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Continued fractions

From the article: "Truncating the continued fraction at any point generates a fraction that provides an approximation for π; two such fractions (22/7 and 355/113) have been used historically to approximate the constant."

Isn't it fair to say that 3, or 3/1, has also been used historically to approximate the constant? It's also a truncation of the continued fraction expansion, albeit a trivial one. It would appear to be used in the Old Testament, for example (1 Kings 7:23-26, 2 Chronicles 4:2-5). This is even alluded to in the article, and discussed at greater length at Approximations of π#Imputed biblical value.

One might object, however, that the number 3 is not commonly regarded as a "fraction", or that it's a matter of interpretation whether the Bible actually uses this approximation. Still, it remains that 3 is a rough-and-ready approximation of pi that has surely been used in many cultures, before they figured out about 22/7.

Here's one (not particularly strong) citation that the approximation pi=3 was used in ancient China: [1] This other link: [2] claims that the ancient Babylonians used pi=3.

Anyway, I can see two sides of this, so I thought I'd bring it to the talk page. What do others think? -GTBacchus^(talk) 17:13, 14 March 2015 (UTC)

Our article on continued fractions defines them: "In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on." citing the Encyclopedia Britannica. The integer 3 would not seem to fit this definition. While I can see a zeroth term argument for 3, I don't think it is important enough, nor helpful to our readers, to include it in our Pi article.--agr (talk) 12:05, 15 March 2015 (UTC)

I don't think this argument is valid. The article has a (not very mathematical) description of the complete continued fraction; mathematically, there would be something about a sequence, and truncation of the sequence would certainly be allowed anywhere, including the truncation to one term. But this is a somewhat separate issue from the question (what is the question exactly?)... the following is certainly true:

• "Truncating the continued fraction at any point yields a rational approximation for π, and the first three of these (3, 22/7, and 355/113) have been used historically as approximations."

Imaginatorium (talk) 13:05, 15 March 2015 (UTC)

A standard result about continued fractions is that every real number, including 3, has an expansion as one, and the expansion is terminating if and only if the real number is rational. I don't know of any mathematical authority that makes an exception for integers, but not counting a one-term sequence as a continued fraction expansion. A whole slew of results would have to be written with special exceptions for short continued fractions if we were to disallow expansions such as [3].

One problem, though, with the suggested edit. The first three convergents of the continued fraction expansion of pi are 3, 22/7 and 333/106. The famous approximation 355/113 is the fourth. The approximation 333/106 has been used historically, but not as often as the others. -GTBacchus^(talk) 19:08, 15 March 2015 (UTC)

Per above, I withdraw my objection to including 3 in something like the language proposed.--agr (talk) 16:59, 16 March 2015 (UTC)

How about:

• "Truncating the continued fraction at any point yields a rational approximation for π; the first four of these are 3, 22/7, 333/106, and 355/113. The second and fourth of these numbers are among the most widely used historical approximations."

Does something like that work? It seems slightly awkward to me... -GTBacchus^(talk) 15:32, 18 March 2015 (UTC)

Looks OK; I would just like to put in a word for the first, 3, which I think is greatly underrated as an approximation for pi. There was a fuss in Japan a few years ago when it was revealed that primary school children were told they could use the value 3 -- this caused outrage that they were not being taught the "correct" value of 3.14. Imaginatorium (talk) 18:02, 18 March 2015 (UTC)

Alright, how about this: "Truncating the continued fraction at any point yields a rational approximation for π; the first four of these are 3, 22/7, 333/106, and 355/113. These numbers are among the most well-known and widely used historical approximations." The only weakness I see there is the implication that 333/106 is well-known or widely-used, and it isn't, particularly. If you believe certain interpretations, it's the Biblical approximation, and that's probably what it's most famous for, besides just being a convergent of pi. -GTBacchus^(talk) 22:10, 18 March 2015 (UTC)

Continued fraction II

Is it known whether the (regular) continued fraction (written in the form [3; 7, 15, 1, 292,...]) of Pi contains all natural numbers?--78.54.83.14 (talk) 23:19, 5 April 2015 (UTC)

No. I'm not sure if it's known whether it has arbitrarily large numbers, but I'm sure it's not known that it has all natural numbers. — Arthur Rubin (talk) 00:29, 6 April 2015 (UTC)

By definition, continued fractions written in that form are sequences of natural numbers, unless I'm very mistaken. Dicklyon (talk) 04:25, 6 April 2015 (UTC)

Yes, but the question is whether they all show up somewhere in the sequence or whether some of them are omitted. —David Eppstein (talk) 05:42, 6 April 2015 (UTC)

Oh, sorry, I sure did miss the intent of the question. Better would have been to ask whether it contains "every natural number". Interesting conjecture; I'd bet that it could be proved false. Dicklyon (talk) 05:51, 6 April 2015 (UTC)

If someone tackles this "challenge" and finds a conclusive proof - no matter if positive or negative, please post here. I just looked up the first 20.000 numbers of the continued fraction - the smallest number absent there is 103. I think if the answer to my question is "no", the smallest counterexample will be some *big* number.--78.54.83.14 (talk) 15:07, 6 April 2015 (UTC)

I think it's likely to be true and likely to be very hard to prove. —David Eppstein (talk) 15:33, 6 April 2015 (UTC)

See OEIS A225802 "Position of first occurrence of n in continued fraction for Pi": "All positive integers <= 49003 occur in the first 15,000,000,000 terms of the c.f. (the first that do not are 49004, 50471, 53486, 56315, 58255, ...". Of course, that does not prove that all natural numbers will eventually appear. --Macrakis (talk) 19:01, 6 April 2015 (UTC)

Requested edit of guy's name

I don't really know how to use talk section. anyhow. it seems the guy who holds the guinnes record for most digits memorized is named Chao Lu not Lu Chao and the link from his name gives you some artist dude who don't know 6 numbers of pi. — Preceding unsigned comment added by Tlaxómat (talk • contribs) 06:40, 26 June 2015 (UTC)

--DYKnapp (talk) 12:03, 14 July 2015 (UTC) Now, the world record was some Japanese guy..... I think he memorized 100,000 digits. I'll look into it. DYKnapp (talk) 12:03, 14 July 2015 (UTC)

Computations

DYKnapp (talk) 12:05, 14 July 2015 (UTC) ¶ I am really ignorant of the details of the calculated number, especially beyond 10 or 20 digits. I can understand that pi is the ratio ... etc., but the measurement of the radius and circumference of an actual circle would yield a fraction that was, at best, an approximation - it being so difficult to make a linear measure of the circle's rim and then so difficult measuring to molecular precision. So pi is APPROXIMATED at being 22/7 or some more convoluted fraction, but this is at best an approximation. When computers churn out pi to fifty or a hundred or a million decimal places, what fraction are they really calculating?? Sussmanbern (talk) 14:55, 17 March 2015 (UTC)

Please note this talk page is for discussing the article, not asking questions about the subject matter. Very briefly: they are not "computing fractions" at all; try reading section 3 of the article, but I'm afraid this is quite mathematically difficult. It really has nothing to do with measuring physical circles -- even 1000 digits is enough to measure a circle the size of the universe to an accuracy of a fraction of an electron size. (if electrons have size??) Imaginatorium (talk) 17:07, 17 March 2015 (UTC)

I Giuseppe Stagno the author of pi=2^54*sin(90/2^54)*2 ask me anything. Thank you, — Preceding unsigned comment added by 99.247.171.90 (talk) 18:22, 20 March 2015 (UTC)

--DYKnapp (talk) 12:05, 14 July 2015 (UTC) Check out the Approximations of pi page. There are lots of ways to calculate pi. P.S. the Approximations of Pi page really needs work. DYKnapp (talk) 12:05, 14 July 2015 (UTC)

Add in popular culture reference

Suggestion to add in life_of_pi book, as the main character's name 'pays tribute to the irrational number which is the ratio of the circumference of a circle to its diameter'

Is there any depth at all to the reference? Or is it just a shout-out? Can you find reliable sources saying so? If not, it shouldn't be in the article. —David Eppstein (talk) 02:48, 23 July 2015 (UTC)

Nontrivial infinitesimal arguments

There is an interesting new addition stating that non-trivial infinitesimal arguments are needed to show that the ratio C/d does not depend on the size of the circle. While this is something that is certainly worth exploring, I don't think the stackexchange discussion is very convincing. At least, it is not a reliable source. I think a better source would be a very good idea.

On a related matter, most modern treatments now define π in terms of the periodicity of trigonometric functions, but the article does not explain why this appears to be preferred nowadays. I have checked a few of my sources, and they also do not explain why they do it this way. (There is an easy reason, that the trigonometric functions are definable using the machinery of calculus and/or complex variables. But it seems like there might be other reasons too.) I think some discussion of this would be valuable. Sławomir Biały (talk) 11:08, 11 July 2015 (UTC)

I see the content was reverted instead of being improved. This is somewhat unfortunate since there is indeed some discussion missing on this very point. Sławomir Biały (talk) 12:26, 11 July 2015 (UTC)

Being an encyclopedia, adding content that is not supported by reliable sources often gets reverted. Those sources only contained some chat on the internet, and we cannot use that. -- [[User:Edokter]] {{talk}} 12:50, 11 July 2015 (UTC)

In principle I agree, but it would be nice if someone tried to find better sources for this, rather than removing it. Sławomir Biały (talk) 13:16, 11 July 2015 (UTC)

I reintroduced my additions and provided a reference to Euclide. Ok I know that this may be considerd as original research (not sure if it is), so it may be no good source. Maybe you find a better one. However removing an obviously correct statement that has some importance is not ok. Even if the 3 links I provided are probably not hte best ones.--Jocme (talk) 14:55, 11 July 2015 (UTC)

The concept is good, but even finding a neat way of stating it may be a challenge. I don't believe WP requires sourcing for uncontroversial statements (and something that could in principle be sourced should not be reverted). Yes, the geometric definition very much relies on the properties of Euclidean space; maybe something more bland like "and for validity requires proof that the ratio C/d is constant" might work. For example, arguments invoking elliptic or hyperbolic geometry provide a counterpoint without any mention of infinitesimals or limits, or even in Euclidean geometry the length of a fractal curve of dimension ≠1 does not scale linearly with a "diameter", but do we really want to introduce such an elaboration? —Quondum 15:09, 11 July 2015 (UTC)

