Talk:Pi
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Has anyone ever held a contest on how many digits of pi one can memorize. I teach high school math and we had a pi day contest on march 14 and we had a student memorize 318 digits of pi. Anyone know what the record is?
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Open Issue on decimal expansion of Pi ...
Granted, it has been proven that the literal decimal expansion of Pi never cycles. However, I am wondering if there is a "deeper pattern" in the expansion which does, in fact, cycle. In other words, consider the sequence:
[012 123 234 345 456 567 678 789 8910 91011 101112 111213...]
Spaces are included above to make the pattern obvious. Without the spaces you would have:
[012123234345456567678789891091011101112111213...]
The digit-sequence above will not cycle. However, there is an obvious "pattern" in the sequence that repeats over and over, and I am wondering if there might be such a repeating pattern in Pi's decimal expansion (perhaps not such a short and simple one, but a pattern nevertheless). Or perhaps, the sequence of Pi is more or less just...random!
Anyway, has the issue I am discussing above been investigated for this wonderful irrational number ?? Thankyou...Mike Keith (lynne_mike@alltel.net)
- I'm open to correction, but as a mathematician I'd say it's highly unlikely. There's absolutely no reason to think it might be and it doesn't look right - not aesthetically consonant with the greater body of mathematics.
- There is no such pattern in the decimal expansion of pi. I have discovered a truly remarkable proof, but this talk page is too small to contain it. ^_~ -- Schnee 15:38, 13 Jun 2004 (UTC)
- While there is no proof, Pi is usually considered to be normal, which would means there are no patterns
- Which would also imply that every finite pattern appears infinitely many times in the expansion? [[User:Sverdrup|✏ Sverdrup]] 21:31, 9 Sep 2004 (UTC)
- I don't see the pattern in the sequence you have. Apparently, you haven't given enough terms to make it obvious (to me). What is the pattern? Normal doesn't mean "no patterns", it has a precise definition concerning frequency of finite sequences, and this definition itself is a "pattern". Asking if the expansion of π has a "pattern" is not well-defined question, until you define what "pattern" means. I don't think any artifically constructed "pattern" would likely occur. Unless it has some possible connection to other ideas in math, there's no reason to think so, and more importantly, virtually no guide to prove or disproof the conjecture. Revolver 04:30, 19 Sep 2004 (UTC)
I removed ...
I removed the following badly garbled sentence:
- The PiHex Project have 1000000000000000, 1^15 bits of pi. (1^30 digits of PI) PiHex,PiHex Site (http://www.cecm.sfu.ca/projects/pihex/index.html)
—Herbee 2004-02-13
Stirling's formula
Shouldn't Stirling's formula contain the asympotically-equals sign (~) instead of the aprroixmatly equals sign. The form is a much more precise statement. I don't know how to edit this in. Dmn
Bailey-Borwein-Plouffe
The article says:
It was shown that the existence of the above mentioned Bailey-Borwein-Plouffe formula and similar formulas imply that the normality in base 2 of <eth> and various other constants can be reduced to a plausible conjecture of chaos theory.
What is the conjecture? Who showed this reduction? Can you give a reference for the papers involved? Or their websites? Even a few keywords suitable for web searching would be very much appreciated!
Bailey and Crandal
I see you've added "Bailey and Crandal in 2000". That's great. It would be good to add a link to their paper (or it's abstract) if it exists somewhere on the web. I've tried searching for it, and haven't had any luck.
But the article does give an URL, and you can find the paper there, in both PDF and PostScript. (The PDF is very badly done, however.) --Zundark, 2001-08-21
You're right. My mistake. I'd missed the URL in the earlier section.
non Euclidian universe
Of particular interest to me is this: if we live in a non Euclidian universe, does that alter the value of pi? Is it possible that a non euclidian universe would render pi a distance dependedt function? Just musing, really.
-- Pi is usually defined as the ratio of a circle's circumference to its diameter in Euclidean geometry. So if the universe was not Euclidean, this ratio would be different, but it would not be called pi.
- The fact that our universe is localy euclidean to a high degree of approximation means that the known value of pi is a very good approximation. Non-locally, the value is not constant because the shape of space is not constant.
- As said in the article, pi is a mathematical constant independent of physical reality. Regardless of whether the universe is Euclidean, or even if it exists or not (not to start a philosophical debate), pi's value will remain constant by definition. Its value is a priori knowledge.
hexagonal world
I define Pi as a function of the distance metric in a metric space: Pi equals half the arc length of the curve created by the locus of points of distance 1 from a given point. In a hexagonal world such as that used in many turn-based video games, Pi == 3. In a geometry with distance metric d((x1, y1), (x2, y2)) == (abs(x2 - x1) + abs(y2 - y1)) such as city blocks, Pi == 4. Of course, the familiar Euclidean distance metric provides a value of Pi just a bit more than 355/113, and nearly all digital signal processing takes place in Euclidean geometry.
In geometries that don't preserve lengths of translated lines (such as the geometry of curved spacetime), "distance 1" is meaningless, and Pi depends on the location and the radius.
