Talk:Pi/Archive 15

A lot of new material has been added under the heading "Fourier series and modular forms". I don't believe most of this is directly relevant to pi. At most, there should be a brief mention of these topics, with links to the main articles. --Macrakis (talk) 17:39, 7 May 2016 (UTC)

It is directly relevant to π. In fact, one of the definitions of π in the literature is half the magnitude of the derivative of the unique complex character that is an isomorphism on the circle group. This definition is of direct importance to the theory of Fourier series (which an editor above requested that we add information on). The application to modular forms is perhaps slightly more peripheral, but it does clearly show the appearance of π in non-abelian harmonic analysis as well. Sławomir Biały (talk) 18:02, 7 May 2016 (UTC)

OK, let me rephrase what I mean by "directly relevant" in the context of the article. There are clearly connections between π and various areas of mathematics. These connections should be mentioned in this article. However, in the context of an encyclopedia like WP, we shouldn't be redundantly writing expositions of those other areas in multiple articles, but rather the minimal text that explains the connection, relying on the main articles on the topics for more detail. --Macrakis (talk) 19:43, 7 May 2016 (UTC)

The written text seems precisely adequate to explain the essential role of π in mathematics. It's not clear to me what part of it could be removed without sacrificing comprehensibility. Much more of the article is dedicated to various algorithms for computing digits of π. What was missing was an explanation of the reason π appears in so many formulas of mathematics. The added text attempts to do that. Sławomir
Biały 19:57, 7 May 2016 (UTC)

I agree that the article has too much material on computing π; most of it should probably be moved to a separate article. But that is orthogonal to this discussion.

I am sure you could do a better job condensing this section than I can, but since I don't seem to have succeeded in communicating what I had in mind, let me try my hand at it.

Fourier series and modular forms

The constant π appears naturally in Fourier series.

In harmonic analysis, periodic functions are often regarded as functions on the circle group T. The Fourier decomposition shows that a complex-valued function f on T can be written as an infinite linear superposition of complex exponentials ${\displaystyle e_{n}(x)=e^{2\pi inx}}$, each of which involves π. The complex exponentials, in turn, can be characterized by their transformation properties under T: they constitute a character of T, that is, a function from T to the group of unit modulus complex numbers, that is also a group homomorphism. Using the Haar measure on the circle group, the constant π is half the magnitude of the Radon–Nikodym derivative of this character. The other characters have derivatives whose magnitudes are positive integral multiples of 2π.^[1] As a result, the constant π is the unique number such that the circle group, equipped with its Haar measure, is Pontrjagin dual to the lattice of integral multiples of 2π. This is a version of the one-dimensional Poisson summation formula.

Modular forms satisfy an invariance property under the modular group ${\displaystyle SL_{2}(\mathbb {Z} )}$ (or its various subgroups), a lattice in the group ${\displaystyle SL_{2}(\mathbb {R} )}$. Like the complex exponentials of Fourier series, modular forms are characterized by their transformation properties. They also involve π, once again because of the Stone–von Neumann theorem.

1. [1]

   Nicolas Bourbaki (1979), Fonctions d'une variable réelle, Springer, §II.3.

I am certain that this can be improved, but my point here is that the text in the pi article should not include expository material about Fourier series or modular forms, but only refer to those articles and mention how they are relevant to the present article. --Macrakis (talk) 22:30, 7 May 2016 (UTC)

That's pretty good, but I think the article should at least include the form of the Fourier series, to show where π appears. Also, the text above does not connect Fourier series with periodic functions on the circle. Finally, there is no mention at all of the Jacobi theta function. Also, I really disagree that there should be no expository material here. There is plenty of expository material about other aspects of π throughout the article. It should be kept to a minimum, but it cannot be eliminated entirely. Sławomir Biały (talk) 11:23, 8 May 2016 (UTC)

In both modular forms and in the Fourier series, π appears only in the context ${\displaystyle e^{2\pi ix}}$, which is equal to ${\displaystyle (-1)^{2x}}$. So it is arguably the identity ${\displaystyle e^{\pi ix}=(-1)^{x}}$ about complex exponentiation that is more fundamental and "explains" the appearance of π. Your thoughts? --Macrakis (talk) 18:12, 8 May 2016 (UTC)

Superficially, yes. A minor quibble is that one needs to be very careful in writing things like ${\displaystyle (-1)^{x}}$, as this doesn't properly make sense due to the properties of the complex logarithm. What is more correct is possibly to say that π is the smallest positive real number for which ${\displaystyle e^{\pi i}=-1}$. Thus π is indeed connected in a fundamental way with the complex exponential.

Fourier analysis is the study of the additive group ${\displaystyle \mathbb {R} /\mathbb {Z} }$ of real numbers under addition, where we identify two numbers if they differ by an integer. Although this is sometimes called a circle group, the definition is not immediately connected with any geometrical idea of "circle" that one might have. One needs to contend with this group throughout mathematics, even areas of mathematics where the concept of a "geometrical circle" is a very remote one (e.g., in algebraic number theory). The standard way groups are studied is via their unitary representations. The quantity π appears as an eigenvalue, without any reference to complex exponentials: ${\displaystyle \pi =|e_{1}'|}$. The equation ${\displaystyle e_{1}(x)=e^{2\pi ix}}$ is a true fact that can be proved, but it's not a definition of ${\displaystyle e_{1}}$, which is more primitive from this point of view. This is the perspective taken by Bourbaki's text.

One might legitimately wonder if there is a version of Fourier analysis that is independent of π. Fourier series are uniquely characterized by certain properties having nothing to do with the complex exponential. The Fourier series associates to a function ${\displaystyle f\in L^{2}(\mathbb {R} /\mathbb {Z} )}$ a square-summable sequence of complex numbers ${\displaystyle {\hat {f}}(n)}$. This association is unitary and extends to a homomorphism of von Neumann algebras of ${\displaystyle L^{1}(\mathbb {R} /\mathbb {Z} )}$ to ${\displaystyle \ell ^{\infty }}$. It is a theorem of Fourier analysis that the Fourier series is the only such transformation. Thus anything that "behaves like" the Fourier series (for incredibly general values of the term "behaves like") must actually be the Fourier series. There is nothing in this description that involves complex exponentials. Yet nevertheless π always appears as an eigenvalue. Sławomir Biały (talk) 19:03, 8 May 2016 (UTC)

Thank you for your explanations, which I admit are beyond what I learned in my undergraduate math degree.... --Macrakis (talk) 21:44, 8 May 2016 (UTC)

It looks like Ozob did I nice job of clarifying the intended logic of things. I have streamlined it a bit more, in a way to clarify the sense in which π is essential to Fourier analysis, at the slight expense of leaving out some of the more familiar formulae of the subject, since these apparently distract from the main point. Sławomir Biały (talk) 20:42, 8 May 2016 (UTC)

I have concerns about the following statement in the article:

Monte Carlo methods for approximating π are very slow compared to other methods, and are never used to approximate π when speed or accuracy is desired.^[1]

It is referenced to Arndt and Haenel (2006) and Posamentier and Lehmann (2004), neither of which is very awe inspiring as a reliable source for mathematical claims like this. Moreover, the assessment of Arndt and Haenel is that Monty Carlo methods converge slowly. This is clearly not even true, since every conventional method already is a Monty Carlo method (with a fairly boring probability distribution). Sławomir Biały (talk) 13:15, 13 May 2016 (UTC)

I take conventional methods to be computations using a series; for example arctangent. Some series converge very quickly. No random number draws are used.

Monte Carlo methods use random number draws with accept/reject criteria. The samples are then averaged. Averages converge slowly; law of large numbers.