To editor Edokter:, you're coming across as somewhat bossy, and you are clearly pre-empting a discussion in progress. You have not had agreement with your perspective and indeed some dissent, so you are in no position to say "Don't [...], period". Your action is not conducive to constructive cooperation. —Quondum 16:44, 11 July 2015 (UTC)

That is just policy; we add information we can source, and not before that. The merit of the subject itself is not relevant. We cannot add statements that are sourced solely by answers from some internet fora. -- [[User:Edokter]] {{talk}} 17:25, 11 July 2015 (UTC)

Tsts ... ok next time I don't add sources ... Seems to be more accepted. When you read the article you will find a lot of unsourced statements.--Jocme (talk) 19:55, 23 July 2015 (UTC)

Far from being "obviously correct" as Jocme asserts, statements of the form "proof of X can only be done using Y" are usually false. Yes, a proof that the ratio is constant in Euclidean geometry is necessary to use this definition, and it is relevant to point out that it is not constant in other geometries. But I doubt it requires infinitesimals in any sense. For one thing, anything that can be done with infinitesimals can be done with limits and no infinitesimals. And Euclid is not a valid reference to a requirement for infinitesimals, because Euclid does not discuss infinitesimals, and a reference is needed for an assertion of this sort. —David Eppstein (talk) 17:31, 11 July 2015 (UTC)

You're right. I should have been more precise: for the definition to be valid it has to be proved that the fraction is constant. The esiest way to prove that is probably using infinitesimal/limit arguments.Jocme (talk) 19:55, 23 July 2015 (UTC)

Miscellaneous thoughts:

1. Limit arguments are still part of the "infinitesimal calculus", as it is occasionally still called, so that doesn't by itself invalidate the claim.
2. However, far from having a reliable source to substantiate the claim, it's not even very clear what the claim means. At this time I agree with Edokter and David that the material should not appear.
3. As for improving rather than reverting, it is my position that it is usually cleaner to just revert atomically while the matter is under discussion, unless it's clear that the new material should stay in some form. That minimizes the "version hell". Anyone who's ever worked with a revision-controlled software development system should understand what I'm talking about.
4. Even if the claim were somehow substantiated, the putative corollary that it would be "more convenient" to define π as the circumference of a circle with diameter 1 is still totally unjustified. More convenient how? You still have to prove that all such circles have the same circumference, so I don't see much benefit in any case.
5. Edokter, I think you're correct on policy, but Quondum's complaint goes more to style. This has come up before and you might take it on board.

That's it for now. --Trovatore (talk) 19:47, 11 July 2015 (UTC)

Some comments. This thread contains two different discussions, both initiated in the first post of Sławomir Biały. The first one is about the reverted edit. I agree with the revert, not only because the edit is poorly sourced, but also because it does not reflect the content of the source. Moreover, the proof that C/d is independent of d, may not, in any case, be qualified as "non-trivial". In fact, the true question is how the "length of the circumference" is defined. At Euclid's time, it was not formally defined, but was computed as the limit of the perimeters of regular inscribed or circumscribed polygons. Thus the fact that the quotient is independent of the size of the circle is simply the formula lim (u_n/d) = lim (u_n)/d, which can hardly be qualified as "non-trivial". In modern formulations, the length of a curve arc is defined by an integral, which is nothing else (by definition) than the limit of polygons approximating the curve. In this context, the proof that C/d is independent of d is essentially the same formula, that is the linearity of the integral operator, which, also may hardly be qualified as "non-trivial", as soon as integration is defined.

Sure, what one considers as trivial depends on the previous proven statements. In this way you can always argue that any statement is trivial since any statement can be derived as a corollary of another statement. So we probably can agree that what one considers as a trivial statement depends on the context (known theorems) and the complexity of the proof from this context. The context that is given for the definition is Euklidean geometry. And it is nontrivial to derive the statement from the axioms. Also Euklid thought that a proof is necessary. However, Edokter thinks that we should not add Euklids opinion here and removed my referece--Jocme (talk) 19:55, 23 July 2015 (UTC)

Where in Euclid is the proportionality of circle perimeters considered? The closest I can find is XII.2 on the proportionality of areas (which is indeed proven using a limiting argument). —David Eppstein (talk) 22:00, 23 July 2015 (UTC)

This leads to the second part of the discussion: why the old definition of π is nowadays replaced by other definitions involving trigonometry. IMHO, it results of the preceding paragraph: this leads to a more elementary definition. However, if sine and cosine are defined from angles, their definition depends on the definition of "radian", which involves the length of an arc. Therefore, in my opinion, the most elementary formal definition of π is through the definition of sine and cosine by a series or a differential equation. However, the intuitive definition through arc length, remains important for explaining the importance of π. D.Lazard (talk) 02:26, 12 July 2015 (UTC)

Although I disagree with the wording, I think the original edit was getting at the fact that there is something not altogether adequate about the definition of π as C/d. It is perhaps not just that one needs to prove that this ratio is independent of the size of the circle, but also that the definition of circumference is rather tricky to do in a way that does not beg the question if this is to be suitable as a definition of the constant. For example, even if one defines the circumference as an integral, it has to be the right integral. The most natural choice is to use the arclength parameterization. But π already appears as half the period of this parameterization, and indeed the integral is just $\int _{0}^{2\pi }r\,d\theta$, so that clearly will not do as a definition of the circumference if we wish to avoid circularity (no pun intended) in our definition of π. One can obtain the circumference in a more suitable way using the integral $2r\int _{-r}^{r}{\frac {dx}{\sqrt {r^{2}-x^{2}}}}$. It is a consequence of the change of variables formula that this is homogeneous of degree one in r, and so has the form $2\pi r$ for some constant $2\pi$. But this is pretty far from what the article currently says.

So I think the point remains, that the definition of π as C/d is not entirely satisfactory by modern standards, and the article should point this out (with appropriate sourcing). I'm not suggesting that we should dwell on the details overmuch in the definition section, but I don't think it is good practice simply to ignore such nuances entirely. Sławomir Biały (talk) 14:27, 12 July 2015 (UTC)

I don't see this as a huge issue. I wouldn't think arc length is standardly defined as an integral, is it? That seems awfully non-coordinate-free for the modern aesthetic. I would expect the definition to go something like, for every δ there is a way of breaking up the curve such that, for every refinement of it, the sum of the distances along that refinement is within δ of L. Or something like that. Completely coordinate-free and almost obviously independent of the paramaterization, with some obvious quibbles.

Anyway, that strikes me as maybe a little bit tangential in an article about π. --Trovatore (talk) 21:13, 12 July 2015 (UTC)

You could also use the Crofton formula as a definition of arc length: it's the measure (with multiplicity) of the set of lines crossing a curve, for the unique (up to scale) affine-invariant measure on lines. That seems pretty coordinate-free to me. —David Eppstein (talk) 21:24, 12 July 2015 (UTC)

Arguing about whether it is coordinate-independent misses the point, I think. A satisfactory definition of the circumference of the circle that does not involve π is missing. Contrary to the widely-held belief, circumference is not a primitive geometrical notion. In fact, $C=\pi d$ is more suitable as a definition of the circumference than it is a definition of π. To treat this as a definition instead of π, we need to agree that there is an agreed-upon notion of circumference. But there is no such agreement. Indeed, there are many candidate ways to define circumference: one where we already know about trigonometric parameterizations, one using rectifiability, one using Hausdorff measures, one using integral geometry, and probably others if we try hard enough. It is not a completely trivial task to prove that these are equivalent to one another for smoothly embedded closed curves. And regardless of what our concept of "circumference" is, surely that must be an ingredient in the definition of π as well, if we adopt the view that π is defined by the $C/d$ formula. It is certainly worth pointing out that, at least in some sense, "circumference" is not a word that is not entirely trivial to make precise. Sławomir Biały (talk) 21:49, 12 July 2015 (UTC)

I disagree that $C=\pi d$ is a satisfactory definition of the circumference. There is no such difficulty defining what a circle is (locus of points at given distance from a point) and the proper definition of circumference is the arc length of this curve, however we define arc length. But we should be using a definition of arc length that applies at least to convex smooth curves (if not to rectifiable curves) not just to circles. —David Eppstein (talk) 00:59, 13 July 2015 (UTC)

Interjecting with a layman's perspective, I agree with Sławomir that it might help to have at least some indication that the assumption that the circumference in the geometric sense is a nontrivial construction, in particular that the intuition of it as a near-primitive concept cannot be assumed. The simple counterpoint of non-Euclidean geometries gives a nice flavour of this. —Quondum 02:01, 13 July 2015 (UTC)

I guess what I think is that most people have an intuitive notion of arc length that, while they may not know how to formalize it, is essentially correct, and for which all faithful formalizations will give the same answer in well-behaved cases.

So to me that makes this issue not particularly urgent. Just the same, this is a big article, and space might be found somewhere for a nod to the fact that, whether or not the notion is obvious, its formalization is not. --Trovatore (talk) 03:49, 13 July 2015 (UTC)

David, I think you are still missing my point. When a student of Euclidean geometry, without the techniques of the infinitesimal calculus, says "circumference", the only satisfactory definition available is $C=\pi d$. We might wave our hands about and talk about bits of string, or rolling circles, or whatever. But there is no other satisfactory definition of this geometrical concept available at that level of mathematical sophistication. So seeing a formula $\pi =C/d$ as a definition of π results in a vicious circle.

I am arguing that the article should make clear that circumference can be defined independently. So far, we have various proposals: using an integral, exhaustion, rectifiable curves, integral geometry, and measure theory. None of these are intuitively evident constructions, all are very different from one another, and all require a relatively high degree of mathematical sophistication (I have listed them approximately in increasing order). One thing they have in common is that they without exception rely on the modern notion of a limit (or, equivalently, a supremum) to make precise. That is essential to making the definition section useful to someone reading about π, to avoid the vicious circle that I referred to. It is a legitimate and non-trivial question how exactly circumference is defined, and we should be clear that this can be done in a way that does not rely on the defining formula $C=\pi d$. Sławomir Biały (talk) 12:19, 13 July 2015 (UTC)

General applications use 40 digits?