That is not the standard definition however: Pi is a well defined real number, and it has nothing to do with geometry. It is always 3.14.. no matter what. Mathematical constants don't depend on physical contingencies. If our world is not Euclidean, then there will be some circles of diameter one whose circumference is different from Pi. --AxelBoldt
not OK to link to the image
I deleted the image at http://www.nersc.gov/~dhbailey/dhb-form.gif from the article. It might be OK to swipe the image and put it on the Wikipedia server, but it's not OK to link to the image itself on the government server. We don't own their bandwidth... --LMS
See bandwidth theft. --Theaterfreak64 23:28, Apr 24, 2005 (UTC)
particularly long value of pi
I've got no problem with showing a particularly long value of pi, assuming it's accurate as far as it goes. It may be curious and totally useless, but it's not vandalism. Eclecticology
- I called it vandalism because it's merely part of a days long pattern from this character. Rgamble
Geometry
Geometry was a purely axiomatic mathematic until Cartesian thought entered. You can't claim that these equations are Euclidean, Euclid wouldn't have recognized them.
- They are certainly not from "plane geometry" since some of them talk about three dimensional objects. The "Euclidean" was there to emphasize that there are other geometries where these formulas are wrong. But that point was made earlier already, so I guess we can just call it "geometry". AxelBoldt 03:08 Oct 1, 2002 (UTC)
Concerning this section, I just was thinking about adding a comment after all the volume/area formulae on the Pi page, kind of "all these formulae are in fact a consequence of the second one, as all of them give the volume of solid of revolution (by the formula \pi x \integral...)". I hesitate, because I know that the "pi" page is really a most public place (and thus should be considered almost "locked" for edits, in some sense).
Any comments? — MFH: Talk 00:03, 11 May 2005 (UTC)
Is there any point in computing many digits of pi?
Is this true?
- pi to 40 places is sufficient to measure the circumference of the known universe with an error less than the width of a hydrogen atom.
If so, what is the purpose of calculating pi to large numbers of digits?Ortolan88
- What is the purpose of climbing the Everest? It is there. Same with pi.--AN
But is the statement true? If so, it, or some variant, should be in the article to give some proportion to this quest. I would also like to know what is so hard about calculating that number of digits. Is it anything more than long, long, long division? In what way does this advance the art of mathematics? These are serious questions by an uninformed member of the encyclopedia-reading public. Ortolan88
IIRC, this BBC programme (http://www.bbc.co.uk/radio4/science/rams/5numbers2.ram) mentions facts relating to this. Mr. Jones 14:23, 8 Mar 2004 (UTC)
Computing many digits of pi is trivial and not mathematically interesting. It is often done to test (super-)computer hardware. I don't know if the statement about the universe is true, but something close to it is probably correct. You don't need more than a couple dozen digits in real word applications. AxelBoldt 21:07 Nov 23, 2002 (UTC)
The specific knowledge of KNOWING the digits of pi is trivial, but the mathematics used to GET more digits is mathematically interesting and not trivial. For instance, many of the advances made in calculating more digits of pi in a shorter amount of time came not from faster computers, but from better identities for pi, i.e. either sums, products, or continued fraction expansions with faster rates of convergence. Coming up with these identities with faster rates of convergence is NOT easy; nor is it a trivial and uninteresting math problem. In fact, PROVING that many of these algorithms for getting more digits of pi are true, will give more (correct) digits, and will give them faster (i.e. proving the rate of convergence), all of these are very interesting questions, and this is not just to get more digits of pi. Often, the ideas behinds these proofs of the identities are related to other problems in math, i.e. an IDEA used to prove an identity used to get more digits of pi might be used to prove another result, which does have other "real-world" applications, or at the very least, other applications within other areas of pure math. Revolver
According to the article on pi, "The most pressing open question about π is whether it is normal, i.e. whether any digit block occurs in the expansion of π just as often as one would statistically expect if the digits had been produced completely randomly." This can only be answered by a mathematical study of the digit sequence. Obviously one wouldn't study it by looking it up in an encyclopaedia, but even so, people at home might want to do a bit of a check for themselves, to see if it sounds plausible. :) -- Oliver Pereira 00:20 Nov 24, 2002 (UTC)
- Good luck. Even looking at the first 10,000 or 100,000 digits isn't enough to give some kind of statistical assurance. The short-term random luck factor is just too high. Anyone who has played poker for any length of time will understand what I mean by this. Revolver
Evidence for or against the normality of pi can obviously never be produced by looking at a finite initial segment of the digit sequence. That's why I said above that computing digits of pi is mathematically uninteresting.