Glrx (talk) 05:03, 18 May 2016 (UTC)

What's going on here is that Monte-Carlo method means different things to different people. Glrx seems to be taking it in the "strict" sense, where you just run a bunch of simulations and count how many of them fall into a certain bucket. Sławomir is using the looser sense, meaning "any technique that uses random or pseudo-random numbers in an essential way". I suspect both viewpoints are defensible and represented in the literature; the article just needs to make clear what it's talking about. --Trovatore (talk) 06:33, 18 May 2016 (UTC)

Update: Apparently we have articles Monte Carlo method and Monte Carlo algorithm that are distinguished one from the other on this basis. Now that I seriously doubt is a distinction made clearly and consistently in the literature. I think that should probably be fixed. --Trovatore (talk) 06:38, 18 May 2016 (UTC)

The distinction is very clear in the literature. On one side one has Monte Carlo methods (which are not algorithms), consisting in running random samples for getting an approximation of the expected value of a random variable. In the article, they are applied to random variables for which the expected value may be exactly computed, and is related to π. It is almost an evidence that such a method is slow for getting an accurate result (that is a high number of digits for π). Moreover, they do not provide any way to certify the number of exact digits that are obtained. I am quite sure that the efficiency of Monte Carlo methods for computing π are discussed in Knuth's The Art of Computer Programming. On the other hand Monte Carlo algorithms and Las Vegas algorithms use auxiliary independent random numbers, that are independent of the input. I have never heard of the use of such algorithms for computing π.

By the way, I do not understand the concern of Sławomir Biały, and I agree with Glrx: Monte Carlo methods, as well as Monte Carlo algorithms use randomness in some way, which is not the case for conventional (?) algorithms for computing π. D.Lazard (talk) 08:59, 18 May 2016 (UTC)

Typically isn't randomness an additional input to an algorithm that can be used for accelerating it? Compare with primality testing, where the fastest algorithms are the probabilistic algorithms. This is the reason Monte Carlo methods are used in the first place. Knuth defines a Monte Carlo method as "Any method that uses random numbers (possibly not producing a correct answer)." This distinguishes "method" from "algorithm" I guess. Anyway, it's trivial to construct a Monte Carlo method (in fact an "algorithm") that is at least as fast as conventional methods: just toss a coin to decide which of two reasonably fast algorithms to use. This does not use randomness in a very essential way that actually accelerates the convergence, but it also certainly doesn't hurt either.

On a related matter, we know that pi can be felt in spectra of random processes like Johnson noise. Is there any practical implementation of this? I'd be curious to know if there is an estimate of the marginal cost, in bits of entropy, to produce a digit of pi. It seems from the random walk example that convergence is very slow. I did another simulation, with comparable performance, that used Brownian local time. Sławomir
Biały 12:42, 18 May 2016 (UTC)

Primality testing (which has a binary result) is a Monte Carlo algorithm rather than a Monte Carlo method. Glrx (talk) 14:22, 18 May 2016 (UTC)

One would think that "method" includes "algorithm", in the sense that every algorithm is a method but not vice versa. But if we want to split hairs about nomenclature, are there any Monte Carlo algorithms for computing π? Do we infer from the text of the article that all Monte Carlo algorithms for computing π are slow compared to deterministic methods? Sławomir Biały (talk) 15:50, 18 May 2016 (UTC)

I remain skeptical that this method–algorithm distinction is really as universal as claimed, but in any case, even if it is standard in computer science, casual readers are not going to know that; it should be explained inline. There's a lot of cases, in science in general, where some group tries to standardize terminology, but it doesn't necessarily work. Algorithm is a case in point — if you look it up in references that try to define it abstractly, they usually impose conditions, especially guaranteed termination, that don't really apply to the concept as encountered in the wild. As people actually use it, the real distinction is not algorithm–heuristic, but rather algorithm–implementation; that is, an algorithm is what's left of a program when you abstract away implementation details. But good luck finding that explicitly written down. --Trovatore (talk) 21:41, 18 May 2016 (UTC)

In the lead section we find:

"The digits appear to be randomly distributed; however, to date, no proof of this has been discovered."

This is highly suspicious information. It is known that pi is infinite and never repeating. Someone needs to clarify this, as it will confuse the reader. EeeveeeFrost (talk) 01:26, 11 July 2016 (UTC)

It is explained, in the section Irrationality and normality - normality being the formal mathematical term for a number with randomly distributed digits. Detailed explanations with references are not needed in the lead section; it is usual to have to look in the body of the article.--JohnBlackburne^words_deeds 01:39, 11 July 2016 (UTC)

I'm not exactly sure what you're talking about. I checked the irrationality and normality section and all I found was "Because π is irrational, it has an infinite number of digits in its decimal representation, and it does not settle into an infinitely repeating pattern of digits."

In the next paragraph, normality is mentioned. However, it is highly misleading to say that "The digits appear to be randomly distributed; however, to date, no proof of this has been discovered." in the lead section! EeeveeeFrost (talk) 01:48, 11 July 2016 (UTC)

Why is it misleading? It is correct, and is explained more fully in the section below, in particular at The conjecture that π is normal has not been proven or disproven. Again, that is how articles are written. The lead is a summary of the topic, so does not go into detail or even necessarily mention everything in the article.--JohnBlackburne^words_deeds 01:59, 11 July 2016 (UTC)

This discussion appears to conflate at least three things. First, having an infinite and never repeating decimal expansion is a necessary, but not a sufficient, condition for having randomly-distributed digits. However, also it's not clear that normality is a necessary or sufficient condition. What does "randomly distributed digits" mean? Does Champernowne constant have randomly distributed digits? Random sequence is not much help in this regard. Obviously, the digits of π are not actually random; they are the digits of π. So "random" actually means that the digits of π pass some statistical test of randomness, like normality. I believe in the article, we have implicitly taken normality as the definition of random, but there are numbers like the Champernowne constant that have a definite pattern, despite being normal in base 10 (and even satisfying other tests of randomness like Kolmogorov complexity). I think someone like User:Trovatore might have some insight into this conundrum. Sławomir Biały (talk) 13:17, 11 July 2016 (UTC)

It is misleading because it has more to do with statistics than mathematics. We all know that pi is infinite and "random" (in a sense that the digits are random). My suggestion is that we simply omit that information from the lead; at least for now. That sentence isn't necessary to make the paragraph coherent anyway. EeeveeeFrost (talk) 13:36, 11 July 2016 (UTC)

I think it's important and worth discussing in the lead, which is supposed to summarize the most significant aspects of the article. But the sense in which the digits are (believed to be) "random" needs to be clarified I think, in the body of the article at least. Sławomir Biały (talk) 14:18, 11 July 2016 (UTC)

Let's not say that pi is random in the lead. In my opinion, that first highlighted sentence above ("Because π is irrational etc...) is good enough to be put in the lead. EeeveeeFrost (talk) 14:30, 11 July 2016 (UTC)

I just restored (and copy-edited) the lead bit about "randomness". Please do not remove it (i.e. preserve the status quo) unless you can get a consensus for this here. This is not likely to happen, because it is fairly clear you have not understood this: the relevant paragraph in the body starts: "The digits of π have no apparent pattern...", and this is what is meant by being "random". Read the paragraph What does "randomly distributed digits" mean? by Sławomir above very carefully... and notice the difference between the informal definition of rational as "not settling into a repeating pattern" and a number like 0.123456789101112131415161718192021... which is clearly irrational, but definitely not "random" in some sense. I think there could be a sensible discussion about whether "random" is the best word to use in the lead, but it is quite appropriate to refer somehow to this property. Imaginatorium (talk) 15:10, 11 July 2016 (UTC)

I decided to just replace that sentence with another one. This new one is perfectly valid, it fits well in the context and there is no need to remove it. It is much more useful than the other one, which is a mere problem related to statistics. EeeveeeFrost (talk) 15:33, 11 July 2016 (UTC)

The new sentence that you added does not clarify the content of the old sentence, so I have reverted it. It is already redundant with the first two sentences of that paragraph of the lead. One can take (for the sake of argument) the definition of "random" to mean that the frequency of any string of digits in the expansion of π depends only on the length of the string. (This is actually the definition of normal in base 10.) With that definition, it is much stronger than "never settles into a permanent repeating pattern". That is probably not a good definition of "random", which I think deserves to be clarified in the body of the article, but what we mean by "random" here should imply this statement about frequency. Sławomir Biały (talk) 15:44, 11 July 2016 (UTC)

@Sławomir Biały: Sure, I understand. Just let me ask you something: Don't you think that the sentence might confuse the average reader? Because when I read it for the first time, I understood that there is a possibility that pi is not just infinitely many numbers that are (or appear to be) random; and that made me feel confused at first. We know that pi is calculated using an infinite sum π/4 = 1 - 1/3 + 1/5 - 1/7..... and that obviously leads to non repeating and "random looking" numbers.