The article states that "Scientific applications generally require no more than 40 digits of π ..." Isn't this excessive? I've changed it to "about 15 digits", which seems more like the "general" precision used in most typical scientific applications. Note that it is probably reasonable to assume that most everyday scientific computations use IEEE 754 double-precision floating-point values, which provide only 16 decimal digits of precision. — Loadmaster (talk) 16:07, 17 July 2015 (UTC)

I believe the "about 40 digits" is in reference to the fact, quoted in the article, that 39 digits will allow you to compute the volume of the observable universe to within one hydrogen atom. As discussed in the article, Arndt and Haenel conclude that a few hundred digits would suffice for any scientific application, although earlier they do say that no more than 10 are needed for "practical calculations". According to this article, NIST regularly uses 32 digits of precision. Borwein and Plouffe indicate that there are applications in scientific computing that use a few thousand digits. There is a rather wide discrepancy here between 10 digits, 15 digits, 32 digits, hundreds of digits, and thousands of digits. Really this should be settled by what is in sources. Sławomir Biały (talk) 16:25, 17 July 2015 (UTC)

In fact, it may be best to echo what the sources say about the typical precision needed for π. There are a few cases where practical computation needs π to very high precision. For example, an algorithm that will compute the sine function for large arguments may require an internal value for 2/π that has 1133 bits of precision, which would be about 340 decimal digits. This problem is hinted at in our sine article which mentions trouble with sin(1022) but doesn't give any details. More info is in a 1992 paper by K.C.Ng. Sun donated a version of their math library called FDLIBM that handled these arguments correctly. Instead of handling large angles properly, some math libraries choose to raise a math exception called TLOSS (total loss of significance). EdJohnston (talk) 17:20, 17 July 2015 (UTC)

Borwein and Plouffe suggest that only a few hundred digits are likely to ever be needed in any scientific application. That seems like a statement that is easy to attribute, even if it seems to be a little on the high side. I think we could, without fear of committing original research, add that significantly fewer digits are needed in most practical applications. Sławomir Biały (talk) 17:50, 17 July 2015 (UTC)

Bear in mind that we are looking for a description of general or typical scientific application computations. While there are doubtless several citable examples using rather high precision, we cannot assume that they represent the most common typical computations. Some reasonable realistic examples might include things like the precision required for GPS satellite signals (which deal with relativistic space-time adjustments) and particle collider experiments. For more extreme precisions, we could mention a few examples as being atypical exceptions to the use of pi in general applications. — Loadmaster (talk) 22:14, 17 July 2015 (UTC)

In principle I agree, and indeed 15 digits seems like quite enough digits for general computations. However, I don't really see that there is much general agreement in sources about this. Arndt and Haenel, for example, mention that 10 is sufficient for many purposes (although I can't recall the precise wording at the moment). I think rather than focus on a number of digits suitable for "general" calculations (do GPS systems really qualify as a "general" use? That seems highly specialized), we should try to write text that is unambiguously supported by some reliable source. That's why I suggested the Borwein and Plouffe source. There is a clear, attributable, statement of a ballpark number of digits that should be adequate for any purpose whatsoever. We do not have a source that says 15 digits are enough for "general" scientific work, possibly because there is no such thing. Sławomir Biały (talk) 23:12, 17 July 2015 (UTC)

It might possibly be worth mentioning here (though maybe too technical and off-topic for the article itself) that for many algorithms in computational geometry, it is essential for the correctness of the algorithm to be able to compute precise combinatorial relations between values derived from the inputs, without approximation, even if those input values represent real-world quantities that are known only approximately. Failure to determine these relations exactly may lead to a crash in an implementation of an algorithm. And since the numbers being compared are often polynomials of the input values, the number of digits necessary to carry out the computation can blow up by a constant factor. So, for example, if you have input point coordinates represented to 10 decimal digits, but your scientific calculation involves computing a convex hull of these points, then avoiding crashes may require using intermediate values that are more like 20 decimal digits. And comparisons of lengths of polygonal chains are even worse because in those cases the numbers involve square roots rather than polynomials and we don't even know how much precision might be required. So claiming that a certain number of digits is sufficient based on the reasoning that it represents the real world more accurately than any measurement error, without taking into account the need for greater precision in intermediate steps, may be an oversimplification. —David Eppstein (talk) 23:38, 17 July 2015 (UTC)

Since it's not that critical for the theme of the article, we perhaps could rewrite the sentence to something along the lines of "10 decimal digits is sufficiently accurate for most physical measurements, but there are computations that require higher precisions up to several hundred digits. Beyond that, however, the primary motivation for these extreme computations is the human desire to break records." I.e., we want to emphasize that at some point the extreme number of digits is physically meaningless even if it is still mathematically interesting or useful. — Loadmaster (talk) 15:53, 18 July 2015 (UTC)

That sounds good to me. Martin Hogbin (talk) 09:00, 22 July 2015 (UTC)

How much precision is needed for various calculations is a technical matter and we should not be making broad generalizations. The ubiquitous x86 processor series supports 80-bit precision and Quadruple-precision floating-point format with 113-bit precision is specified in IEEE-754-2008, Furthermore editorializing about the motives of people computing Pi to high precision seems highly inappropriate.--agr (talk) 14:27, 24 July 2015 (UTC)

Tau redirect

[Note: This was originally on my talk page, copied from there by Helloholabonjournihaonamastegutentag to Talk:Tau (mathematical constant) after he failed to get any support from me, and then copied again by Helloholabonjournihaonamastegutentag (minus one or two comments about where would be best to hold the discussion) to here. —David Eppstein (talk) 01:18, 26 April 2015 (UTC)]

I did copy this from your talk page, because you said that this would be the best place for this discussion. Helloholabonjournihaonamastegutentag (talk) 05:48, 27 April 2015 (UTC)

Hi. I'm helloholabonjournihaonamasetgutentag. I noticed that you made the Tau (mathematical constant) page into a redirect into Pi. Why? I think that Tau should have its own page as more and more people are using it. Please check out [this video]. Thanks for considering! Helloholabonjournihaonamastegutentag (talk) 15:44, 23 April 2015 (UTC)

Because this has been discussed to death at Talk:Pi and Wikipedia:Articles for deletion/Tau (mathematics) and I didn't see any evidence that whoever created the new article had paid any attention to those discussions or brought anything new to them. —David Eppstein (talk) 17:23, 23 April 2015 (UTC)

I created that article. I understand that Pi is much more commonly used, but Tau has many advantages. Even if you prefer Pi, which is fine and is your choice, don't you think that Tau should at least have its own article? It seems better to have a Pi article and a Tau article. Helloholabonjournihaonamastegutentag (talk) 20:00, 23 April 2015 (UTC)

Even in your latest message, you are completely ignoring both my pointer to past discussions on this issue and my response itself, which said nothing about what my own preference is. If you are really asking why I did what I did, this is a strange way to respond to the answer. If your question was rhetorical and your goal was to try to convert me into a tauist, you are misguided, in that I don't think my personal opinion on the issue is relevant for deciding how we cover the subject on Wikipedia. —David Eppstein (talk) 20:12, 23 April 2015 (UTC)

Sorry if it sounds like I am trying to convert you. I'm fine if you're a piist, but Wikipedia should be neutral. True, this isn't an ad, but I really think that we should have both Pi and Tau. Helloholabonjournihaonamastegutentag (talk) 15:05, 24 April 2015 (UTC)

My actual opinion on this issue is that you have to make a more or less arbitrary choice in defining the notation for this number, the choice we have already made may not be perfect but is good enough, and that the cost of switching to a new notation would vastly outweigh any benefit from greater convenience of that notation. But again, I don't think it should be relevant. Neutrality does not mean covering all the crackpot opinions in equal proportion to the widely-held mainstream position, which is what you seem to be arguing for here. —David Eppstein (talk) 16:07, 24 April 2015 (UTC)

For what it's worth, despite my personal preference for use of tau rather than pi (even at a switching cost), David is correct about how tau should be handled in WP. A separate article is not justified, and the topic is already adequately covered in related articles (Pi, Turn (geometry)). If the real-world notability of tau changes, we can update WP accordingly. —Quondum 17:04, 24 April 2015 (UTC)

By the way, I also think the benefits of switching have been somewhat overstated. I find myself writing π significantly more frequently than 2π (when talking about angles), mostly because in most situations where one could write 2π, writing 0 is simpler and better. —David Eppstein (talk) 18:03, 24 April 2015 (UTC)

I'm surprised. C = 0r? ${\hat {f}}(\omega )=\int _{-\infty }^{\infty }f(t)\ e^{-0it\omega }\,dt$? You deal more with specific angles than with angular displacements or formulae? —Quondum 18:42, 24 April 2015 (UTC)

Integrals like that don't tend to come up in my research. And if they did, maybe I'd think of ${\sqrt {\pi }}$ as more fundamental (e.g. in Stirling's formula and in the normalizing factor for Gaussians). But for example yesterday I was working on a paper that involved chains of circular arcs on a sphere, and I needed to avoid some special cases where the general results were invalid, so I wrote: "Given a closed spherical linkage $C=\{c_{1},...,c_{k}\}$ where the $c_{i}$ are the vertices, in order, embedded on a sphere in some configuration (i.e., with angles $\alpha _{i}\notin \{0,\pi \}$ at each vertex $c_{i}$ for $i=1,\dots ,k$ and with $0<\operatorname {arc} (c_{i},c_{i 1})<\pi$ for each arc), ..." —David Eppstein (talk) 19:00, 24 April 2015 (UTC)

Sorry, I couldn't resist responding to 2π ≡ 0 (mod 2π) making 2π less important than π (maybe just my warped sense of humour about circularity). A debate would only be a discussion of aesthetics. —Quondum 21:11, 24 April 2015 (UTC)