But people should definitely be encouraged to experiment with the digits. So we should tell them how to generate the digits for themselves, so that they have them in a format which allows all sorts of statistical tests. Right now, using our data to find out how often the digit sequence "33" occurs among the first 10,000 digits would require a perl program that's not much simpler than directly generating the digits from scratch. AxelBoldt 20:42 Nov 24, 2002 (UTC)
- Axel is absolutely right here -- if we want to know something for an infinite number of cases, (all natural numbers, all digits of pi, etc.) just because we only have a finite # of cases, doesn't mean it's worthless. Looking at the known finite # of cases allows one to make reasonable conjectures. It doesn't prove anything, of course, but it keeps us from wasting our time going down alleys that (almost certainly) dead-ends. Revolver
- Yep, I agree - except about the simplicity of writing a perl script to compute the value of pi! :) Since we are writing an encyclopaedia for general use, most people referring to it will just be ordinary people without any programming ability. Some of whom genuinely believe that pi is exactly 22/7! Such people might want to see a concrete demonstration of just how random the digits of pi are. Plotting bar charts of the frequencies of digits might be an amusing mini-project for a schoolchild who is starting to learn about statistics. Well, maybe not, but I expect there are many more plausible suggestions that people could come up with. I just mean that members of the public might be curious to see the digits of pi and play with them, even if they don't have the ability to do anything serious with them. I'm thinking of a past version of me when I was at school, for example. -- Oliver Pereira 21:38 Nov 24, 2002 (UTC)
- The problem with plotting frequency of digits to test normality is it's not just single digits, it's all finite sequences of digits, and the distribution of these, the more digits you get, becomes much more susceptible to short-term luck. Revolver
width of a hydrogen atom
I removed this:
- It is said that pi to 40 places is sufficient to measure the circumference of the known universe with an error less than the width of a hydrogen atom.
Who says that? If somebody has done the calculation, we can simply omit the qualifying "it is said that" which essentially renders the whole paragraph pointless. AxelBoldt 20:00 Nov 25, 2002 (UTC)
- I'm confused about it myself. Surely the accuracy of the result is dependent on the accuracy of the figure we have for the diameter of the universe, as well as the accuracy of pi. -- Tarquin 20:07 Nov 25, 2002 (UTC)
- It isn't pointless. It gives a sense of proportion to the whole thing. 22/7 is sufficiently accurate for pi to get the circumference of a can of corn. With a circle of one mile diameter 3.14 yields 16572.2 feet circumferences, 3.142 gives 16589.76, 3.1416 gives 16587.648, and 3.14159 yields 16587.595, which is where my calculator poops out, but, obviously, the more digits of pi are adding accuracy. If one of you cosmo brains would do the same arithmetic for some galactic measure or other (instead of silently deleting my interesting, if unproven statement about hydrogen molecules) using, say 40 places for pi, then maybe we could put this million digits of pi business into some kind of proportion. What is confusing or pointless about that? This is an encyclopedia. My whole intent is to give readers some idea of the value of the additional digits of pi.Ortolan88
- Okay, i'll have a go... for a universe 12 billion light-years across, that's about 1.135e26 m. An atom of hydrogen is about 1e-10 m across, meaning you'd need about 36 decimal places in pi to get error levels below the diameter of hydrogen. So 40 does it admirably. Graft
- It isn't pointless. It gives a sense of proportion to the whole thing. 22/7 is sufficiently accurate for pi to get the circumference of a can of corn. With a circle of one mile diameter 3.14 yields 16572.2 feet circumferences, 3.142 gives 16589.76, 3.1416 gives 16587.648, and 3.14159 yields 16587.595, which is where my calculator poops out, but, obviously, the more digits of pi are adding accuracy. If one of you cosmo brains would do the same arithmetic for some galactic measure or other (instead of silently deleting my interesting, if unproven statement about hydrogen molecules) using, say 40 places for pi, then maybe we could put this million digits of pi business into some kind of proportion. What is confusing or pointless about that? This is an encyclopedia. My whole intent is to give readers some idea of the value of the additional digits of pi.Ortolan88
- The observable universe currently has a radius of about 50 billion lightyears, because of the past expansion, but your estimates still work. Of course, if you want the volume of the observable universe precise to the volume of a helium atom, you need about 270 digits of pi. AxelBoldt 23:17 Nov 30, 2002 (UTC)
- Cool, so I assume you'll add this to the article? Ortolan88 05:57 Nov 29, 2002 (UTC)
- PS - OK, no takers, some time this weekend I'll add it to the article. It isn't just whim, folks, it is something the average encyclopedia reader deserves and will only get from the Wikipedia. I may move one of the more interesting paragraphs up a bit too. And, mathematicians, I read in a rival encyclopedia [1] (http://www.encyclopedia.com/html/p1/pi.asp), that Euler came up with a connection between pi and natural logarithms that isn't mentioned in our article. Ortolan88 15:40 Dec 7, 2002 (UTC)
- Sure it is -- take Euler's identity, take the log of both sides, you get pi = (-i)(log (-1)), where we take principal branch of log function. At least, I assume that's the relation they had in mind. Revolver
Thats goign to take more than 1 Terabyte of memeory!!! :-S
- Regarading the rival encyclopedia article: it says "the famous formula ei 1, where i 1 ." now unless that's my browser skipping characters, they've made a complete pig's ear of Euler's identity... oh yes, by the way... we have a full article on that equation. nananana! ;-) -- Tarquin 22:22 Dec 7, 2002 (UTC)
these approximations
- These approximations were once useful to the applied sciences; the more recent approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers.