Anyway, if you don't think that the sentence might be misleading, then I'll just give up and leave the way it is. EeeveeeFrost (talk) 15:58, 11 July 2016 (UTC)

There certainly is a possibility that the digits of π are not random (in whatever reasonable sense one might define random, which I think still should be clarified). For example, it is possible that the sequence "1423509876723" never actually appears as a sequence of digits in the decimal expansion of π, and that there is, therefore, a cosmic conspiracy to exclude this particular pattern of digits from appearing somewhere. The digits would then not be random. Sławomir Biały (talk) 16:01, 11 July 2016 (UTC)

That's stretching random. By random, (I believe) we mean that there is no way to correctly predict a series of digits without actually calculating them. —Compassionate727 ^(T·C) 16:05, 11 July 2016 (UTC)

Random implies normality, by the Borel-Cantelli lemma. So it isn't stretching random. It would, however, be nice if there were a clear mathematical statement. For now, I think we should link to the article random sequence. At least that gives a little context. Sławomir Biały (talk) 16:12, 11 July 2016 (UTC)

Responding to Sławomir's ping — it's a complicated question what "random" means for an individual sequence. In some sense I think it's misleading to speak of the concept at all without elaborating. A sequence can be drawn from a random distribution, but any particular sequence has the same probability as any other (namely zero), and no sequence can be completely random, because it is the sequence that it is.

That said, there's a quasi-hierarchy of stronger and stronger notions of what it means for a fixed infinite sequence to be "random". All of these notions (as far as I'm aware anyway) can be expressed by identifying a family of measure-zero sets, and saying a sequence is "random" if it's not in any set in that family. In this context, the most common meaning for calling a sequence "random" without further elaboration is 1-randomness, where the family of measure-zero sets is "G-delta sets determined by a constructive null cover".

The decimal representation of π, considered as a sequence of digits, is computable, so it's not even close to 1-random.

As for normality, it does have this general form of a randomness criterion, because you can come up with a collection of measure-zero sets (actually just a single one in this case), and say a sequence is normal if it's not in any of them. But on the hierarchy of randomness notions, normality is either near the bottom or not on it at all, depending on how you look at it. I do think it's a bit misleading to refer to the decimal representation of π as "random", at least without inline elaboration. --Trovatore (talk) 18:20, 11 July 2016 (UTC)

It seems like there is some notion of statistical randomness, that should imply normality (among other things), that roughly captures the notion that the distribution of digits "appears random". Word frequencies seem like a good way to do this. But I would like to get some clarity on exactly what sense we are entitled to say that the digits of π are random. Sławomir Biały (talk) 19:42, 11 July 2016 (UTC)

I think normality is the strongest thing you're going to get that's limited to "distributions of digits". Stronger properties are going to involve subtler tests. So going that route, I think we should just talk about normality, not randomness.

That said, normality is, if not a very weak kind of randomness, at least some sort of approximation to randomness. I can't see that statement being particularly controversial. And we would probably be remiss if we didn't say something about that, both because randomness is a lot more familiar to most readers than normality (even if they don't have a clear understanding of what it means), and because it's sort of the reason that we expect π to be normal, absent a reason why it shouldn't be.

But I'm not happy with lies to children, and we also have to avoid original research, and between those two constraints it's going to be tricky to come up with appropriate wording. --Trovatore (talk) 19:48, 11 July 2016 (UTC)

Meh. The empirical fact is that the digits of π are "random" in a way that the digits of other normal numbers do not seem to be. This is certainly not intended as a mathematically rigorous statement, but it does seem to be empirically true. I don't think many readers are likely to be misled by the statement as it currently exists in the article. Most readers are unlikely to go into deep philosophical waters, like trying to figure out exactly what "random" actually means. That's well outside the scope of this article.

However, I think the statement in the article is not misleading in the "telling lies to children" sense that you are concerned with. Whatever the word "random" should mean in this setting, it must at least imply that the digits of π are normal. (Think of the word "random" really as an equivalence class of predicates, each of which is a normal sequence.) The distribution of digits of π being "random" is at least consistent with the article as stated. Now, since π is not known to be normal, it is also not known to have random digits in any sense in that equivalence class. So, we don't actually need to know what "random" means in order to make sense of the statement in the article. Sławomir Biały (talk) 21:50, 11 July 2016 (UTC)

Well, my concern is that random-full-stop is too strong, because its default meaning in this context is 1-random, which π is definitely not.

I don't think we should use the r-word directly and without elaboration as being a possible property of π. "Normal" is fine, and we can probably work in a defensible explanation of how normality is related to, or a weak version of, or an approximation to, randomness. --Trovatore (talk) 21:55, 11 July 2016 (UTC)

By the way, this is a little off-topic, but you seem to be interested in a criterion that would separate π from Champernowne's constant. I thought about it a little bit and I think I have one.

In Champernowne's constant up to the 10ⁿth digit, successive blocks of n digits are highly correlated. If you keep n fixed and let the number of digits go to infinity, the correlation goes away, of course. But if you look at the sequence r_n equals the correlation between successive blocks of n digits up to the 10ⁿth place, that sequence is going to have very different asymptotic behavior in Champernowne's constant than in a random sequence, and presumably π will be more like the random sequence than like CC.

But I doubt that can be sourced. --Trovatore (talk) 22:08, 11 July 2016 (UTC)

I'm not certain I agree that 1-random is "the" default meaning of random here, but I'll accept that it might be in some circles. I've clarified the language in a way that hopefully settles that objection. Sławomir Biały (talk) 23:04, 11 July 2016 (UTC)

There used to be an HTML comment at the top of the article saying "Please note that Wikipedia is not the place to store millions of digits of pi." Somebody removed it. Why?? Georgia guy (talk) 23:19, 11 July 2016 (UTC)

I don't know, but if no one's done that recently, then maybe we don't need the note. It's not like it's a lot of trouble to revert when it happens, as long as it doesn't happen frequently. --Trovatore (talk) 23:39, 11 July 2016 (UTC)

The comment hasn't been in the article for more than four years. It was never effective, and the article was indefinitely semiprotected instead. Sławomir Biały (talk) 12:15, 12 July 2016 (UTC)

The current text on spigot algorithms says that no such algorithm is known for decimal digits of pi.

Yet there's this article 'A Spigot Algorithm for the Digits of Pi', by Stanley Rabinowitz and Stan Wagon, to be found here: http://www.mathpropress.com/stan/bibliography/spigot.pdf that says how to do it.

Is this research discredited or should we alter the text on spigot algorithms? Dr. Crash (talk) 15:15, 10 October 2016 (UTC)

Please put new talk page messages at the bottom of talk pages. Thanks.

That does not look like a proper wp:reliable source in the Wikipedia sense. Book? Article in an established and relevant journal? - DVdm (talk) 15:32, 10 October 2016 (UTC)

The article states: For example, one may compute directly the arc length of the top half of the unit circle given in Cartesian coordinates by x² + y² = 1, as the integral:

${\displaystyle \pi =\int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}.}$

But at the same time, in abuse of notation it is explained: A related abuse of notation occurs when integrals like ${\displaystyle \int {1 \over x}\,dx}$ are written as ${\displaystyle \int {dx \over x}}$, as if ${\displaystyle dx}$ were a term multiplied into the integral's ${\displaystyle 1 \over x}$ argument.