This isn't meant to discuss the math behind why you like one or the other. I strongly believe that both Pi and Tau should have a page. Please provide your reasoning why you think that there should only be a Pi age and not a Tau one. Helloholabonjournihaonamastegutentag (talk) 05:33, 25 April 2015 (UTC)

Quote:A separate article is not justified, and the topic is already adequately covered in related articles (Pi, Turn (geometry)). The topic is covered in related articles such as Pi, but not much at all. As User:Rdococ pointed out, Google has Tau in its system, and even some textbooks are starting to use Tau. Helloholabonjournihaonamastegutentag (talk) 23:05, 25 April 2015 (UTC)

There has been a bunch of editors with some kind of bizarre objection to Tau. I see no special merit in its use but it a subject that is referred to in many reliable sources and is therefore notable enough to have its own article. Martin Hogbin (talk) 16:07, 27 April 2015 (UTC)

The most recent substantial discussion on this is here: Draft talk:Tau (proposed mathematical constant)/Archive 3, which closed with the outcome "The result of this RfC is that this article is not yet ready for mainspace", and I don't see any evidence that consensus has changed since. Certainly this stub provides nothing new in the way of sources or content to suggest it is more notable than last time it was discussed. If anything the interest in it seems to have died down in the last couple of years, as the press now having covered it and used it for their odd news of the week already are not interested in doing so again.--JohnBlackburne^words_deeds 16:30, 27 April 2015 (UTC)

I added a tau image to the Pi page where Tau was mentioned, thinking that if editors didn't want Tau to have its own page, I could at least improve its coverage in the Pi page. It was immediately reverted by User:David Eppstein with the following message: Undid revision 659773343 by Helloholabonjournihaonamastegutentag (talk) fringe notation-revision, almost as crackpot as the duodecimalists. Can't we at least have some more Tau in the Pi page? Also, by the way, if Tau is almost as crackpot as duodecimals, why is there this page but not one for Tau? Helloholabonjournihaonamastegutentag (talk) 20:41, 29 April 2015 (UTC)

In my estimation that revert was maybe a bit harsh: in this section, it would merely be illustrating the text, which is about its occurrence and history, not about it as a mathematical object. —Quondum 23:05, 29 April 2015 (UTC)

For all of you, Helloholabonjournihaonamastegutentag, talk, JohnBlackburne, David Eppstein, and Quondum, the subject of tau isn't just some "crackpot" opinion made up by idiots, it has been a subject of debate amongst the more progressive mathematicians. It appears on TED and on math channels. Legitimate arguments come up frequently on the subject of its simplicity (for example, tau/4 corresponds to 1/4 the unit circle, or pi/2, since 2pi = tau). Criticize the argument, not always those who made it up. Oh, and look at this video (https://www.youtube.com/watch?v=jG7vhMMXagQ). Dandtiks69 (talk) 06:45, 25 May 2015 (UTC)

For anyone else who is hesitant to open youtube links with no description of what they are, or who wonders why people make long talking-head polemic videos when it would be much easier to read the same argument in written form than to sit through it, this is just Vi Hart's old video on the subject. Which (as usual with Vi's videos) I find quite entertaining, but in this case not particularly persuasive. —David Eppstein (talk) 07:27, 25 May 2015 (UTC)

You would still have to sit through written text like in a video,David Eppstein, lol; it's really a matter of preference. I don't see how this isn't persuasive, though (I admit she talks a bit silly). It's a summary of some of the arguments that have come up in preference to tau. Dandtiks69 (talk) 23:37, 25 May 2015 (UTC)

I very much agree with Dandtiks69, tau or pi is a matter of choice (though I personally prefer tau), but I really believe that tau at least deserves its own Wikipedia page. All sorts of crackpots things have Wikipedia articles about the, and tau is not even that crackpot. David Eppstein, please at least consider our arguments, rather than just knocking them straight back with the same argument over and over. Thanks! Helloholabonjournihaonamastegutentag (talk) 06:09, 3 June 2015 (UTC)

Nothing "deserves" a Wikipedia article; an article is created if it is warranted, which is determined by meeting one or more of Wikipedia's notability guidelines. Are there new sources since the last RfC that would cause some change in consensus towards the notability of the subject? I think that's currently in this article is plenty of coverage given what the sources show. - Aoidh (talk) 06:56, 3 June 2015 (UTC)

I've said this before, but I think not so often as to be a pest, so I'm just going to restate my view here. I don't think tau deserves a mathematical article. That is, it would be just silly to reproduce the material in the pi article, but translated into tauish. The tauists have some plausible arguments that might convince me if we were to redo mathematical notation from scratch, but that's irrelevant because we aren't going to do that, and if we were, it would not be Wikipedia to lead the way.

However, I do think the cultural phenomenon of tau probably is notable enough for a separate article. I don't think it's even a particularly close call. I'm a bit surprised at how hardened the attitudes against that option seem to be; I can only surmise that it's an overreaction to the solution that some tauists might actually like, the one I reject in the previous paragraph. But that isn't really a good reason not to have the cultural article. --Trovatore (talk) 07:28, 3 June 2015 (UTC)

The issue is that, unless there are new sources I'm not aware of, the sources that have been discussed don't reflect that opinion at all. - Aoidh (talk) 08:20, 3 June 2015 (UTC)

Multiple significant mentions in reliable sources? Easily meets that standard, and has for a long time. Obviously there are judgment calls involved, but if you're not thinking of it as a math article per se, I think it's easily notable enough for a separate article. -Trovatore (talk) 04:59, 4 June 2015 (UTC)

That is certainly your opinion. Consensus, however, disagrees with you. The "it's not about the math subject but the following around it" avenue was brought up, that was quickly shot down, because that has even less notability than the actual subject does, and comes across as an attempt to appease a group determined to create an article about Tau, whatever the form. This is quickly becoming a perennial proposal and, as nothing has changed since the last few times this was brought up, I don't see any reason why this discussion needs to continue ad nauseam. - Aoidh (talk) 01:05, 23 July 2015 (UTC)

I have no interest in "appeasing" anybody, certainly not aggressive tauists, who are indeed a bit annoying. I think that annoyance has tainted your objectivity, because the topic rather clearly meets notability requirements. --Trovatore (talk) 18:01, 23 July 2015 (UTC)

Unless something has changed since the previous RfC, consensus is that the topic clearly does not. - Aoidh (talk) 21:46, 23 July 2015 (UTC)

Consensus, as they say, can change. I don't recall the RfC of which you speak, but my guess is that it must have been influenced by an overreaction to the (justified) annoyance at the tauists. Because on the face of it it doesn't make sense at all. The "news splash" someone else mentioned, back in 2011 or whenever it was, is the sort of thing that is generally enough to establish notability (multiple significant mentions in independent reliable secondary sources).

Look, I completely understand the annoyance, and I understand that WP must not be used to push neologisms. But reporting on a neologism is not pushing it; it's an important distinction. There was a point at which someone tried to make bright link to an article on a certain current of the "New Atheism"; that was a NEO violation and could not stand. But we have an article on the Brights movement and a link to it from the bright dab page, and that's just fine. --Trovatore (talk) 22:00, 23 July 2015 (UTC)

I've come across many consensuses (what a strange sounding pluralization to say out loud) that I was dumbfounded by, and yes I completely agree that consensus can change, but as WP:CCC points out, that's usually when new or previously undiscussed arguments have arisen. What's being presented here is exactly what was presented at the previous RfC; that if Tau itself cannot have an article, surely there must be enough sources for an article about the Tau movement...but consensus was against that. I'm not suggesting there can't be another RfC on the subject, only that if one was started with these same arguments again and again, I wouldn't bet on anything changing. - Aoidh (talk) 08:02, 24 July 2015 (UTC)

I do not think that we should just reproduce everything in the pi article into tau form, we should have an article saying that some people prefer tau for these reasons, other prefer pi for these other reasons. We can mention some equations, but not too mathematical (unless tau becomes mainstream, of course). Thanks! Helloholabonjournihaonamastegutentag (talk) 15:13, 3 June 2015 (UTC)

I'm with Trovatore on this. What would such an article have as title? Not "Tau", IMO. "Tau movement"? I could live with that, though I am happy with the coverage of the topic in Pi and Turn (geometry). The down side of a separate article is that people would take it as a licence to insert their favourite expositions (though I suppose that's perennially the case in all WP articles). —Quondum 15:44, 3 June 2015 (UTC)

It seems to me that the Tau redirect should go to Turn (geometry), or Turn (geometry)#Tau proposal since that is where the topic is covered in Wikipedia. Any objection?--agr (talk) 13:46, 4 June 2015 (UTC)

Sounds reasonable. I would propose the same for Tau (2π) as well. Tkuvho (talk) 14:11, 4 June 2015 (UTC)

Seems fair. We should link to the redirect from the mention in this article. There are three redirects that I'm aware of: Tau (2π), Tau (mathematical constant) and Tau (number). —Quondum 16:41, 4 June 2015 (UTC)

As long as it remains focused on the tauist movement and doesn't try to become a content fork of this article with π replaced by tau, I'm ok with changing the redirect target. —David Eppstein (talk) 17:44, 4 June 2015 (UTC)

I think that we should keep the mention of the tau movement there (in Turn (geometry)#Tau proposal) very restricted; I understand this proposal to be dealing only with the redirects. If someone wants to write an article on the tau movement, that would be separate. It would have to fly on its own merits as an article. —Quondum 00:01, 5 June 2015 (UTC)

I've changed the three Tau redirects mentioned above to target Turn (geometry)#Tau proposal. I did not change the mention in this article, which already links to turn. I have no objection if someone else wants to. People concerned might want to keep an eye the Turn article.--agr (talk) 15:05, 5 June 2015 (UTC)

Excellent, Helloholabonjournihaonamastegutentag, Quondum, and agr, though I think in the redirect a slightly larger discussion of the advantages of using tau would be helpful in understanding why anyone could have thought of a seemingly unorthodox mathematical proposal. The redirect already has the advantages of the notation of a turn in terms of tau (like a half revolution would be tau/2), but I believe a small discussion on the advantages on the sine function (like in this old Vi Hart video https://www.youtube.com/watch?v=jG7vhMMXagQ) would very much strengthen the argument. It doesn't have to be large. Or maybe that could appear on the sine page. — Preceding unsigned comment added by Dandtiks69 (talk • contribs) 06:11, 8 June 2015 (UTC)