I don't understand the above sentence.
- Of what use were "these approximations"?
- Why aren't they useful any more?
- Are the recent approximations entirely useless, or just the extra digits?
Without answers to all the above questions, I'm inclined to delete the quoted sentence. But I hope someone who knows a lot more about math than I do (like Axelboldt) can help out here. --Ed Poor
- it means this: newer approximations have more digits. They don't make the old ones obsolete. An old approximation of, say 50 digits is useful. A 10,000 digit is pointless; the extra accuracy is irrelevant because we never need it. It's hust the extra digits that are irrelevant. The 50th digit (for example) has not changed since it was first determined -- that's something for an article on approximations to explain: it is normally possible to know how good an approximation is: we don't just calculate 50 digits, we know that those 50 digits are right, and even if we went further, we'd still get them. hope that helps -- Tarquin
This entire debate is patently ridiculous and should be deleted. The children with their pocket calculators should go off and let the rest of us work in peace.
First, using pi calculations to calibrate supercomputer speeds is absolutely meaningless. Supercomputers are meant for real work, not moronic games. Calibrate speed with real work.
- I don't know where you got the idea that pi calculations are used to "calibrate supercomputer speeds" nor do I know what that even means. Pi calculations (and prime number calculations) are often used to test new supercomputers. You let them compute digits of pi for a week with two different methods and then compare the results, to check for hardware bugs. AxelBoldt 05:13 Nov 26, 2002 (UTC)
Second, nobody needs to know pi to more than five digits. Period. End of discussion. Anything more than that is simply compensation. Something so delicate as to need outrageous values of pi is a toy, that's all.
Whatever. Revolver
Third, every resource spent analysing pi is a resource wasted. Spending time on pi is exactly as fruitful as spending time with the human genome.
- Every resource spent writing symphonies is a resource wasted. Every resource spent pondering the meaning of existence is a resource wasted. In short, every activity YOU think is a waste of time is a resource wasted. Revolver
- Some people think the human genome will someday in the future produce a useful medicine or treatment. Some people also think that pi may have a similar surprising usage. Psi 12:37 Dec 2, 2002 (UTC)
- Who cares? Investigating properties of the expansion is an interesting question, even if nothing "useful" comes from it. Revolver
- Every resource spent whining that every resource spent analysing pi is a resource wasted is a resource wasted. If you don't get that, don't read math articles. ;-)
- —Herbee 22:36, 2004 Mar 5 (UTC)
the physics formulae
I am inclined to question the inclusion of the physics formulae. Surely the appearance of pi in these is simply a quirk of the definition of the physical constants such as Plank's constant and the gravitational constant. The significance of a physical constant tends to be recognised early in the development of the theory in which it features. When different derivations are made from the theory sometimes a factor of pi will appear in a formula, and sometimes not, for detailed mathematical reasons (eg the inversion of a fourier transform).
- But I believe no matter how you redefine the physical constants, pi will always show up in your fundamental equations, just in different locations. For instance, if you redefine G to get rid of pi in Einstein's equation, it will then show up in Newton's law of Gravity. So we may as well list the locations that pi shows up in our accepted system of physical constants. AxelBoldt 23:17 Nov 30, 2002 (UTC)
- This is true. The same issue arises in electromagnetism and is further exacerbated by the contrast between MKS and CGS units. In the Coulomb force law for electrostatics (at least in SI units) Pi does appear, which, in turn, results in its not appearing in Maxwell's equations. In quantum theory an explicit reference to Pi can be made to appear or disappear from an equation by changing from h to h-bar. Perhaps this is worth explaining in the article.
- If we want a sample equation from physics where the appearance of Pi is less arbitrary then I think the period of small oscillations of a pendulum in a uniform gravitational field would be a better candidate. -- Alan Peakall 12:25 Dec 2, 2002 (UTC)
Also I believe the mnemonic linked to Isaac Asimov was coined by James Jeans -- Alan Peakall 12:00 Nov 29, 2002 (UTC)
statisticians pi
In a long ago and fruitless sojourn into the land of entry-level statistics, I seem to remember that statisticians use a wholly different pi that stands for some variable or another. The statistical use probably doesn't deserve a whole article, but there should be a mention that the same Greek letter is used in statistics too, if anybody knows exactly what it stands for. Tokerboy 23:03 Dec 7, 2002 (UTC)
- Oh they use the symbol pi to represent profit in economics. -- Mark Ryan
See pi (letter) for various usages of the Greek letter in different fields. SCCarlson
pi as a number container
I understand that pi is infinitely long when expressed as a number. I also understand that it never repeats. I have also heard that all possible finite sequences of numbers are contained within it. I can see that the first two statements don't imply the latter, ie
<math>0.1121231234...<math>
is a counter example. However, I have heard this asserted on a (mostly) serious radio program (http://www.bbc.co.uk/radio4/science/5numbers2.shtml). Any thoughts?