Am I missing something? --bender235 (talk) 23:06, 19 November 2016 (UTC)

I have no problem with that notation, and so the abuse of notation article seems wrong here. This is not some odd, non-standard ways of writing integrals but is common, clear and unambiguous. It appears in other places in this article and seems properly sourced.--JohnBlackburne^words_deeds 23:14, 19 November 2016 (UTC)

This edit request has been answered. Set the |answered= parameter to no to reactivate your request.

We need to update the record number of digits, which has increased significantly in November. See http://www.pi2e.ch/ which is linked at your reference 3. The new values are: 22 459 157 718 361 decimal digits 18 651 926 753 033 hexadecimal digits There is an error in saying trillion is 10^13. It is 10^12.

The original text is:

extended the decimal representation of π to, as of 2015, over 13.3 trillion (10^13) digits.[3]

The new text should be"

extended the decimal representation of π to, as of November 2016, over 22.4 trillion (10^12) digits.[3]

A new footnote reference to http://www.pi2e.ch/ might be added, even though it is linked at reference 3.

agb — Preceding unsigned comment added by 143.43.206.244 (talk) 20:29, 22 December 2016 (UTC)

 Done  Paine Ellsworth  ^u/c 01:25, 26 December 2016 (UTC)

Content like this should require reliable secondary sources. It was weakly sourced before too, so I don't think reversion is appropriate, but the sentence should be rewritten in a way that does not rely on the primary self-published literature. Sławomir Biały (talk) 03:07, 26 December 2016 (UTC)

I went ahead and generalized the statement in the article, since I could not find any news sources that have yet reported on the new figure. It might show up in the December issue of the American Mathematical Monthly, and possibly in The Conversation as well.  Paine Ellsworth  ^u/c 22:03, 27 December 2016 (UTC)

Interesting solution. I like it. Sławomir Biały (talk) 23:12, 27 December 2016 (UTC)

Thank you, Sławomir Biały! I could be wrong, but it might be better to leave the specifics out of the lead. If and when the new record appears in a news source, it can be included in the lower content of the article.  Paine Ellsworth  ^u/c 11:38, 29 December 2016 (UTC)

This looks like it might be some good WP:RS. I don't think I saw everything from it in this in the article, but I didn't look closely. Was trying to figure out how old the concept of PI is (not necessarily its computation -- it is after all a ratio, so you don't have to necessarily calculate to use it (i.e. if you have one circle of diameter d, and circumference c, then you can estimate the circumference of a circle of 2d to be 2c by proportion without ever calculating pi, but understanding that the ratio is fixed). I think our article is incomplete on dealing with the concept of Pi, but I'm no expert in this subject. --David Tornheim (talk) 14:56, 27 March 2017 (UTC)

That looks like a student paper for a class, so not WP:RS. Also, based on the third paragraph, I would say that this source is factually very suspect. Sławomir Biały (talk) 16:56, 27 March 2017 (UTC)

Ah, you are right, it is probably a student, not faculty. Pretty good writing for a student! Maybe a grad. student?

I'm not clear: What is wrong with the third paragraph? Are you saying that the source "Blatner, David. The Joy of Pi. Walker Publishing Company, Inc. New York, 1997" or the source "Tsaban, Boaz and David Garber. "On the Rabbinical Approximation of pi." Historia Mathematica 25, Article HM972185. Academic Press, 1998." is suspect or that writer speaking about what was in one or both of these sources was suspect? If so what specifically do you find suspect? I did not check the quote to the Bible or think deeply about what was said in the paragraph and compare it to what is in the rest of our article and its sources. I just don't know enough about the history to PI to know what is or is not suspect in that paragraph. Maybe I missed something glaring? --David Tornheim (talk) 21:49, 29 March 2017 (UTC)

That has been discussed before, and excluded per WP:FRINGE if I recall. In particular, the invocation of the gematria is particularly suspect. Sławomir Biały (talk) 22:41, 29 March 2017 (UTC)

Re this diff.

If my memory is correct, which it may not be, this was added by User:Sławomir Biały. At the time, I really didn't understand what it was supposed to mean, but I figured that was probably because I hadn't thought about it hard enough.

So now that it's come down to a showdown — Sławomir, what did you mean exactly? It sounds interesting. Would be good to have a cite. --Trovatore (talk) 08:57, 14 April 2017 (UTC)

As detailed in the section on spectral characterizations, π is the smallest eigenvalue for a vibrating string with fixed endpoints. It is the best constant in the isoperimetric inequality, which is associated with eigenvalues of the Laplace operator and the Dirichlet energy. These two applications are detailed for example in Volume 1 of Courant and Hilbert. It appears as the best constant in the uncertainty principle. This can be obtained as a Rayleigh quotient for the symmetric form ${\displaystyle \int x[{\overline {f}}f'+{\overline {f}}'f]}$, which the view taken in appendix I of Weyl (1931) "The theory of groups and quantum mechanics".

Also, it is the unique constant making ${\displaystyle e^{-\pi x^{2}}}$ equal to its own Fourier transform (which is referenced to the paper by Roger Howe on the Heinseberg group). Sławomir Biały (talk) 10:26, 14 April 2017 (UTC)

This explanation is far too technical to appear in the lead. Moreover the point is not that π appears often as eigenvalues. The point is the WP:OR assertion that this is the cause of the ubiquity of π ("Because of its special role as an eigenvalue, π appears in areas ..."). IMO, this assertion is wrong, and the appearance of π as eigenvalue has the same cause as its appearance in so many scientific areas, namely its relation with every periodic phenomenon. Therefore, I'll replace the above phrase by The number π appears generally in the mathematical analysis of any periodic phenomenon; it appears therefore in areas .... D.Lazard (talk) 12:07, 14 April 2017 (UTC)

I'm not suggesting that we give this explanation in the lead of the article. But there is an entire section dedicated to the role of π as an eigenvalue. These do not involve periodic phenomena, at least not directly (for example, in the central limit theorem, Gaussian functions, Wirtinger's inequality, and Heisenberg uncertainty principle). Anyway, seeing the edit in full, I suppose it's a reasonable compromise. Sławomir Biały (talk) 12:25, 14 April 2017 (UTC)

I've left in the role in periodic phenomena. But the role of π in periodic phenomena arises exactly because π is an eigenvalue of the Sturm-Liouville problem. So I still think that deserves to be mentioned. Sławomir Biały (talk) 12:34, 14 April 2017 (UTC)

I'm not really familiar with the notion of an "eigenvalue of a problem". What I see here is that −π² is an eigenvalue of the second-derivative operator, and one that corresponds to a solution with the desired boundary values. Can we expect readers to know what an "eigenvalue of a problem" is? Is it explained in our eigenvalue article, and should there be a link to a specific anchor?

