 digits after the decimal point and the info box shows 25. Should we show more?

SFU link. https://www.cecm.sfu.ca/organics/papers/borwein/paper/html/local/billdigits.html

3.1415926535 8979323846 2643383279 5028841971 6939937510

 5820974944 5923078164 0628620899 8628034825 3421170679
 8214808651 3282306647 0938446095 5058223172 5359408128
 4811174502 8410270193 8521105559 6446229489 5493038196
 4428810975 6659334461 2847564823 3786783165 2712019091
 4564856692 3460348610 4543266482 1339360726 0249141273
 7245870066 0631558817 4881520920 9628292540 9171536436
 7892590360 0113305305 4882046652 1384146951 9415116094
 3305727036 5759591953 0921861173 8193261179 3105118548
 0744623799 6274956735 1885752724 8912279381 8301194912

Vanguard10 (talk) 04:20, 16 February 2026 (UTC)

Don't see much encyclopedic benefit to more digits. That is the kind of data that is best provided to readers via a URL to an external web site. If twenty digits is good enough for NASA's calculations, it is good enough for an encyclopedia. Noleander (talk) 04:31, 16 February 2026 (UTC)

Totally agree with user:Noleander. Just for the record, if you calculate the circumference C of the Earth's orbit around the Sun using r = the average Sun-Earth distance and C = 2*pi*r, the error from rounding your pi value to 50 decimal places is a few nonillionths of an Ångström, or about 0.0001 Planck length. For the Sun's orbit around the centre of the Milky Way, the error would be a few sextillionths of an Ångström, and for the circumference of the known universe, the error is a few quadrillionths of an Ångström. There is no particular reason to stop at 15 decimals, or 50 decimals, or 500, or 1000000, but as long as we link sources that give loads of decimals, anything beyond 15 decimals (say) is just idle because-we-can nitpicking pedantry, irrelevant in a general encylopaedia. Nø (talk) 16:40, 16 February 2026 (UTC)

This article is so dense that I cannot find where it shows pi to 20 digits.

Regarding Nø's comments, I disagree with "there is no particular reason to stop at 15 decimals, or 50 decimals, or 500, or 1000000". Wikipedia uses citations. We shouldn't make up an arbitrary limit but use only cited figures. I see that there is 50 digits shown along with a citation. I believe that is good.

Regarding "irrelevant in a general encylopaedia" per Nø, this whole article is almost entirely beyond a general encyclopedia. The article has way too much detail for a general encyclopedia but sufficient detail for Wikipedia.

Conclusion: now that I see there is 50 digits listed along with a citation (which I didn't add myself, digits nor citation), I am satisfied with that unless someone has better logic. Vanguard10 (talk) 23:32, 16 February 2026 (UTC)

We could easily substitute a source giving 50 decimals with an equally good source giving other favoured numbers of decimals like 2, 4, 6, 10, 15, 20, 100, 1000, etc. (but probably not exactly 37 decimals, say). Nø (talk) 10:28, 17 February 2026 (UTC)

31 digits, obvs. Or 314, or 3,142. NebY (talk) 11:02, 17 February 2026 (UTC)

For what it's worth. I googled the most precise physical constant and the Rydberg Constant is known to 1ppt (one part per trillion, 1E-12, error÷value). So, you could argue that 12 digits is good. I've seen the claim that NASA uses 20 digits; but is that true? My guess is that the need for precision is greatest for the Horizons' ephemeris data where JPL give data for hmmm, iirc, about a century (a half-century?). Whether that directly involves the use of pi, IDK. But having used it, I found that it takes about 28 digits of precision to get things reasonably accurate when doing some interplanetary (fictional) spaceflight. The problem is iteration, and how invisibly small errors grow as the number of iterations grows. There's a principle of engineering that you should never let the math be a significant contributor to the error of the result. This suggests that numerical constants should be as accurate as need be, and that the needs vary. I suspect certain applications use 50-digit precision calculations, I'm certain some use 20-digits. OTOH, what I think WP should do is mention two things: 1. What's the number of digits of pi known (published?) and 2. Where can the reader easily find pi reported to many decimal places. What I think WP shouldn't do is post the value to a bazillion digits - as others have said.(But reporting the first 50 or 100 wouldn't take up that much space, imho)~2026-92463-4 (talk) 06:43, 16 March 2026 (UTC)