I agree with Dandtiks69, we should have a bit about the advantages like sine and radians, but this is not exclusively to promote tau (and yes, I'm a tauist and I MUCH prefer tau). For that, we can have a link to the tau manifesto. We should also, to keep it neutral, include some info from the pi manifesto. I personally think that the pi manifesto's points are terrible, but so be it. :) Helloholabonjournihaonamastegutentag (talk) 17:12, 8 June 2015 (UTC)

You must not promote tau at all on Wikipeia. I have always supported having a Tau article but it must not be used to promote the idea. Martin Hogbin (talk) 17:28, 8 June 2015 (UTC)

That putting it in nutshell very neatly; I second that principle. —Quondum 18:21, 8 June 2015 (UTC)

There is a fine line between promoting and elaborating tau. Dandtiks69 (talk) 00:27, 11 June 2015 (UTC)

So, what do you all (David Eppstein, Quondum, Martin Hogbin, JohnBlackburne, Dandtiks69, Aoidh, Trovatore, and agr) think? To me (please correct me if I'm wrong), it generally sounds like Wikipedia should have a small article not promoting but providing reasons why and why not some people like to use tau. It would have little if any mathematics. You have mentioned plenty of names (Tau Movement, Tau (2π), Tau (mathematical constant), Tau (number), etc.). What is the general consensus for a name? Helloholabonjournihaonamastegutentag (talk) 20:28, 21 July 2015 (UTC)

"Not promoting but providing reasons" is oxymoronic. If we have an article, it should be centered on the movement and its history, not on their bullet list of talking points. Some properly sourced description of why the people in the movement believe what they believe would be worthwhile, but it would be equally worthwhile to balance that with sourced material from critics of the movement. —David Eppstein (talk) 20:48, 21 July 2015 (UTC)

The neutral point of view policy explicitly rejects the idea of a content fork in which we extol the virtues of using τ over π; see WP:POVFORK. The "movement", such as it is, is already covered at turn (geometry) and here at pi. I don't see much evidence that there is anything that would merit a separate article. Sławomir Biały (talk) 23:54, 21 July 2015 (UTC)

An article on the movement would not (or at least should not) be a content fork, because it would not be covering the mathematical content per se, but only using it to explain the reasoning of the proponents. Counter-arguments would be covered as well the main one, and for me the definitive one, is of course, "why bother?", which may not be directly sourceable, but we should be able to find something along those general lines. I think there's plenty of source material for a separate article, and it's not a particularly good fit for either of the articles you mention, which are mathematical articles, whereas this would be an article on a social phenomenon. --Trovatore (talk) 03:07, 22 July 2015 (UTC)

I'm with Trovatore: Tau movement could absorb the content from Pi and Turn (geometry), and those could then have their content on the topic reduced to a mere mention and link. This would be a cleaner than the current situation. I don't see merit of more than a mention in the latter two articles (or even only in Turn (geometry)), but the topic of the movement does deserve a paragraph or two; where better than in a short article? Such an article should not extol the virtues of anything, but should only describe the phenomenon and the position of involved parties. —Quondum 04:20, 22 July 2015 (UTC)

"Not promoting but providing reasons" is what WP does, or should do, all the time. We can, for example, state Hitler's or Stalin's reasons for doing the things that they did but we must be careful to make clear that they are Hitler's or Stalin's reasons and that they are not generally accepted. Such reasons must never be presented in WP's voice. It can be a very fine line which is easily crossed, see veganism for example.

From a mathematical point of view it is hard to think of a less interesting subject but it seems that some people, on both sides, do have bizarrely strong feelings on the matter. I am genuinely baffled by the strength of feeling against having an article on a bunch of people who support a pointless minority cause. Please do not worry, they are not going to achieve anything, still less spoil mathematics in some way.

Of course we should have an article on Tau, just as we have articles on all sorts of crazy topics, but is should not be a showcase the Tauists cause or an place to promote the case against them. Tau movement would be a good title and I suggest that we start it now. Martin Hogbin (talk) 09:19, 22 July 2015 (UTC)

I guarantee you that if such an article were created, the AfD would be created almost immediately after. There are no sources to support such a topic, as the previous RfCs showed, and unless some vast plethora of sources have popped up since then it would be deleted. - Aoidh (talk) 10:02, 22 July 2015 (UTC)

It is not true that there are no sources to support the Tau movement. A talk on the subject was given at the University of Oxford and, as is pointed out below, there are two refs on the subject in this article. Martin Hogbin (talk) 18:47, 22 July 2015 (UTC)

This was discussed during the previous RfC, and the consensus was against this being a topic. "Two refs on the subject", especially the type of references that have been presented, do not warrant an article. Claiming that there's a "movement" for something is an extraordinary claim, and would need extraordinary sources showing that there is a movement. A few sources making vague claims that some people promote something does not constitute an article, as everything is promoted by someone, that doesn't mean everything needs an article, especially a "movement" article that tries to circumvent the lack of notability for a subject, and that's all a "tau movement" article would be. - Aoidh (talk) 22:07, 22 July 2015 (UTC)

I think Quondum and Trovatore misunderstand me. The tau movement is notable and it is already covered on Wikipedia, here at pi and at the article turn (geometry) (in slightly more detail, but also questionably sourced to self-published media). I don't think anyone here is actually claiming that the content at turn (geometry) has gotten so long that it needs to be split out as a separate article. Certainly, in principle a separate article could be written if we had enough good content to warrant one. But we don't, and attempts to produce enough content for a separate article have all been POV forks.

So, let's just recap what the movement has consisted of. Palais wrote an article in the intelligencer, Hartl self-published a "manifesto", some media mentioned it, some talking heads made YouTube videos (Vi Hart and Khan Academy, maybe), and some people thought it would be funny to celebrate June 28 as "tau day".

Now, to me this does not seem to be enough for a standalone article. As I see it, the only thing a standalone article would accommodate is the various arguments that these people have made in favor of tau. That seems like what Hogbin, Quondum, and Trovatore are all saying. But how could such an article maintain a neutral point of view? It's all well and good to say that we'll balance that with the response from the "mainstream" mathematics community, but there really has not been a response, because the mainstream mathematics community has largely ignored the "movement". So a standalone article, while in principle could exist, really couldn't say much more than we already do at pi and turn (geometry), while maintaining a neutral point of view. Needless to say, I have been unimpressed with the standalone articles that have been proposed here to date, as these have mostly been advocacy pieces. Sławomir Biały (talk) 12:05, 22 July 2015 (UTC)

You maintain a NPOV by properly attributing the arguments to their proponents. So we would say things like, 'Supporters of Tau say that Tau is better than Pi because...', and, 'Other mathematicians point out that...'. We can list the claimed benefits of Tau but we must make clear that these are the benefits claimed but its supporters, we musy not give them in WP's voice or suggest that they are accepted by mainstream mathematicians, indeed we must point out that Tau has made no significant progress in mainstream mathematics. Martin Hogbin (talk) 18:47, 22 July 2015 (UTC)

What you seem to be proposing is an article attributed entirely to primary sources written by tau proponents, and that such an article would be neutral as long as every view presented is prefixed with the phrase "According to X". But this is just a weird caricature of NPOV. Just because views are attributed does does mean that they are given due weight. The lack of counteracting critical sources, or indeed any reliable secondary sources, makes it seem highly unlikely that a neutral article is possible. It is like suggesting that an article presenting only creationists' views in neutral because we attribute every statement. That's just silly. Sławomir Biały (talk) 19:10, 22 July 2015 (UTC)

What a standalone article would cover is the social phenomenon, which I think is easily sourceable; there were, at the very least, a number of newspaper articles about it. These are not ideal sources but there are lots of articles appended to reeds that thin. The arguments would, of course, appear (I believe they were covered in the news articles).

It is true that the mainstream math community has mostly yawned, so there might not be much in the way of "counter-arguments", but I don't see that as really a very big problem. In my view there isn't really any mathematical argument to be had; it's an argument about conventions. All we really need to point out is that the tauists' preferred conventions have not achieved widespread adoption (which I hope is sourceable somewhere though I don't really know).

You might make an analogy with, say, artificial gender-neutral pronouns for English (zie and hir and what have you). Does anyone bother to make counter-arguments? Probably not often; mostly they just ignore them. That isn't an impediment to writing a neutral article on them, as long as the article itself doesn't assert the proponents' arguments as persuasive. --Trovatore (talk) 19:20, 22 July 2015 (UTC)

Right, but what aspect of this social phenomenon of lasting encyclopedic import is not already covered by the present articles under discussion. It would be absurd to have an article consisting of a list of every time someone's blog mentioned tau. We could only note that the there is such a movement, who the main proponents are, and why they believe this. We already do that in a neutral way, with appropriate context and weight. Martin appears to want an article where the arguments are all hashed out, but "neutral" because every sentence carries the disclaimer "According to Hartl..." I do not think that this is what you are suggesting, but what your are suggesting is what we already have. Sławomir Biały (talk) 20:10, 22 July 2015 (UTC)

What I'm saying is, it's not a particularly good fit for mathematical articles. I think it should in an article where the "movement" itself (for lack of a better word) can be covered.

Look, I don't think this is a really big deal, in either direction. It's no great tragedy if WP doesn't have this article. But I don't see any great valid objection to it either, and I think it would be useful to have a dedicated article for people who want to know about it. --Trovatore (talk) 21:08, 22 July 2015 (UTC)

[Edit conflict]Sławomir, I do understand what you are saying, that is exactly what is happening at carnism and what has happened at March against Monsanto, where I have been fighting against the very thing that you are worrying about. however, the problem with those articles is that they are being written almost exclusively by supporters of the subjects. It is quite clear that in this case we have plenty of editors willing to ensure that the article is written in an encyclopedic and neutral style. We should not avoid having an article just because it might possibly be badly written. Modern flat Earth societies is well written and Hollow_Earth#Hypotheses presents some pretty crazy ideas in an encyclopedic manner.