MrJones 10:54, 19 Oct 2003 (UTC)
- Don't believe everything you hear. "it never repeats"? it repeats right there on the third decimal when the numeral 1 turns up (3,141...). If you mean that it is not cyclical, that is true, but not really that much connected with pi itself, but more to do with the properties of cyclical decimal expansions... -- Cimon Avaro on a pogostick 21:35, Oct 22, 2003 (UTC) & Revolver
- I should have said recurs, perhaps, as in non-recurring decimal expansion. You knew what I meant, though. Can you name some other numbers that don't recur in their decimal expansion? I don't believe everything I hear, that's why I asked the question. MrJones 00:52, 25 Oct 2003 (UTC)
- Numbers with a non-recurring decimal expansion are irrational.
- —Herbee 22:36, 2004 Mar 5 (UTC)
more Bailey-Borwein-Plouffe
Recently there has been a post to Usenet, under the name of Simon Plouffe, which states that he (Plouffe) discovered the formula given in the Bailey-Borwein-Plouffe paper, and that Bailey in particular arrived on the scene after that formula was already discovered. (I forget at the moment what he said about Borwein.) See: "The story behind a formula of Pi", sci.math, Jun 23, 2003 by simon.plouffe@sympatico.ca, also "Sur l'histoire entourant la d écouverte d'une formule de Pi.", fr.sci.maths, 2003-06-24, by plouffe@math.uqam.ca.
Taking this account at face value, it seems that crediting the discovery to Bailey primarily -- "David H. Bailey, together with Peter Borwein and Simon Plouffe, discovered a new formula..." -- is unjustified. I wonder if one could get a comment directly from Plouffe via email. Hmm, comments from Bailey and Borwein would also be interesting.
The Bailey, Borwein, & Plouffe paper itself does not clarify the discovery of the formula. It just says "we" discovered it. A later paper by Borwein (a summary of pi computing history) says "it was discovered". I've been unable to find anything by Bailey or Borwein which states a direct attribution for the discovery of the formula.
dropping toothpicks
I have read that it is possible to get an approximation of pi by dropping toothpicks on a floor. Specifically, drop toothpicks on a grid of squares with the squares' sides equal to the length of the toothpicks. Count the times a toothpick intersects a square's side. Then pi should equal 2 times the number of toothpicks dropped divided by the number of intersections. Is this true?
- Err... never mind. I found Pi through experiment.
fundamentals using pi
The formulas involving pi under the physics section involve costants that are commonly considered to be less fundamental than if you absorbed pi into them. for instance Newton's law of gravitation can be put in the form
<math>F = \frac{4 \pi GMm}{4 \pi r^2}<math>
which has the interpretation that the field(force over the mass of the object it acts on in this case) per unit area(since the field posesses spherical symmetry) is 4 π GM.
in Einsteins field equation we are dealing with a field density, thus we do not include 4 π r2 and the 4πGM turns up, as the more fundamental constant.
The 2π in the uncertainty principle arises from considering frequency, rather than the arguably more fundamental angular frequency, in the definition of planks constant; h dived by 2pi is used more often than h itself.
64.161.172.140 01:48, 13 Feb 2004 (UTC)
- I agree completely. Idem for the formula μ = 4 π 10-7 As/Vm (which is 1 in other units).
- IMHO, this comment should be added in the main page. MFH 14:02, 14 Mar 2005 (UTC)
Pi's old name
Am I the only person who gets Main Page edits when following the page history link for π ? I think that the ampersand character may be fucking with the dynamic linking... Perhaps the page should be moved back to Pi? Matt gies 01:24, 5 Mar 2004 (UTC)
How I want a drink, ...
How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics! How I want a drink, alcoholic of course, after the heavy chapters involving quantum mechanics!
Those two sentences to help remember the decimal sequences to π are attributed by various webpages to one of several individuals: George Polya, Martin Gardner and Issac Asimov. What's the true attribution? - Bevo 22:52, 22 Mar 2004 (UTC)
- I'm quite sure it existed well before M Gardner, at least. I remember an article in "Scientific American" where he (or was it Ian Stewart?) cited this (and many others), probably with hints on sources. [Somebody could put the exact reference here.]
- You should have put here a link to the web pages you refer to, or to a page listing such pages (with "attribution" information, also for other mnemo texts). Thanks in advance. MFH 14:22, 14 Mar 2005 (UTC)
"Value"
Please note, "value" usually has a certain connotation in mathematics, and to say that π has a "value" is a bit mathematically misleading. π is defined to be a specific REAL NUMBER (a constant); π is not defined to be a constant function which takes on "values". Furthermore, a constant IS a number, a constant does not "have a value which is a number". I don't mean to be nit-picking a tedious point, but the phrasing "the value of π" sounds awkward and silly to my mathematical ears.