For what it's worth, I do not think the article should lean too heavily on the familiar claim that only x digits of pi are 'needed in science.' JPL does have a sourced outreach piece saying that its highest-accuracy interplanetary navigation calculations use pi to 15 decimal places, and that 37 decimal places would suffice for a circle of radius 46 billion light years to hydrogen-atom accuracy. But that is a statement about those examples, not a general theorem about all numerical work. Required precision depends on the conditioning of the problem, error propagation, iteration, discretization, and the arithmetic being used (e.g., Feynman integrals can require enormous numbers of digits). So I would rather the article avoid turning one rule-of-thumb into a universal claim. It would be more useful to readers to say how many digits are currently known, and to link to an external source where many digits can easily be found. That having been said, I don't think there is any value to having a hundred or more digits here, but this should probably be rooted in editorial consensus and Wikipedia style arguments rather than specious claims about how many digits may or may not be required for some particular task. Sławomir Biały (talk) 08:56, 16 March 2026 (UTC)

Geometric method

Thread retitled from The value of pi is wrong.

The geometric method of calculating pi is laughable. It is flawed to equate the perimeter of circumscribed and inscribed polygons to that of a circle. Donato Porpora (talk) 11:31, 2 March 2026 (UTC)

Do you think the existing Pi#Polygon approximation era section should be changed in some way, or are you just noting that you consider it to be outdated and amusing? Belbury (talk) 11:35, 2 March 2026 (UTC)

Well, the above comment is just my personal opinion. Pi is certainly no "transcendental" and it not 3.14159.. That will be proven soon. However that section certainly has other issues. If you read the first paragraph of this paragraph, it states clearly that archimedes "probably" wrote this text. This is the text that contains the referenced geometric "proof" involving circumscribed and inscribed polygons. Why does this article assume that it is a certainty that archimedes is the author? In fact there is evidence that proves its more likely that the great archimedes did not write this amateurish treatise. Donato Porpora (talk) 11:46, 2 March 2026 (UTC)

For more information, please refer to the first chapter titled "the evidence" from the treatise "on the exact measure and quadrature of the circle". Again, it is false to attribute the polygon bounding method to Archimedes. Historians only assume that he wrote "measurement of a circle". They are not certain that he did. We must display the same level of honesty and integrity and not pretend we know better than historians. Donato Porpora (talk) 11:58, 2 March 2026 (UTC)

The work you cited is Mathematical crankery, not a RS, and historians are not mathematicians. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:46, 2 March 2026 (UTC)

The scholarly consensus is that Archimedes wrote such a work. As I understand it, the controversy about the text that we have today is that it is an edited extract of Archimedes' original, with some parts missing (which we know because people cited them, and we still have the citations). Here's Pappus: "For Archimedes has proved in the book On the Circle that of every circle the perimeter is greater than triple the diameter (by) less than a seventh part, but (by) greater than 10/71; and that the rectangle contained by the perimeter of the circle, as a straight line, and the (line) from the center of the circle is double of the area of the circle." Archimedes himself quotes the core result in his Method "... every circle is equal to a triangle having as base the periphery of the circle, and (its) altitude equal to the (line) from the center of the circle."

Beyond that though, inre "Pi is certainly no 'transcendental' and it not 3.14159..", Wikipedia (including talk pages) is not a venue for publishing new contributors' new ideas, see Wikipedia:No original research and Wikipedia:What Wikipedia is not § Wikipedia is not a publisher of original thought. –jacobolus (t) –jacobolus (t) 17:55, 2 March 2026 (UTC)

Then it's a good thing that the geometric method does not equate them. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:46, 2 March 2026 (UTC)

"Half-circle constant" listed at Redirects for discussion

The redirect Half-circle constant has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2026 March 4 § Half-circle constant until a consensus is reached. –Deacon Vorbis (carbon • videos) 15:50, 4 March 2026 (UTC)

Typo

"Also, there are many occurrences of π in many branches of mathematics that are completely independent from geometry, and in modern mathematics, the trend is to built geometry from algebra and analysis rather than independently from the other branches of mathematics."

built => build

Maybe? Perhaps also truncate the sentence at "analysis"? ~2026-15330-73 (talk) 22:14, 10 March 2026 (UTC)

Done. Noleander (talk) 23:31, 10 March 2026 (UTC)

Has been done here: Special:Diff/1342839779 CiaPan (talk) 08:10, 13 March 2026 (UTC)

On main page later today

In honor of pi day, this article will be on English Wikipedia's front page soon. It should appear around midnight in London, or 6pm in New York. The article is semi-protected so most IPs should not be able to vandalize it. Noleander (talk) 13:24, 13 March 2026 (UTC)

Is there a place to put the first 100/1000 digits of pi in the article?