Whatever people may think of it there is a Tau movement and the fact that some editors here do not much like it is not a reason not to have an article on it. Martin Hogbin (talk) 22:44, 22 July 2015 (UTC)

I believe WP:NEO directly applies here: "Articles on neologisms are commonly deleted, as these articles are often created in an attempt to use Wikipedia to increase usage of the term." There is some wiggle room for terms in wide use when there are secondary sources, but tau is hardly used anywhere as a synonym for 2pi. Adding "movement" doesn't save it. A handful of people is not notable enough for a Wikipedia article. I did a Google search on "Tau" and found just 9 hits with the 2pi meaning, most from 2011 when it made a small news splash. I think the coverage in Turn plus the redirects meet the needs of our readers. If someone wants to expand that section a bit, with sources, I have no objections.--agr (talk) 22:37, 22 July 2015 (UTC)

The Tau movement is not a neologism, neither is it a handful of people. It is a small and insignificant movement but talks on the subject have been given at a top world university. — Preceding unsigned comment added by Martin Hogbin (talk • contribs)

How is it not a neologism, "a newly coined term, word, or phrase that may be in the process of entering common use, but that has not yet been accepted into mainstream language"? Tau as a synonym for 2pi is a perfect example of a neologism. And top world universities have talks on lots of novel ideas. That's not enough to get all those ideas their own Wikipedia article.--agr (talk) 13:26, 23 July 2015 (UTC)

The proposed article is not on 'tau' but on 'the tau movement'. The movement has been going for some years and we do not ban organisations and movements from WP just because they are relatively recent. The tau movement has been going for longer than UKIP for example. Martin Hogbin (talk) 17:04, 23 July 2015 (UTC)

This has nothing to do with how recent it is. The "Tau movement" solely exists to promote the neologism tau, meaning 2pi. There is often someone or a group promoting neologisms, so our policy against articles about neologisms would be meaningless if adding "movement" to the new term bypassed the policy. The parent policy Wikipedia:Wikipedia is not a dictionary sums it up well: "In Wikipedia, things are grouped into articles based on what they are, not what they are called by. In a dictionary, things are grouped by what they are called by, not what they are." The thing in question here, a 360 degree rotation, already has an article Turn (geometry) we don't need another article on a proposed different name for the same thing.--agr (talk) 14:47, 24 July 2015 (UTC)

It doesn't make the neologism policy meaningless at all. Reporting on the advocates is completely different from using their preferred language.

Let's go back to the "bright" case. If we were to report that "Richard Dawkins is a bright", that would be using Wikipedia to promote a neologism. "Bright" with that meaning has not successfully entered the larger language. I would add, fortunately, but my feelings about it are of course not the point.

But there is nothing at all wrong with reporting that Richard Dawkins is one of a group of people that seek to promote the use of the word "bright" in that sense, and even having a standalone article on the phenomenon.

As for the dictionary argument, it's absolutely true that we report on the thing rather than its name. However, names for things are, themselves, things, and are sometimes things worth having articles about, when the name itself is the thing being discussed. --Trovatore (talk) 18:21, 24 July 2015 (UTC)

I agree that there is nothing at all wrong with reporting in our article about Richard Dawkins, notable for other reasons, that he is one of a group of people that seek to promote the use of the word "bright" in in a new sense, however in no way would we have a standalone article on the "phenomenon." That is what WP:NEO says.--agr (talk) 23:10, 24 July 2015 (UTC)

Well, we do have it. It's called brights movement. Should it be deleted? I don't think so. To be clear, I do not wish the movement success. I hope they fail. Just the same, I think they're at least marginally notable, and I don't see any reason we can't have an article on them.

My understanding of the intent of NEO is that we shouldn't start using neologisms in articles, including article titles. But talking about them does not constitute using them. --Trovatore (talk) 23:15, 24 July 2015 (UTC)

Whatever one thinks of the article brights movement, Chris Mooney, Sam Harris, Christopher Hitchens, Daniel Dennett and Richard Dawkins aren't exactly random people on the internet. All of these are well-known philosophers, skeptics, and humanist authors. There are notable individuals that have views on "brights", both for and against the moniker, publishing in places like the Gaurdian, the New York Times, and prominent skeptical publications. In contrast, the "tau movement" seems to be about a manifesto self-published by Michael Hartl, and an editorial of no importance published in an obscure non-journal by Robert Palais. No one has ever heard of these people. There was some press about this around tau day a few years back, and a scattering of one or two other slow-news reports, but really nothing much other than to support a sentence or two, a minor footnote in a section on pi in popular culture. Sławomir Biały (talk) 13:00, 25 July 2015 (UTC)

Estimate for pi from Liu Hui and Zu Chongzhi

The correct value of pi is 3.141592653 etc.

This section of the article says:

The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that π ≈ 355/113 (a fraction that goes by the name Milü in Chinese), using Liu Hui's algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value of 3.141592920... remained the most accurate approximation of π available for the next 800 years.[53]

Point #1 is that the value given is not correct for the seven first decimal digits, only for the first six.

Point #2 is that the page for [Chongzhi] gives a value that is correct for the first seven decimal digits: "His best approximation was between 3.1415926 and 3.1415927"

Point #3 is that the page for [Chongzhi] first claims that he used a "24,576 (= 213 × 3) sided polygon". In a later section, however, it says "Zu used the Liu Hui's π algorithm described earlier by Liu Hui to inscribe a 12,288-gon. Zu's value of pi is precise to six decimal places".

So it's not clear what the correct values here should be.

However, I can say that my understanding of the Liu Hui method shows that using a 12,288-gon yields pi > 3.141592517 (correct to 6 decimal digits), and using a 24,576-gon yields pi > 3.141592619 (correct to 7 decimal digits).

Regards, yoram bauman [email protected] (Sorry, I've forgotten my wikipedia login.)

98.125.187.91 (talk) 21:14, 8 August 2015 (UTC)

You can find some details of how Zu Chongzhi (probably) arrived at the approximation at Liu Hui's π algorithm#Later_developments, using a 12288-sided polygon. The Arndt and Haenel book states that it isn't known for certain how Zu arrived at the approximation, but also asserts that it was probably a 12288-sided polygon.

Regarding Point 1, 3.141592920 has correct decimal digits 3,1,4,1,5,9,2,(9). There are seven of these decimal digits, unless I am mistaken. The Arndt and Haenel source also reckons this at seven. I don't have a source for the "best approximation" referred to at the Zu Chongzhi article because that article is poorly referenced. I'm inclined not to give much credence to the statement, unless it can properly be supported. Sławomir
Biały 12:20, 9 August 2015 (UTC)

Thanks for this. I was assuming (perhaps correctly, perhaps not) that the "decimal digits" are only the digits to the right of the decimal point. See also Microsoft. Obviously it's not all that important. Regards, Yoram. 98.125.187.91 (talk) 18:24, 9 August 2015 (UTC)

The easy way out is to say accurate to n digits rather than n decimal digits, but the article uses "decimal digits" throughout to mean base-10 digits (distinguish hexadecimal digits) rather than digits to the right of the decimal point. Glrx (talk) 14:26, 13 August 2015 (UTC)

Estimate for p divided by e by Dr newray`s equation

File:P&e.png

pi divided by e presented by an Iranian physician Dr Goal.a. Newray

i found Interesting equation showing pi divided by e— Preceding unsigned comment added by غلامعلي نوري (talk • contribs) 07:34, 14 August 2015

That’s just a trivial numerical observation. Anyone with some time and basic arithmetic skills could have come up with it or something similar. And it does not need to be presented as an image, which takes up too much space and is harder to read than plain text. I have reduced it in size accordingly.--JohnBlackburne^words_deeds 20:02, 14 August 2015 (UTC)

Semi-protected edit request on 19 September 2015

From (Use Ctrl F to find this in the article)	...According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmological calculations, because that is the accuracy necessary to calculate the volume of the known universe with a precision of one atom...
Change to	...According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmological calculations, because that is the accuracy necessary to calculate the volume of the Observable Universe with a precision of one atom...

That is way more accurate. Thanks. Dan6233 (talk) 23:41, 19 September 2015 (UTC)

Done -- Orduin ^Discuss 16:04, 20 September 2015 (UTC)

Perhaps more accurate, but is it actually true? The observable universe has a comoving radius that is estimated on the order of $R\sim 10^{26}$. The volume is therefore on the order of $V=4\pi R^{3}/3\sim 10^{78}$ (assuming a flat spacelike metric, in agreement with the current cosmological evidence). So it seems to me that one needs more than 100 digits to get the volume of the observable universe to within the volume of even the largest atom. The figure quoted in Arndt and Haenel appears to take R as the age of the universe, so $R\sim 10^{10}$. Then we have $V\sim 10^{30}$, and it then seems much more plausible that forty digits of π would be sufficient. (Plausible, but still not actually true, since the error is of the order of $10^{-10}$, which has the order of the Bohr radius, rather than of the volume of any known atom, the largest of which would be several orders of magnitude smaller). Sławomir
Biały 19:01, 20 September 2015 (UTC)

@Slawekb: I'm not good at math, but there are several sources that use "observable universe". More famously, there is this numberphile video [3] Dan6233^(talk) 14:48, 21 September 2015 (UTC)

The article currently says that only 39 digits of π are needed to compute the volume of the observable universe to within a single atom. That requires a lot more digits than computing the circumference, which the numberphile video indicates. That statement actually seems accurate. I will correct the article to reflect this. Sławomir
Biały 14:57, 21 September 2015 (UTC)

@Slawekb: Oh good. I was thinking that should be changed too, but wasn't sure enough to request. Dan6233^(talk) 15:18, 21 September 2015 (UTC)

Table/List of Pi World Records

I think a section should be added containing world records of Pi, the year accomplished, and the name of the person/people who achieved the world records. A lot of the earlier records were discussed in detail, but no information is present on a lot of the later ones, especially over the past 50 years. I think the best way to do it would be to put a table in "Modern quest for more digits" in a new subsection after "Spigot algorithms." — Preceding unsigned comment added by 2601:5C0:C000:EA21:D0CC:5C2:5C7A:C07A (talk) 20:28, 12 October 2015 (UTC)

I think compiling our own list is a very bad idea, without reliable secondary sources to back it up. Sławomir
Biały 20:38, 12 October 2015 (UTC)

Well, clearly, there would have to be sources. A table/list would be a good way of organizing the list of records without wasting space talking about the insignificant ones. It would be similar to the records lists for sporting events seen on various other Wikipedia articles. — Preceding unsigned comment added by 2601:5C0:C000:EA21:D0CC:5C2:5C7A:C07A (talk) 23:35, 12 October 2015 (UTC)

I do not think that this is the right article for listing "insignificant" π records. If some important algorithms are missing from our current treatment, they should be added to the text, with references to peer-reviewed secondary sources. If they are not significant, and talking about them is a waste of space, then I should think that already answers the question of whether they should be here. If you want to start a pi records article, no one is stopping you. This article is about the mathematical constant, not the sporting event. Sławomir
Biały 23:59, 12 October 2015 (UTC)

mention of angles

In the article it doesn't say that $\pi$ is a sum of three angles as a triangle. — Preceding unsigned comment added by 199.119.233.244 (talk) 16:20, 21 November 2015 (UTC)

The angles of a triangle add up to half a turn. In degrees they add to 180°. No relation to $\pi$.