I'm also a little skeptical that something as specific as Sturm–Liouville theory should be identified as "the" reason that π comes up in so many contexts. Is that really the consensus view in mathematics in general? --Trovatore (talk) 21:50, 14 April 2017 (UTC)

I don't think the lead should point out that π is a Sturm-Liouville eigenvalue. But the Dirichlet boundary value problem for the derivative on the unit interval is the simplest eigenvalue problem that there is. This eigenvalue problem can be formulated in many ways: as a differential equation (i.e., "trigonometry"), variationally where π becomes associated with the best constant in Wirtinger's inequality (also the Heisenberg inequality), or group theoretically in terms of the spectrum of the group ${\displaystyle \mathbb {R} /\mathbb {Z} }$ (Bourbaki). But I suppose "mathematical analysis of periodic phenomena" as a stand-in for "Fourier analysis" might be better in the lead. Sławomir Biały (talk) 22:52, 14 April 2017 (UTC)

You still haven't said what an "eigenvalue of a problem" or a "Sturm–Liouville eigenvalue" is. I only know eigenvalues of operators, not problems. To keep it on topic, I'm concerned that the article explain this notion, as I think a lot of readers may be similarly confused. --Trovatore (talk) 23:22, 14 April 2017 (UTC)

One eigenvalue problem is ${\displaystyle u''+\lambda u=0}$ for u a function in the Sobolev space ${\displaystyle H_{0}^{1}[0,1]}$. This is formulated variationally, and we have ${\displaystyle \pi =\inf \|u'\|_{L^{2}}/\|u\|_{L^{2}}}$, the infinimum being over all nonzero ${\displaystyle u\in H_{0}^{1}[0,1]}$. Alternatively, another is the periodic eigenvalue problem ${\displaystyle u'+\lambda u=0}$, with ${\displaystyle u\in H^{1}(\mathbb {R} /\mathbb {Z} )}$. Sławomir Biały (talk) 23:54, 14 April 2017 (UTC)

So in your first example there, exactly what operator is π an eigenvalue of? --Trovatore (talk) 00:11, 15 April 2017 (UTC)

${\displaystyle -\pi ^{2}}$ is the largest eigenvalue of the second-derivative on the Sobolev space ${\displaystyle H_{0}^{1}[0,1]}$. ${\displaystyle \pi }$ itself is the smallest singular value of the derivative operator, on that same Sobolev space. The general subject of eigenvalue problems (for both Dirichlet and Neumann conditions) as typically understood in differential equations and calculus of variations is described in Courant and Hilbert. One can state these problems in terms of eigenvalues of some explicit operator on a Hilbert space, but this less common in the variationally formulation of these problems. Sławomir Biały (talk) 00:23, 15 April 2017 (UTC)

I do not see anymore why this technical discussion is relevant for this article. After reading this discussion, I oppose to mention eigenvalues in the lead. In fact, most readers ignore what is an eigenvalue, and the link between π and eigenvalues is obscure for most of the others, including two experimented mathematics Wikipedia editors who have participated to this discussion (Trovatore and myself).

IMO, this discussion should be focused to the sections "Spectral characterization" and "Gaussian integral", which suffer of several issues. Firstly they appear as subsections of "Fundamentals", when they should appear in section "Use". Secondly, they present things from a non-neutral point of view, consisting in presenting eigenvalues as the most fundamental concept to which everything to be linked. This appear already in the title of the first section: Is there really any spectral characterization of π? Does exist any source talking of a "spectral characterization of π"? Thirdly, although the applications presented in these sections are fully relevant for the section "Use", the tentative to present them as specific cases of a general eigenvalue problem is a fundamental error. In fact, eigenvalues are not an object of study by themselves, there are a tool for studying (and classifying) various problems. Thus the fact that π occurs as an eigenvalue is not a property of eigenvalues, but a property of some underlying problem. I agree that this is a rather subtle distinction, but explained why a point of view is not neutral, is rarely easy.

My conclusion is that these sections should be moved, the first one should be renamed, and they should be rewritten for much less focusing on eigenvalues. D.Lazard (talk) 11:14, 15 April 2017 (UTC)

There are a number of spectral characterizations of π in the literature: the first is Bourbaki, which defines π via the spectrum of the group ${\displaystyle \mathbb {R} /\mathbb {Z} }$. The second is Rudin, who defines π as the smallest length of a nodal set of an eigenfunction of the one-dimensional Laplacian. But I do agree on reflection that the content is slightly non-neutral in its current titling and placement in the article, so I have attempted to correct this by implementing your suggestion and moving it to the "Uses" section, with a few edits. However, I also feel that it is a grave omission not to mention the fact that the definition of π actually has nothing to do with geometry, and is instead related to the "spectrum" (we can quibble over "eigenvalue") of the group of real numbers under addition. Sławomir Biały (talk) 18:29, 15 April 2017 (UTC)

It is said that${\displaystyle \pi }$ is not algebraic, but by using Taylor series,the inverse trigonometric function for Sine you can obtain${\displaystyle \pi }$. It's not mentioned in the article or is it new?

${\displaystyle {\frac {\sin A}{\sin C}}=a}$

${\displaystyle {\frac {\sin B}{\sin C}}=b}$

${\displaystyle {\frac {\sin C}{\sin C}}=c}$

199.119.233.200 (talk) 12:12, 22 May 2017 (UTC)

The equality ${\displaystyle \pi =2\arcsin 1}$ is well known, and its proof does not implies Taylor series. The result you are talking of is unclear, but, for such elementary questions, there is absolutely no hope to find any new result. D.Lazard (talk) 13:00, 22 May 2017 (UTC)

I think it should also be pointed out that the arcsine is not an algebraic function. Sławomir Biały (talk) 15:10, 22 May 2017 (UTC)

I hope the following example will clarify the problem,and they are all angles in term of cos and sin.

For Lengths of a=x where x,y,z are the lengths of the sides of the triangle in terms of a,b,c.

${\displaystyle x={\frac {\sin(180-\arccos {\sqrt {\frac {(c-b)}{c}}}-\arccos {\frac {a}{c}})}{(1-{\frac {a}{c}})\times {\sqrt {\frac {(a+c)}{(c-a)}}}}}}$

For Lengths of b=y

${\displaystyle y={\frac {\sqrt {\frac {b}{c}}}{(1-{\frac {a}{c}})\times {\sqrt {\frac {a+c}{c-a}}}}}}$

For Lengths of c=z

${\displaystyle z={\frac {(1-{\frac {a}{c}})\times {\sqrt {\frac {a+c}{c-a}}}}{(1-{\frac {a}{c}})\times {\sqrt {\frac {a+c}{c-a}}}}}}$

${\displaystyle \sin((180-\arcsin {\sqrt {\frac {b}{c}}}-\arcsin((1-{\frac {a}{c}})\times {\sqrt {\frac {a+c}{c-a}}})))}$

${\displaystyle a<b<c}$

199.119.233.150 (talk) 18:52, 22 May 2017 (UTC)

Can you add a topic on the digits of π The first 141 digits of π(not including the first 3) is: 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172 Mug1wara (talk) 08:05, 2 June 2017 (UTC)

Pi#Approximate value gives 50 digits. That seems sufficient. PrimeHunter (talk) 10:51, 2 June 2017 (UTC)

The section Pi#Cauchy distribution and potential theory says

The Hilbert transform H is the integral transform given by the Cauchy principal value of the singular integral

${\displaystyle Hf(t)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {f(x)}{x-t}}.}$

Am I correct in assuming that this should have a "dx" at the end? Loraof (talk) 16:39, 2 July 2017 (UTC)

This edit request to Pi has been answered. Set the |answered= parameter to no to reactivate your request.

At the beginning of the article, after the name of pi and the equivalent greek letter π, it would be useful to mention, as an addition, that "this letter is the initial of the greek word περιφέρεια, which stands for periphery, and that this was first introduced by Leonhard Euler in the 17th century". I don't think any reference or documentation is needed here - it is straightforward. Thanks. 37.6.21.2 (talk) 23:16, 10 August 2017 (UTC)

 Not done Perhaps some documentation is needed after all since you've gotten some of your "facts" wrong. See the history section of this article and in particular the subsection on the adoption of the symbol π. The lead of this article is already long and this factoid doesn't really add anything vital to the understanding of this concept, so I don't think that the correct version of this should be placed in the lead.--Bill Cherowitzo (talk) 23:46, 10 August 2017 (UTC)

This edit request to Pi has been answered. Set the |answered= parameter to no to reactivate your request.