Revolver 17:21, 12 Apr 2004 (UTC)
Clarification
Let me clarify a bit more, since I realise there are some cases where "value of π" is warranted, but this is for a specific reason. One correct situation in which "value of π" is appropriate is when referring to a numerical approximation or numerical estimate of π, but it should be realised in this case that this is not the same thing as π -- the number π has many different numerical values or approximations:
- "The value of π was given as 22/7 by ..."
- "A useful numerical value for π is 3.1415..."
- "We can find a numerical value for π by measuring the circumference and diameter of a large circle"
These are ideas based on estimation, measurements, approximations, and expansions. But when talking about purely mathematical properties of π, e.g. its irrationality or transcendence, it's more correct to simply say "π is irrational" or "π is transcendental", very few people would say "the value of π is irrational or transcendental".
Another case where "value of π" might be appropriate is when you are using the term "π" to refer to the symbol π, and you are assigning a value (number) to this symbol, then in this case, π does have a value, because it's the value represented by the symbol π. For example,
- "The state legislature defined the value of π to be 3"
- "If everyone in the world were allowed to vote, would it turn out that people might elect the value of π to be 22/7?"
- "Given any metric space, we define a constant π depending on that space, and we define the value of π to be the ratio of the arc length of the set of points of unit distance from a given point"
These are the only times I can see how "value of π" is not redundant. Revolver 17:58, 12 Apr 2004 (UTC)
- Concerning the last definition ("ratio" should be replaced by "half" and "two dimensional" should be added), a nice exercice:
Prove that the value of π is in between 3 and 4, and that all these values are possible.
- A hint: π equals 4 if the unit ball is a square (sup norm),
- and π = 3 if the unit ball is a regular hexagon. MFH 14:30, 14 Mar 2005 (UTC)
Pi culture
I move that the section on pi culture be moved to a separate article. This article is already getting a bit cluttered and will undoubtedly have more material added to it in the future. As the focus of this particular article is on pi as a mathematical constant, not various mnemomics or "pi day", this may work better at another article. A similar thing was done at the article trigonometric function when the section on mnemomics for the trig functions became rather lengthy and distracting. (I'm not saying to eliminate the section, maybe briefly mention pi day, mnemomics, etc., then give a link.) Discussion?
Revolver 00:10, 13 Apr 2004 (UTC)
This paragraph is getting weird and weird. The
(...). This also leads to some rather interesting adaptations of popular songs such as "Rock Around the Clock".
and
A random, somewhat strange joke involving pi: | pi aren't square, pi are round! | -Oh, the Irony.
(while the previous version (pi r squared...) made it easier to understand...)
My vote is : vanity, move to WP:BJ... — MFH: Talk 07:20, 14 May 2005 (UTC)
Ramanujan's equation
Looking at Ramanujan's eqaution for 1/π given in the article,
- <math> \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k!)(1103+26390k)}{(k!)^4 396^{4k}} <math>
it looks like it's possible to simplify the fraction in the infinite sum by removing a k! factor. I think perhaps (4k)! was intended for (4k!). Can anyone confirm or disconfirm this guess? Eric119 01:09, May 28, 2004 (UTC)
Yep, it is (4k)! (looked in the library). Its also already been changed Mrjeff 12:54, 30 Jul 2004 (UTC)
Pi article title
When did the title start displaying as π ? - Bevo 23:26, 5 Jun 2004 (UTC)
That's even worse. Now I see a completely odd, non-Latin, non-Greek character. RickK 02:01, Jun 6, 2004 (UTC)
Shanks
Shanks' famous calculation of π is listed twice in the history corresponding to two different years, 1853 and 1874. What's going on there? 4pq1injbok 19:01, 2 Jul 2004 (UTC)
- I think the 1853 date is wrong (according to other sources). --Bubba73 17:40, 3 Jun 2005 (UTC)
BCE vs BC
Don't switch existing BCE usage to BC (or vice versa) without very good reason. This is the same as the issue of American vs. British spelling for words, and the precedent (leave usage as it is originally written) is the same. There are good arguments on either side (BCE is the academic/scholarly standard, even in British textbooks... and BC is more widely known, particularly outside North America) but until there is a Wikipedia standard on this issue (and there currently isn't) leave BCE/BC usage alone. --Wclark 18:38, 2004 Aug 29 (UTC)
- Well here are some very very good reasons for switching all BCE/CE to BC/AD, but first let me deal with your opinions. This is not the same issue as AE vs. BE. There is no controversy in usage of language versions, just personal preference usually based on country of origin, so the precedent 'leave as originally written' doesn't apply here. In fact, because there's no Wiki standard, not very much does apply, apart from NPOV. BCE is the academic/scholarly standard... Is it? I don't think so. It's certainly making inroads, particularly in internet writing originating from America, but it's virtually unknown in the rest of the world and is certainly not a standard in British text books.
- So why change?
- 1)Quite simply because use of BCE/CE is 'POV'. The de-facto global standard is BC/AD. A minority in the USA are trying to push their politically correct alternative (and having some success) and to use it in Wikipedia is to support this minority standpoint. Maybe BCE/CE will eventually become the de-facto global standard. If it does, then Wikipedia should adopt it, but in the mean time it use here is, in effect, supporting a political objective. Wikipedia should not promote this usage.