If (and only if) you invent an angle system, such as radians, where $\pi$ is part of their relation to turn, do you find $\pi$ becoming part for the angle sum for a triangle. Andy Dingley (talk) 13:47, 23 November 2015 (UTC)

Well, I don't really agree — radians are the natural unit of angle, and in the normal usage of mathematicians, they don't need to be mentioned. The sum of the interior angles of a triangle are π, period; there is no need to say "π radians". Just the same, I don't think that fact is really about π per se, and I don't see any need to add it to the article. --Trovatore (talk) 18:31, 23 November 2015 (UTC)

If there were a natural place to add it, I would go ahead and do it. But I couldn't see a very natural way to sneak it into what is already written here. I basically agree, it's a property of triangles, not really a property of π. Sławomir
Biały 18:41, 23 November 2015 (UTC)

Radians are the natural unit of angle in trignometry, but not implicitly in geometry. Andy Dingley (talk) 21:40, 23 November 2015 (UTC)

digits needed in science applications

> Practically all scientific applications require no more than a few hundred digits of π

I take issue with this line. I would argue that only a dozen or so digits are needed for the overwhelming majority of scientific and engineering applications. Can anyone name an application that requires more? In fact, I believe there's a quote out there somewhere that says 15 to 20 digits are good enough to compute the volume of the Universe with an error less than the volume of an atom. Polihale (talk) 12:04, 9 January 2016 (UTC)

What you say does not contradict the line in the article. But one issue is that the precision of the output of a scientific calculation is different from the precision needed in the intermediate values of that calculation in order to get that output. Another is that some geometric algorithms will crash unless the intermediate values they compute are guaranteed to have the correct sign, regardless of how close they may be to zero, and this requirement often blows up the intermediate precision that is needed by a moderate factor. —David Eppstein (talk) 06:55, 8 January 2016 (UTC)

The "volume of the universe" comment (Polihale, forgot to sign) cannot be right; Observable universe gives 10^26 m, and atoms are another 10 or so orders of magnitude smaller, so I guess the number of digits needed for this calculation is something like 50; a lot less than "a few hundred" anyway. But equally, David Eppstein's comment doesn't really respond to the original point. Yes, if you are calculating sin(very-large-number), you need to divide by pi very accurately. But this is a special case: I surmise that "practically all scientific applications" are carried out using floating point with a precision of not more than 128 bits (see Floating point#IEEE 754: floating point in modern computers, i.e. under 40 decimal digits. If none of the other numbers has more than 40 digit precision, it is hard to see how having pi more precisely would help. There is a simple fact here (which I don't know): in standard systems, what value is the constant pi held as? (Write Java program, assuming there is a "pi" constant, and print it) But in the end, I think the original suggestion is true: something like 10-20 digits would indeed be enough for "practically all" cases; it would be useful to add the precision used in "standard"? double-precision, which I guess is most used. Imaginatorium (talk) 08:14, 8 January 2016 (UTC)

If you implement computational geometry algorithms (even as simple as finding the convex hull) using floating point, they will crash, because they cannot guarantee combinatorially-correct results, only numerically-close ones, and that's not good enough. —David Eppstein (talk) 08:22, 8 January 2016 (UTC)

OK, so how are such algorithms implemented? Assuming they involve a numerical representation of pi, they cannot be exact*, because there is no exact numerical representation of pi. So how do they represent pi? Any anyway, the point is that "practically all" calculation is not computational geometry; but of course it would be useful to mention this as an exception. (* I mean "exact", in the same way that there are systems for representing rational numbers to indefinite precision.) Imaginatorium (talk) 08:51, 8 January 2016 (UTC)

I agree with Imaginatorium that "10-20 digits would indeed be enough for "practically all" cases", but this is not because a higher precision is needed in the remainding cases. It is because, when 10-20 digits are not sufficient, it is usually difficult to predict which precision, if any, allows to get the correct result. I have put "if any", because of the well known very simple iterative sequence $x_{n 1}=f(x_{n})$ (I forgot its exact definition) that mathematically converges to 6 and numerically converges to 100, whichever precision is chosen for the floating point computation. I do not know a similar example with π. Nevertheless, 20 digits are not sufficient to prove by computing that the transcendental number $e^{\pi {\sqrt {163}}}$ is not an integer. IMO, when the standard double or quadruple precision are not sufficient, this means that other methods of computation are needed, which do not use any approximation of π. D.Lazard (talk) 11:00, 8 January 2016 (UTC)

Triangle and circle

If a sphere is part of a circle and the sum of three angles of a triangle is part of pi,is pi a measure of time and age?Is this the reason why people in Africa and in central America built pyramids and is it their legacy?Is my hand a knowledge of the ancients and does it have a real significant.Has anyone studied the hand and why it is important in the bible.Trenteans123 (talk) 21:13, 20 January 2016 (UTC)

Pi in the Bible

Would it be worth adding to the history section where the Bible alludes to the value of pi?

1 Kings 7:23 He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.
2 Chronicles 4:2 He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.

Translation from the New International Version. I shall leave explaining the origins of these books of the Bible to people better versed in that than myself. 82.128.190.85 (talk) 04:38, 24 November 2015 (UTC)

This approximation is already mentioned in Approximations of π. Murray Langton (talk) 07:33, 24 November 2015 (UTC)

You can't measure pi in cubit,It can't be done. — Preceding unsigned comment added by 199.119.233.154 (talk) 08:23, 24 November 2015 (UTC)

Well, not in cubits but the ratio of cubits. A diameter "ten cubits from rim to rim" and "a line of thirty cubits to measure around it" suggest a value of their ratio around three. Further, if you allow that "a line of thirty cubits" could be a rope or piece of string a bit longer than 30 cubits, you can form the impression that the author knew his circles within about one digit of precision.Gil (talk) 05:02, 14 February 2016 (UTC)

pi

pi een getal dat bestaat uit miljoenen cijfers het begint met 3.141592653589 — Preceding unsigned comment added by 141.134.6.76 (talk) 16:48, 17 February 2016 (UTC)

The close descendant of Newton also works on Pi

The article already mentions both Isaac Newton and Richard Brent. Richard Brent is descended from Isaac Newton's half-sister (Newton himself did not marry). Is this worth putting in? — Preceding unsigned comment added by Reynardo (talk • contribs) 06:58, 13 March 2016 (UTC)

No. It is not relevant to the subject of the article. Sławomir
Biały 12:39, 13 March 2016 (UTC)

Facts about Pi

1. Pi day is celebrated on March 14 at the Exploratorium in San Francisco at 1:59 a.m. PST which is 3.14159. This time will be extra special on 3/14/15.

2. There are no occurrences of the sequence 123456 in the first million digits of pi.

3. In the Greek alphabet, pi is the 16th letter. In the English alphabet, p is also the 16th letter, showing that pi is the same in every language.

4. Pi has been studied for 4,000 years. Longer than:

- Evolutionary biology

- Impressionist art

5. Albert Einstein was born on Pi Day. The fact that people find this mind-blowing is completely relative.

6.Pi has 6.4 billion known digits that would take approximately 133 years to recite without stopping. — Preceding unsigned comment added by 82.20.150.130 (talk) 19:32, 15 March 2016 (UTC)

Resolving the typical precision required for scientific calculations

Note previous discussions: 2007 2015 2015 2015 ★NealMcB★ (talk) 18:53, 21 March 2016 (UTC)

It would be nice to resolve this. I quote the most recent edit comment: "The source does not support this. In contrast, from Borwein et al, 'There are certain scientific calculations that require intermediate calculations to be performed to significantly higher precision than required for the final result')" Although I believe this quote (Borwein) is reliable and correct, there is a logical fallacy in using this to support the revert. So I am reverting back to the JPL-quote supported version. Here are what I see as the most salient facts:

1: In most ("vast majority" etc) scientific calculations, using regular floating point, means around 15-20 sig figs.

2: It is very easy to illustrate pathological functions (f(x) = sin(10¹²⁷ * x) for example) which require indefinitely high precision if evaluated naively. Probably Borwein et al are referring to a slightly more plausible example.