2001:E68:540D:9DDD:99E:F815:EBCB:D24D (talk) 12:49, 21 August 2017 (UTC)

 Not done. No request actually made. --Deacon Vorbis (talk) 13:01, 21 August 2017 (UTC)

There is currently a discussion at Wikipedia:Administrators' noticeboard/Incidents regarding an issue with which you may have been involved. The thread is Pi page. --Shirt58 (talk) 08:16, 17 September 2017 (UTC)

It has always bothered me that the definition of pi does not state that it is defined using the common metric. In other metric spaces the value of pi defined as the ratio of the circumference of a circle to its radius will vary. For instance, in the metric space used to measure distance in a city with regularly spaced orthogonal streets, the value of pi is 2*sqrt(2). If the coordinates are shifted by 45 degrees the metric becomes d = max(|x1-x2|,|y1-y2|) and we get a rational value for pi of 4. PetersRet (talk) 16:50, 2 August 2017 (UTC)

π is a number, not a parameter of metric spaces. It has definitions that have nothing to do with distance, for instance in the solution to the Basel problem. It is incorrect to talk about "the value of pi in other metric spaces"; regardless of what metric space one is talking about, π is the same number. Also, not all metric spaces are two-dimensional and allow one to define the circumference of a disk, and not all two-dimensional geodesic metric spaces (the ones in which you can measure the lengths of curves) have a constant ratio of circumference to radius. But in the ones that do, you should still not make the mistake of calling the ratio of circumference to radius pi. —David Eppstein (talk) 18:09, 2 August 2017 (UTC)

That is true, but look at the first paragraph of the article in which it says pi is the ratio of the circumference to the radius. I was merely pointing out that this is not true in all metric spaces. PetersRet (talk) 18:21, 2 August 2017 (UTC)

It's not really good to define pi as a ratio of geometric quantities. There are several more fundamental approaches covered in the article. That pi appears as an isoperimetric constant is a consequence, e.g., of best constants for Sobolev inequalities (which also make sense in general metric-measure spaces). Sławomir Biały (talk) 21:02, 2 August 2017 (UTC)

For the above reasons, I'll edit the first sentence for saying that is is the original definition, but that it has now many equivalent definitions. D.Lazard (talk) 12:48, 18 September 2017 (UTC)

Seems to me that the edit didn't solve the problem. The edit is confusing. "has many equivalent definitions" is ambiguous. Equivalent to WHAT?? The most obvious interpretation is "equivalent to the geometrical definition" which just returns us to the problem of what curvature the space we are considering has. The Euclidean geometry definition is just as valid as any other, of course. So, the problem is NOT that the geometrical definition is wrong, rather its context isn't clear (isn't explicit). I suggest an edit to include some qualifying term such as Euclidean, flat-2d, historically, or classically.

I'd also like to see more than 6 significant digits in the lead!! (Which states that the ancients knew it to 7.) I'm also slightly uncomfortable with the claim that it can't be exactly expressed as a fraction. It can be exactly expressed, as well as exactly defined, as a recurring fraction (which is not a ratio of two whole numbers). A minor point.98.21.70.161 (talk) 18:26, 24 October 2017 (UTC)

Strongly disagree with Anon. The lead should be clear and simple for non-mathematicians. Euclidean space is reasonably considered the "default" space. Six digits is fine in the lead. "A fraction" clearly does not refer to continued fractions. --Macrakis (talk) 19:16, 24 October 2017 (UTC)

¶ Not quite related to this, but a few years ago on some other place on the internet (I have forgotten where), I was whining about the precision of calculations made with a value of pi that only went to ten decimal spaces, and someone responded that (and I probably have misremembered the exact details) that the value to ten decimal places was sufficient to calculate the volume of the entire galaxy to within a cubic inch, or something like that. Whatever the demonstration of the accuracy of a relatively short value of pi might be, I would be very appreciative if someone knowledgeable would add the appropriate statement and citation to the article. Sussmanbern (talk) 17:37, 6 November 2017 (UTC)

It's already there, in the "Motives for computing π" section. —David Eppstein (talk) 17:47, 6 November 2017 (UTC)

Thank you, Mr. Epstein, for your prompt response and especially for not calling me a jerk. Sussmanbern (talk) 04:05, 7 November 2017 (UTC)

"Digits" of pi has to include all digits, including the leading three. Thus the current article text:
"Decimal: The first 50 decimal digits are 3.14159265358979323846264338327950288419716939937510..."
"Hexadecimal: The base 16 approximation to 20 digits is 3.243F6A8885A308D31319..."
need to be changed to either

• count all digits
  

"Decimal: The first 51 decimal digits are 3.14159265358979323846264338327950288419716939937510..."
"Hexadecimal: The base 16 approximation to 21 digits is 3.243F6A8885A308D31319..."

• use amended wording
  

"Decimal: The first 50 decimal digits to the right of the decimal point are 3.14159265358979323846264338327950288419716939937510..."
"Hexadecimal: The base 16 approximation to 20 digits to the right of the decimal point is 3.243F6A8885A308D31319..."
—DIV (120.17.34.237 (talk) 13:29, 19 October 2017 (UTC))

Agreed. Currently, all four approximations (decimal, binary, hexadecimal, and sexagesimal) ignore the leading digit '3'. — Loadmaster (talk) 21:41, 19 October 2017 (UTC)

something like ${\displaystyle 1/2}$ is not right, ${\displaystyle {\frac {1}{2}}}$ is right 165.255.71.229 (talk) 09:01, 7 December 2017 (UTC)

This is an issue for MOS:MATH, not this specific article, but ${\displaystyle 1/2}$ (or more usually just 1/2) is usually preferable for inline formulas. For displayed formulas,

${\displaystyle {\frac {1}{2}}}$

would be more likely to be correct, but even there the horizontal slashed form is often a better choice. —David Eppstein (talk) 09:24, 7 December 2017 (UTC)

Hi there, it seems to me that many users might come to this page looking for the first few digits of the decimal expansion of pi. To find these it is necessary to scroll to the sixth section, where they aren't particularly well sign-posted. Would it make sense to have more than the current 5 decimal places in the first paragraph? 82.150.96.2 (talk) 10:01, 4 January 2018 (UTC)

I have added more digits to the infobox. Sławomir Biały (talk) 11:13, 4 January 2018 (UTC)

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In the Approximate value section, would someone change "approximation" to "truncation" in several places? These are not the value of pi rounded to the specified number of places, but the value of pi truncated to the specified number of places. Please consider also adding "3.8Ti0l" as an alternate sexagesimal representation. 64.132.59.226 (talk) 19:25, 14 February 2018 (UTC)

Actually, should we have 50 places in the fractional part for each representation? These truncations would be:

1. Binary: 11.00100100001111110110101010001000100001011010001100
2. Hexadecimal: 3.243F6A8885A308D313198A2E03707344A4093822299F31D008
3. Sexagesimal: 3.8Ti0lPr7OvaHh4T7A3fHqaCEaipoFX7Nx9DmMCLjMuldiSbwNL

64.132.59.226 (talk) 19:32, 14 February 2018 (UTC)

I don't see that any of these changes would improve the article. A truncation, just like a rounding is an approximation. The values to various bases are given to roughly the same precision, and the numeric format for sexagesimal is surely more readable than yours... (I'm not Babylonian, so I may be missing local knowledge here.) Imaginatorium (talk) 06:04, 17 February 2018 (UTC)

There's no harm in referring to the truncations as truncations. But there's also no reason to include lots of sexagesimal, hexadecimal, or binary digits. Sławomir Biały (talk) 13:11, 17 February 2018 (UTC)

I think there is a (small) harm: using "truncation" to refer to the thing that has been truncated, rather than the process of truncation, is not very natural. "Approximation" (which is also correct) is much more usual, particularly in terms of readability for non-technical readers who, though they are native speakers unlike you, do not have your mathematical fluency, and would have to cogitate slightly to see the meaning. I agree with you about non-decimal representations; frankly I do not think they are notable at all. Imaginatorium (talk) 14:35, 17 February 2018 (UTC)

This is a standard usage of the English word "truncation": in the same way that "approximation" can refer both to the process of approximating a result, so may "truncation" refer both to the process and result. Approximation suggests rounding rather than truncation. Sławomir Biały (talk) 15:14, 17 February 2018 (UTC)

"Truncation" is too technical; "approximation" is both accurate and understandable.