- 2)Wikipedia should not use terminlogy that most of the world does not understand. An article on pi is relevant to everyone the world over. Readers should not have to ask the question '.. what's all this BCE stuff?'. Arcturus 22:08, 29 Aug 2004 (UTC)
- BCE is not American Political Correctness -- it goes back several hundred years (the 15th century, I believe) and originated with Rabbinic scholars. BCE/CE and BC/AD are both POV, there's no escaping it. Nonetheless, BCE/CE is the academic standard in the English-speaking world (try to find a history textbook -- even a British one -- published in the last decade that still uses BC/AD). However -- in Wikipedia articles we still accept Standard units of measurement (feet, miles, gallons) despite the fact that the academic standard (even in the US) is to use Metric. So like I wrote earlier, there are good arguments on either side. Until there's a consensus (and the Pi article isn't the place to form it), it's best to leave the BCE/BC usage as the original author wrote it. (And for the record, I personally use ISO date formats.) --Wclark 04:28, 2004 Aug 31 (UTC)
I admit that I'm not North American, but also, I didn't know what the heck this BCE/CE stuff was about. Then I found it on WP, via List of acronyms. Now, just some google hit counting:
881 for "before christian era". 9,100 for "before common era". (1st hit is a good reference: www.radix.net/~dglenn/defs/ce.html) 81,300 for "common era" -"before common era". 92,200 for "common era" 269,000 for "before christ" 278,000 for "anno domini"
although maybe not needed, 3 further comments
- the "AD" is probably underestimated, because so well known that explanation/spelling out is probably less frequent w.r.t. occurence
- I also agree with "leave as is" if there's not a really undisputable reason for changing
- "(before) common era" seems to me much more POV than "before christ".
— MFH: Talk 12:28, 14 May 2005 (UTC)
Historical values
The first two historical values for pi given are both in bold, and so were (presumably) both world records. However, the first one is actually closer to pi than the second. What's going on there?
- They were both in 20th century BC, so maybe it's because we don't know which one came first? ugen64 02:29, Sep 12, 2004 (UTC)
I don't understand the recent suppression of
|mid 6th century BC||1 Kings 7:23||3
by Rossnixon, with "explanation"
Bible ref would be internal, not external circumference of the vessel
The text is: [3] (http://www.biblegateway.com/passage/?search=1Kings%207)
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 [p] to measure around it.
As far as I understand, this clearly means: circumference = diameter × 3. Although this is certainly not a candidate for a world record, it might be noteworthy, allows people to discover on-line details of the bible, and besides this, there are other more urgent things ("random joke"...) to delete on this page, imho. — MFH: Talk 12:02, 14 May 2005 (UTC)
- Well, removing it makes sense. There's nothing in the passage that says pi is any specific value, or that it intended to convey a value of pi. It doesn't say, "Solomon wanted to compute pi, so he made a large circular device, and the ratio came to about 3." This information could still be mentioned, just not in a table of computed values. Eric119 19:47, 14 May 2005 (UTC)
- More explanation for my deletion, see especially the last paragraph of the article at http://www.uwgb.edu/dutchs/pseudosc/pibible.htm RossNixon 20:41, 14 May 2005 (UTC)
transcendental in 1885
the book i'm looking at (Chaos Theory, Peitgen Juergens Saupe) says that Lindemann proved pi transcendental in 1885, not 1882.
Def of constructible
The definition of a constructible real number is a number which lies in a field gotten by taken a finite sequence of quadratic extensions of the rationals, i.e. it's an x such that there are m1, m2, ..., mi, such that x is in Q[sqrt(m1), sqrt(m2), ..., sqrt(mi)]. (Sometimes the definition is taken to be geometric, but it's not much to show that this definition is equivalent.) Every constructible number is then "expressible as a finite number of integers, fractions, and square roots", (in fact, this is a defining property), so every constructible number is "expressible as a finite number of integers, fractions, and nth roots", and all of these numbers are algebraic. However, not all algebraic numbers can be expressed this way, (Galois theory). So, constructible ==> "exp. in finite # of int., frac., nth roots" ==> algebraic, but none of these implications is reversible. Revolver 08:48, 7 Oct 2004 (UTC)
Pi and physics
The reason it [π] occurs so often in physics is simply because it's convenient in many physical models.
- Isn't one of the major reasons π occurs so often in physics simply because π occurs so often in Euclidean geometry and most of classical physics is based on models using Euclidean geometry? Revolver 02:50, 2 Nov 2004 (UTC)
- I don't agree to either of the above (somehow depending on definition). The π in (almost?) all physics formulae is not 3.14159... but the ratio of the circonference to the diameter. (Thus, only the value of π comes from Euclidean geometry.)
- So, for me, the reason is rather the cylindrical or spherical symmetry in most physical "models"(sic...), or, put otherwise, the basic concept of isotropy of space (while I cannot really understand the signification of being "convenient in many models" - it's not a matter of convenience; once we defined the quantities and units (ok, this could bring in or avoid some π in some places, but most probably at the cost of reappearences elsewhere), we don't have any choice).