But no-one has yet made any plausible suggestion as to the implications of fact 2. It certainly isn't that "a few hundred" is enough, because there's the next obvious pathological function. It would suggest that you have to use symbolic calculation techniques and so on. *If* there is a source that names a field in which commonly some number >20 and not more than "a few hundred" is used, then we haven't seen it. And Yet even given such a source, do Borwein et al (price around 30 pounds) claim that more than a tiny proportion of calculations are of this type? Imaginatorium (talk) 17:44, 21 March 2016 (UTC)

Essentially all algorithms in computational geometry (convex hulls, etc) require intermediate calculations to be done to a precision significantly higher than the final result (such as two or four times as many bits), because otherwise inconsistencies in the combinatorial decisions made on the basis of those calculations will crash the algorithm. —David Eppstein (talk) 18:10, 21 March 2016 (UTC)

I agree with Imaginatorium - very well put. I note that the latest reference (http://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need) which says JPL only uses 15 decimal places (i.e. presumably IEEE), is written from the engineering perspective that we're talking about here, whereas both Arndt and Borwein are written from mathematical perspectives. And computational geometry is more about math than about science as discussed here. I also note that the quotes in Arndt and in Borwein are almost the same, and thus might be related. There is a reference numbered "[13]" in Arndt referred to on page 17 (visible online at [4]), but I don't know what it is. It could help a lot. ★NealMcB★ (talk) 18:20, 21 March 2016 (UTC)

I don't think the distinction between "maths" and "engineering" is really germane. Any of this hard calculation is computation; as I've tried to point out, mathematically for any n it is trivial to construct a function which requires more then n (or 10^n) digits. But if all we're talking about is a factor of 2 or 4, as David Eppstein suggests, then well, probably most calulations need 3 or 4 digits accuracy in the result, so, um 4 x 4 = 16 digits precision for a pi constant. But yes, of course, if 8 digits are required in the answer, then ordinary floating point isn't enough. But we are trying to suggest how many digits are actually required for the majority of science and engineering calculations, not something else. Imaginatorium (talk) 19:13, 21 March 2016 (UTC)

Surely numerical calculations in routine, day-to-day, application to, say, problems in dynamics require floating point values having more than 16 significant digits. Arbitrary precision packages exist because, even if all we actually care about is the value to the nearest mm, errors can accumulate depending on the size of some Lyapunov exponent. What we had was, at least, sourced to people who know a lot about numerical calculation, and also as a published, reliable, peer reviewed source. This is in contrast to the blog post that has been suggested is a better source. I will revert the added source again. The person adding the content needs to explain clearly why (apparently) a blog post is assigned more WP:WEIGHT than a peer reviewed secondary source (that is, normally the kind of source one uses to write encyclopedia articles, as opposed to entertainment news articles, for instance. I realize it is becoming increasingly hard for Wikipedians to tell the difference, but here it actually is rather clear-cut. Thanks, Sławomir
Biały 18:55, 21 March 2016 (UTC)

Sorry, but no, the claim that "not more than a few hundred are required..." has never been sourced. We have a source, surely reliable enough, from JPL, commenting that they do ordinary calculations with ordinary floating point, which is around 15-20 sig figs. Then we also have reliable sources pointing out that "you have to be careful", because there are some (something between a "tiny" and "large"(surely not?) proportion of scientific calculations require *significantly more* places in intermediate results. No-one disputes this, but there is no evidence that "significantly more" must mean a factor of (n100 / 20). The sort of thing that would really help is a reference to how a real-life "indefinite precision" system copes with pi. Does it really have, say, n00 (means "several 100", OK? much easier in Japanese), or n000, or n(0)^p for some p sig figs of a value? Or does it (surely much more likely), calculate sufficient digits for a particular calculation on demand? Imaginatorium (talk) 19:07, 21 March 2016 (UTC)

I did not argue that the status quo was necessarily ideal, but that one should not therefore seek out bad sources that actually do contradict our good sources. If the text does not adequately summarize the reliable source, then propose a rewrite of the text that does summarize that source. The text as currently written is as permissive as possible given our knowledge from available sources. There does not seem to be consensus on how many digits are required, but we can (and do) give an upper bound that is consistent with the one given in the cited reliable source. Also, it seems rather obvious that one would need a lot more digits at intermediate stages of a calculation than one would require in the final result. I feel like this point was dismissed rather unfairly. Is there any evidence that this is actually not true, as has been stated, for example in atmospheric modeling? Here is a preprint reporting the use of thousands of digits to simulate the Lorenz equation, for example (but that is probably the most extreme example that one could find, naturally no True Scotsman would ever use so many digits). Sławomir
Biały 19:20, 21 March 2016 (UTC)

To get you started, here is what the Borwein, Borwein, Borwein, Plouffe article says: "There are certain sci-entific calculations that require intermediate calculations to be performed to significantly higher precision than re-quired for the final results, but it is doubtful that anyone will ever need more than a few hundred digits of for such purposes. Values of ~- to a few thousand digits are some-times employed in explorations of mathematical questions using a computer, but we are not aware of any significant applications beyond this level." Emphasis added. This is where the "few hundred" figure in the article comes from. Sławomir
Biały 19:33, 21 March 2016 (UTC)

First, note that I see little disagreement re what I wrote about "the vast majority of scientific applications" using IEEE Floating point. What fraction of scientific calculations world-wide do you think use more than double-precision? And I also said "Some applications may require intermediate calculations to use significantly higher precision" which covers a lot of ground and would include "hundreds" of digits if they were "significant".

Sources discussing the world of dynamics would be greatly informative. I think we want better sources, by scientists and engineers, including those working on algorithms for use in practical applications. The Arndt and Borwein sources have only offhand mentions in materials written by and for mathematicians. The official JPL Educational post at http://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need is designed for teaching, is better informed than most of the secondary sources that we accept, and is hardly a blog post. To broaden the discussion, here's a blog post that also seems pretty high-quality to me, and adds a helpful data point about quadruple precision use at NIST: http://blogs.scientificamerican.com/observations/how-much-pi-do-you-need/ It may lead to even more appropriate sources. ★NealMcB★ (talk) 19:30, 21 March 2016 (UTC)

We have peer-reviewed secondary sources. We don't need to accept these poorer quality sources here. They might be fine in an entertainment news type article, but not here. Sławomir
Biały 19:33, 21 March 2016 (UTC)

Imaginatorium, Re the question of rare examples of using lots of digits, here is one very rare one. It isn't clear whether or not the extra digits they needed involved pi (which after all is the point of the section in question), but the work noted here [5] and available here [6] says "To address these differencing problems we systematically increased the number of decimal digits used for only the phi part of the calculation up to a maximum of 160 decimal digits." ★NealMcB★ (talk) 21:02, 21 March 2016 (UTC)

Broken Scripts

scripters some scripts broken and not showing the right content in this page for ex:$ \pi =\int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}. $ pls fix this for i have no experience with scripts or inserting math symbols thanks Hanz24 (talk) 17:17, 5 April 2016 (UTC)Hanz24

It works for me, in math tags as it’s meant to be used. E.g.

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

What platform are you on, e.g. operating system and browser? --JohnBlackburne^words_deeds 18:06, 5 April 2016 (UTC)

chrome os acer chromebook 11 Hanz24 (talk) 17:04, 6 April 2016 (UTC)Hanz24

The other relevant question is: if you go to your Wikipedia preferences, in the "math" box of the "appearance" pane, what do you have it set as? Likely choices would be "PNG images" or "MathML with SVG or PNG". But if you have it set to "LaTeX source (for text browsers)", you are going to see the source code for the math rather than a displayed formula, much as you describe. —David Eppstein (talk) 17:45, 6 April 2016 (UTC)

Ok that was that was the problem i set to latex and saw source code thanks for the help :) Hanz24 (talk) 16:41, 7 April 2016 (UTC)Hanz24

precision of common applications

It might be interesting (useful?) to know the precision of the stored value of pi as used by common calculators, MS excel. 210.253.93.229 (talk) 00:46, 10 April 2016 (UTC)

Reason why pi appears in many formulas

I always assumed the reason pi and e appear in so many formulas is that only sine and exponential functions survive integration and differentiation without change in shape. For example, if one were to study the harmonics of a squarewave, the term pi quickly appears even though there is nothing spherical about the original function. Is this correct, and worth a mention? 210.253.93.229 (talk) 00:46, 10 April 2016 (UTC)

I would say that yes, it is correct. The constant π is an eigenvalue of the oscillator representation of the Heisenberg group, and appears whenever this oscillator representation is used (i.e., in all of Fourier analysis). It is no accident that π also appears as an isoperimetric constant in the Heisenberg group (this is the classical Dido problem). The article would benefit from an explanation. (It does not even mention the word eigenvalue, as of right now!) Sławomir
Biały 13:46, 10 April 2016 (UTC)

This seems correct. However the concepts that are used are too elaborated for a convincing explanation. I would say the following.

The differential equation y'' y = 0 is undoubtedly the simplest differential equation that has periodic solutions, and every solution of this equation has π as half period. Thus is it not surprisingly that π occurs almost always when studying periodic functions. This is the case of all of Fourier analysis, and of the study of the circle (a moving point on a circle has a periodic movement).

However, this question is an instance of a really intriguing more general question, which, as far as I know, has never been clearly answered: why so many fundamental concepts and results, which have been elaborated in some part of mathematics, appear later to be also fundamental in completely different parts of mathematics? An example of this is the discrete Fourier transform, which is presently fundamental for fast integer multiplication. There are a lot of similar examples. D.Lazard (talk) 15:44, 10 April 2016 (UTC)

But I think the simpler explanation misses the "spectral" aspect. Here, we look for eigenvalues of the differentiation operator on the circle group, equipped with its Haar measure. These eigenvalues are imaginary integral multiples of 2π. They are the characters of the circle group. The Fourier series is the eigenfunction expansion in $L^{2}(S^{1})$. To me, this looks a lot more "universal" than solutions of $y'' y=0$ (one could argue that this differential equation isn't really very natural, because the units of the variable of differentiation must be selected to normalize constant k in the more sensible equation $y'' kx=0$.) The characters on $S^{1}$ are then related to the periods by Poisson duality. Sławomir
Biały 16:40, 10 April 2016 (UTC)

Also missing is some explicit discussion of the isoperimetric inequality. Sławomir
Biały 19:40, 10 April 2016 (UTC)

commonly approximated as 3.14159?

"Commonly" what? Are you serious? Walk the street and ask people the "value" of pi --93.71.24.22 (talk) 06:07, 23 April 2016 (UTC)

I too find the phrase "commonly approximated by" to be strange here. We could say, "It is approximately 3.14159." However, I fear that the number of digits would become a bone of contention. It seems that there is always some smartass who wants to write 50 or more digits everywhere a decimal approximation is used. So if we change the wording, we should get a clear consensus for the change, as well as the number of digits displayed. Sławomir
Biały 13:31, 23 April 2016 (UTC)