I also don't see the point of the non-decimal representations. --Macrakis (talk) 15:16, 17 February 2018 (UTC)

What if we change the likes of "The base 2 approximation to 48 digits" to "The first 48 binary digits"? To me the former (incorrectly) says "rounding" where the latter (correctly) says "truncation". 64.132.59.226 (talk) 15:05, 20 February 2018 (UTC)

Yes, it is possible to write something like "the first so many digits". But does it actually help the typical (not very mathematically fluent) reader? The claim that "approximation means rounding (and truncation is not rounding)" is plainly false: another term for truncation is "rounding down". And anyway, the difference between truncation to 20 digits and (some particular form, e.g. "banker's rounding") of rounding to 20 digits is much smaller than the difference between showing 20 digits or 18 digits. And this is supremely irrelevant to the point, which is simply to show a number of approximations (as they are) to pi in different bases. Imaginatorium (talk) 18:32, 20 February 2018 (UTC)

If we're going to argue that there are different approximations of a number to a certain number of digits, then surely the definite article "The base 2 approximation to 48 digits" and "The base 16 approximation" is not in accord with this viewpoint. Sławomir Biały (talk) 19:02, 20 February 2018 (UTC)

I get that there are levels of understanding and that we wish to avoid drowning a non-technical reader with details that are too technical. But it is even better if we can achieve that AND also avoid making false statements, even statements that are false merely on a technicality. If "The first 50 digits are" and "The approximation to 50 digits is" are indistinguishable to the novice, but the latter is objectionable to some technical readers then let us go with the former. 64.132.59.226 (talk) 19:39, 20 February 2018 (UTC)

The animation use the circle's diameter (and not the the radius) and finishes by labeling the full rotation of the circle as being pi and (not 2 times pi) — Preceding unsigned comment added by NicolasLefebvre54 (talk • contribs) 04:06, 20 March 2018 (UTC)

The GIF is correct: it shows the diameter as 1, and as the circle rolls we see the circumference is pi. Imaginatorium (talk) 04:17, 20 March 2018 (UTC)

The article states that: differential calculus typically precedes integral calculus in the university curriculum, so it is desirable to have a definition of π that does not rely on the latter. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following: π is twice the smallest positive number at which the cosine function equals 0.

I have looked into one of the citations "Walter Rudin - Principle of Mathematical Analysis" (p 182-183), and found out that the existence of such constant is proved using an integral. Is there another proof that doesn't uses integral calculus at all? Otherwise this statement seems to be false. — Preceding unsigned comment added by 109.67.101.82 (talk) 08:53, 23 March 2018 (UTC)

There's a difference between using a definition and proving that the definition is well-defined. The former doesn't require integral calculus while the latter most certainly does. Elementary calculus students do not typically have the mathematical maturity to explore any rigorous proofs of the existence of pi from any of these definitions (even for the arc length formula, they need to know how to prove that it works for continuously differentiable curves).--Jasper Deng (talk) 09:17, 23 March 2018 (UTC)

The cosine can be defined as the solution of a differential equation ${\displaystyle C''(x)+C(x)=0}$ with the initial condition ${\displaystyle C(x)=1}$. This avoids using an integral. A proof of periodicity that completely avoids integration can be found in the second volume of Bourbaki's topology. Sławomir Biały (talk) 09:53, 23 March 2018 (UTC)

I'm curious though; is there an elementary proof that the cosine function has roots? Establishing periodicity isn't enough; for all we know, it could be periodic without ever crossing the x-axis.--Jasper Deng (talk) 09:58, 23 March 2018 (UTC)

π is defined there in terms of a period. But they also relate the periods to the zeros of the cosine. (Though I wouldn't call Bourbaki's treatment "elementary".) An elementary argument using only the Mean Value Theorem can be found in Ahlfors. Sławomir Biały (talk) 10:19, 23 March 2018 (UTC)

(edit conflict) (all this discussion was published when I wrote the following answer): Landau's definition supposes that one has a definition of the cosine that is independent from π. Therefore I would prefer the (almost equivalent) definition: π is half of the smallest period of any solution of the differential equation ${\displaystyle y''+y=0.}$ This definition does not uses anything else than derivative. I do not know whether there is an elementary proof that the solutions are periodic and have all the same period, but I guess so. Here are some hints. At least, the solutions are bounded (hint: at a point where the function is positive and increasing, the derivative decreases, and decreases more and more quickly when y increases, so, the derivative must reach zero). As translating and scaling a solution gives another solution, there is a solution such that y(0)=1 and y'(0)=0, and this solution is unique, as the values of the of y and y' at 0 define the Taylor expansion at any order. If one call cosine this solution, it must have the value 0 for some x, because the function is bounded between two local extrema, the second derivative, and thus the function must have a zero. It remains to prove that cosine is periodic, but the existence of Landau's constant is proved (up to technical details). D.Lazard (talk) 10:22, 23 March 2018 (UTC)

Question; Why does this keep getting deleted for "unsourced"? It is clearly stated that this is engraved in stone at the Washington Park MAX station (Portland, Oregon). I took that picture. What else would make it "sourced"?

As for "trivial", a wrong value engraved in stone at a very public location is "trivial"??? Then, what about the one right above it about Knuth's silly version numbers? Would you delete that as well? Williampfeifer (talk) 21:24, 14 April 2018 (UTC)

It needs a reliable source. See finding reliable sources for a guide, but by any criteria a photo taken by an editor is not a reliable source. And I agree it is trivial. An embarrassing mistake by someone but, like other similar mistakes where someone spells a name wrong in a sign, hardly significant. Finally the text you added to explain it, "The digits displayed are digits 1 to 10, 101 to 110, 201 to 210 and so on", did not make sense on their own. You cannot assume someone can see the image – someone might be using a screen reader or be browsing WP over a connection to slow to load images – so article text should make sense without seeing the image.--JohnBlackburne^words_deeds 21:39, 14 April 2018 (UTC)

The article should not contain a lengthy decimal representation of a number that isn't π. In fact, the added text

3.1415926535 821480865144 288109757245 870066330572 703698336733 620005681271 420199561150 244594555982 534904897932

has more digits of non-π than the article has of the digits of π itself! Sławomir Biały (talk) 22:33, 14 April 2018 (UTC)

Would this count as a reliable source?: https://blog.plover.com/math/wrong-pi.html
I agree, a simple mistake is trivial. What makes it non-trivial is that it's engraved in stone in a very public place, for about 20 years now, and isn't been fixed. (Yes, the city knows about it; there was an article in the Oregonian newspaper about it back then, but I can't figure out how to access old articles on that paper)
I'm aware that someone could not see that picture. That is why I added the numbers as text. However, this text addition itself is being objected to. So I'm caught between a rock and a hard place. Williampfeifer (talk) 01:44, 15 April 2018 (UTC)

If a reliable source can be added, I am somewhat neutral on the image. I do not think the text of the engraving should be included in the article however. The caption can be clarified to resolve JohnBlackburne's concern. Sławomir Biały (talk) 01:59, 15 April 2018 (UTC)

How about something like this:
At the Washington Park MAX station (Portland, Oregon) are granite plaques honoring various attributes of Oregon. One plaque, intending to honor the science sector, supposedly bears the engraved value of pi. However, the digits are incorrect, beginning at the 12th place after the decimal^[1]. Instead of 3.14159265358979323846..., it shows 3.1415926535821480865144..., and so on. In fact, the digits, as engraved on the plaque are pi's digits 1 to 10, followed by digits 101 to 110, then the digits 201 to 210 and so on. Williampfeifer (talk) 02:55, 15 April 2018 (UTC)