Might be a silly question, but given the article is about pi, shouldn't some parts of pi, like at least 100-1000 digits, be included somewhere? VidanaliK (talk to me) (contributions) 00:29, 14 March 2026 (UTC)

The actual digits of pi really aren't that notable. Human efforts to compute lots of digits are notable, and the practice of memorizing digits is notable. Both of these are covered in the article. ~2026-16209-89 (talk) 00:41, 14 March 2026 (UTC)

I believe notability is only required to have a specific article about it. I'm asking if there is a place in the existing pi article where the digits can be put. VidanaliK (talk to me) (contributions) 00:43, 14 March 2026 (UTC)

This question has been asked several times over the past 2 decades. After discussion, the consensus has always been to not display more than 25 digits or so. Readers who want more digits can use any of the "external links" at the bottom of the article to get more digits. Noleander (talk) 00:46, 14 March 2026 (UTC)

Did you try clicking the link in "(see OEIS: A000796)"? That page links to e.g. https://web.archive.org/web/20140225153300/http://www.exploratorium.edu/pi/pi_archive/Pi10-6.html –jacobolus (t) 01:39, 14 March 2026 (UTC)

I don’t really think that it would be necessary to include over 20 digits of pi Cenchros Roseus (talk) 20:39, 23 March 2026 (UTC)

The GIF makes the text difficult to read

It is very hard to read text with an animated GIF constantly moving in one's field of vision. The GIF ought to be moved down, at the very least.— chbarts (talk) 02:57, 29 March 2026 (UTC)

Can you give more details... Are you saying the gif is covering up text? Or you saying the movement simply distracting? Gifs are fairly common on the Internet these days, I think most people are accustomed to seeing them in many websites and apps. Noleander (talk) 03:31, 29 March 2026 (UTC)

MOS:ANIMATION requires that animations either run shorter than five seconds and then stop (not loop), or have a control to stop them. They can be very distracting for some people, as the OP states. I have added a stop control, modeled after the one in Four-dimensional space and added there after a similar discussion, Talk:Four-dimensional space § Animation fails standards. —David Eppstein (talk) 04:40, 29 March 2026 (UTC)

Thanks for doing that. While you are looking at this, I think it would be more explanatory if the animation ran just a bit slower - maybe 2/3 the speed. Currently it is all over before you have time to see what is happening. Imaginatorium (talk) 05:30, 29 March 2026 (UTC)

That's not something that's easy for me to do. Someone else more experienced in retiming animated gifs might find it easier. We do have several versions of this animation on commons including one with "slow" in its name but they all appear to run at the same speed. —David Eppstein (talk) 06:49, 29 March 2026 (UTC)

I have had a go - if you open a gif with GIMP, each frame is a layer, and the duration is embedded in the name of the layer. Please see if you think this one would be better: https://imaginatorium.org/private/Pi-unrolled-720a.gif

I don't know how the animation controls work, but it might be better to have a non-looping version, with a "Replay" button. Imaginatorium (talk) 18:30, 3 April 2026 (UTC)

The point of MOS:ANIMATION is that some users of WP want to have video controls (stop/pause/replay) so they can proceed thru an informative video at their own pace. So MOS:ANIMATION specifies two options:

1. Decorative GIF - under 5 seconds: video controls not required
2. Informative GIF - over 5 seconds: video controls are required

The infoBox should have a catchy, decorative 5 second GIF: clean, tidy, no video controls. Optionally, the article can include a second, longer, informative version in the article body, with video controls

Can someone create a 5 second version of that GIF? Even if we don't use it in this article, it would be good to have a 5 sec version in Wiki Commons. Noleander (talk) 19:02, 3 April 2026 (UTC)

I don't know about you but I would find a version of that gif that runs for five seconds and then stops to be incredibly annoying. I want to watch it repeatedly until I understand it, and five seconds is barely enough time for a single iteration. And repeating the same image twice in different versions, once to be decorative, is a waste of space and viewer attention, also highly annoying. What we have now is an informative gif with a control, meeting the requirements. Why do you think that controls must be omitted? And why do you think there should even be an infobox? (The sidebar navbox is not an infobox.) —David Eppstein (talk) 19:19, 3 April 2026 (UTC)

"3.14159265358979323846264338327" listed at Redirects for discussion

The redirect 3.14159265358979323846264338327 has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2026 April 23 § Long PI redirects until a consensus is reached. Squawk7700 (talk) 09:24, 23 April 2026 (UTC)