Again, this sort of thing is common, where an error somewhere along the line leads to a sign with an embarrassing error. It’s carved into stone, but they could pay someone to do it again – there is always a cost to fix something like this. A blog is not a reliable source. If as it seems there are no reliable sources it confirms it is too trivial for mainstream coverage. Wikipedia is based on reliable sources, we cannot base an article on a blog or on an editors personal observations.--JohnBlackburne^words_deeds 03:10, 15 April 2018 (UTC)

It was covered in the mainstream The_Oregonian paper back then. Mostly because someone happened to notice it and call the paper. They were astonished and wondered how many people would have noticed that. They contacted the artist, who admitted the mistake and promised he would fix it at his expense. However, nothing has happened so far. Unfortunately, I cannot get at the article in The Oregonian. They don't appear to allow access to 20 year old articles any longer. Oh, well, at least I'm learning about Wikipedia formatting and conventions. Williampfeifer (talk) 03:59, 15 April 2018 (UTC)

It's pure trivia. HiLo48 (talk) 03:33, 15 April 2018 (UTC)

Ah, yes. And the one about Donald Knuth's silly version numbers is very deep and profound. Williampfeifer (talk) 03:59, 15 April 2018 (UTC)

Please read Wikipedia:Other stuff exists. HiLo48 (talk) 05:41, 15 April 2018 (UTC)

Fact is

• There is an engraving in stone, involving up to the 910-th decimal of π, which suggests a wrong value of π.
• It is hard to argue that a sequence of twelve or whatever decimals is not from π.
• The claim "... then the digits 201 to 210 and so on" is wrong as its stands, since after the decimals #901-910 the decimals #11-15 are engraved (I gave a scenario reasoning this in my edit).
• This memorial is not for an algorithm calculating selected digits, but is intended to celebrate π, in a similarly superficial way as Pi-day.
• Referring to Knuth's version numbering refers to this article, as would be "Indiana Pi Law".

My opinion is

• You have not the slightest chance to establish this petitesse in WP against the opinion of at least FOUR WP-editors in good standing, and almost no chance if it were important.
• It is of no importance that I would want to have a trimmed version in the article, that requesting additional sources and mentioning WP:OR here is ridiculous, and that I consider personal valuations of what is trivial as non-neutral POV (every proven theorem is a (?trivial?) tautology).
• Walking away might be the best solution.

Sorry, Williampfeifer. Purgy (talk) 07:54, 15 April 2018 (UTC)

You're right; will do. I think i'll also just walk away from trying to contribute anything to WP. Wikipedia:Other stuff exists shows that one cannot use existing precedence of what is acceptable and what is "trivia"; one is completely at the mercy of the pettiness of the in-crowd. Williampfeifer (talk) 20:12, 15 April 2018 (UTC)

Please drop the attacks on other editors. I personally am hardly a part of any "in-crowd". Until a couple of weeks ago I hadn't contributed to Wikipedia for four years. I am not familiar with the work of any of the other contributors to this discussion. HiLo48 (talk) 20:50, 15 April 2018 (UTC)

This edit request to Pi has been answered. Set the |answered= parameter to no to reactivate your request.

No formula was derived to calculate pi in 21st century until 2017. In 2017 a young Indian Mathematician Karthikeya Gounder alias Karthik(18 years old) had derived a new formula to calculate the value of pi upto infinite digits in the paper " π-The Transcendental Number"[1]. In that paper,Pi is expressed in a new way rather than a series converging to the exact value. It relates the mathematical constant pi with Trignometric tan.

References 1.Karthikeya Gounder: π-The Transcendental Number

MAT-KOL(BanjaLuka)
ISSN:0354-6969(p),ISSN:1986-5228(o)

Vol.XXIII(1)(2017),61-66 DOI:10.7251/MK1701061G

Paper link: http://www.imvibl.org/dmbl/meso/mat_kol_23_1_2017/mat_kol_23_1_2017_61_66.pdf

Additional reference 1.http://www.academia.edu/32137897/Karthikeya_Gounder_π-THE_TRANSCENDENTAL_NUMBER_MAT-KOL_XXIII_1_2017_61-66 Karthikeyagounderr (talk) 05:08, 29 April 2018 (UTC)

 Not done @Karthikeyagounderr: This is nothing remarkable. The Leibniz series and associated trigonometric formula has been known for ages; Ramanujan found many numerically faster formulae decades ago. Your method is also nothing new, see Area of a circle#Archimedes' proof. And finally, you committed clear and blatant plagiarism of the lead section of the Wikipedia article. Also your formula is basically useless; to do calculus with degrees rather than radians one must multiply the argument by ${\displaystyle \pi /180}$ in any case. Using this we obtain for your expression, with ${\displaystyle t=10^{-x}}$, ${\displaystyle \lim _{t\to 0}180\tan({\frac {\pi }{180}}t){\frac {1}{t}}=\pi }$. This is a true statement but one not useful for calculations, since the numerical value of ${\displaystyle \pi }$ is itself involved in calculating values of the tangent function for arguments that are rational numbers when expressed in degrees. And how do we calculate tangent? Taylor series or other formulae using series, my friend, so you're not really avoiding the use of infinite series this way. Even I, an amateur, can do better than this: ${\displaystyle \pi =4\arctan(1)}$ involves the evaluation of just one trigonometric function rather than many. And above all, even disregarding how poorly written this is, this looks like your own self publication and original research, and therefore cannot be included. If you think my comment is harsh, then you should do your research and get some more formal training before you attempt to publish papers.--Jasper Deng (talk) 05:32, 29 April 2018 (UTC)

There is a new type of ${\displaystyle \pi }$ calculus in Regular rule published in 2018 by Amir Forsati. its proof is written on that page too.

${\displaystyle \lim _{n\to \infty }n\tan({\frac {\pi }{n}})=\pi }$

Please not that the ${\displaystyle \pi }$ in the tangent is 180 degrees.

proof confirmation by StackExchange geeks. — Preceding unsigned comment added by Pw2iur (talk • contribs) 18:31, 10 April 2018 (UTC)

Neither of your links is a reliable source, and your formula has nothing to do with pi (plug any other number in place of pi on the left hand side and you will get the same number on the right hand side). —David Eppstein (talk) 19:03, 10 April 2018 (UTC)

Thank you for your review. First of all this proof is not a series typed proof. Have you please read the ${\displaystyle \pi _{n}}$ in that article for regular polygons? I think this should be a new and exiting declaration of ${\displaystyle \pi }$ for calculating area of regular polygons and circles by ${\displaystyle S=\pi _{n}r^{2}}$ (${\displaystyle n}$ the count of the sides in polygon). I suggest read ${\displaystyle \pi }$ test in Regular rule.

I'm the author of this article; The source urls truly are not reliable and because this theory is new and is in the way of being an academic article. But i think the best sources for this method of ${\displaystyle \pi }$ proof is Math that has a unit logic and reliable. if you couldn't put urls to this article, you can add math proof to it. regards.— Preceding unsigned comment added by Pw2iur (talk • contribs)

@Pw2iur: What do you mean by "the ${\displaystyle \pi }$ in the tangent is 180 degrees."? I have rejected a similar thing in the "Pi Formula in 21st Century" section below, on the grounds that this is nothing groundbreaking (see Area of a circle#Archimedes's proof). Please also read WP:NOR.--Jasper Deng (talk) 07:25, 29 April 2018 (UTC)

Thank you for reply. I've changed the reference to article. please read the true Regular rule article @Jasper Deng:.

Does anybody think it would be appropriate to include a section with pi written out to 1,000 decimal places? DangleSnipeCelly (talk) 20:43, 4 May 2018 (UTC)DangleSnipeCelly1

There has been a lot of discussion in the archive. The consensus is that it is not appropriate to include lots of digits in the article. Sławomir Biały (talk) 20:48, 4 May 2018 (UTC)

OK. Thank you. DangleSnipeCelly (talk) 23:48, 4 May 2018 (UTC)DangleSnipeCelly1