- And, b.t.w., the choice of units is not really a matter of physics (laws and relations would hold (to the same approximations) with or without the existence of Earthlings), but of engeneers or of labels on measuring devices. — MFH: Talk 18:18, 10 May 2005 (UTC)
Pronunciation
While the original Greek letter for pi was phonetically equivalent to the English letter p, it has now evolved to be pronounced like the word pie in most circles.
Surely this should be up near the beginning of the article, not in the properties section. Furthermore, it is very poorly phrased: what it means to say is something like "Although in Greek the name of the letter π is/was pronounced something like the name of the english letter p, the standard English pronunciation is identical to pie." Furthermore, I'm not entirely sure that the Greek pronunciation is all that relevant to this article. I would make these changes myself, but frankly I'm having trouble coming up with a phrasing that isn't totally clunky and awkward. Someone else want to take care of this please? --68.78.77.224 04:29, 22 Nov 2004 (UTC) Iustinus
- Be bold. Fix it.
- I don't agree. For such kind of mathematical symbols (after all, that's what it is), it is quite useful to give the "standard" (pie) and the "truely right" pronounciation (pee), ASAP after it's introduction. Your critics comes from the fact that for you, the knowledge of the pronounciation is trivial and maybe not important, but think of other cases ("ess" or "integral"? "S" or "sigma" or "sum"? "ex" or "cross" or "times"? "U" or "union"?, without even speaking of amalgam product, etc.) — MFH: Talk 18:32, 10 May 2005 (UTC)
Greek word for π?
I read somewhere that in modern Greek, what we call π is called something else, but I can't for the life of me remember what it was. Does anyone else know, perchance? Gus 04:31, 2005 Jan 3 (UTC)
Maybe something like "number of the circle" in Greek (like the German de:Kreiszahl), or you may think of "perifereia" (="periphery"), that's where (the initial) π comes from. (could be mentioned in the introduction...) — MFH: Talk 00:21, 11 May 2005 (UTC)
Break-up of digits?
Someone today just switched the digits of pi to a breakup of 3 digits each rather than the former 5 ie.
- 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58
- 3.14 159 265 358 979 323 846 264 338 327 950 288 419 716 939 937 510 58
I think this second format is harder to read (personally I can't stand it), any input? Additionally the code is messed up with s NitrogenX (Michael Hines) 05:22, Feb 25, 2005 (UTC)
- Agree. Can we also get rid of half of the digits? 30 digits or so should be more than enough. Oleg Alexandrov 16:37, 7 May 2005 (UTC)
- The 3-digit grouping is the most common format for engeneers and everyday people, for evident reasons (at least, but not only, for digits in front of the decimal point). As a mathematician with 5 fingers on each hand, I might also prefer the 5-grouping, but most of such preferences are/should be somehow suppressed in this encyclopedia. I think the overwhelming majority of earthlings rather tend to the 3-grouping of decimal digits.
- In what concerns the 2nd point, why not leave "one line length" (i.e. at least 60 characters) of digits (i.e. such that it does not take more space on the page layout when viewed with some reasonable browser, euh... lynx, euh... reasonable resolution, I mean); maybe this way some young talented Indian guy passing by this place will notice "the pattern", which he would not if only 30 digits were present... — MFH: Talk 17:48, 10 May 2005 (UTC)
- Well, books such as Abramowitz and Stegun Handbook of Mathematical Functions, CRC Standard Mathematical Tables, and textbooks usully group digits after the decimal point in groups of 5. I vote for 5. --Bubba73 03:57, 6 Jun 2005 (UTC)
π? Pi?
Maybe I missed something, but why not fix up the crummy Pi title, and move the lot to a proper "π" page?
Roy da Vinci 05:25, 13 Mar 2005 (UTC)
Inconsistency with "irrational number" and "finite length" claims
This article claims that pi is both an irrational number and at the same time having a finite length of 1.3511 trillion digits. Both of these claims can not be true, since finite length implies pi being a rational number, which can be expressed as a ratio of two integers:
- <math>\pi = \frac{3 \; 141 \; 592 \; ...}{10^{1 \; 351 \; 100 \; 000 \; 000} } <math>
--Fredrik Orderud 00:44, 9 May 2005 (UTC)
- I think you're misinterpreting the article. I'm going to remove the word "total" which I hope will make it clearer. Eric119 02:03, 9 May 2005 (UTC)
- Thanks, that's a lot better! But my main reason of worrying is the Yasumasa Kanada page, where profesor Kanada, who computed all the digits, claims that "pi has a digit expansion of 1.3511 trillion and does not expand indefinitely as previously assumed". This implies that the 1.3511 trillion digits are all the digits of pi. --Fredrik Orderud 08:50, 9 May 2005 (UTC)
Kanada's claim of finding 1.3511 trillion digits of pi in 2004 was pure fiction, based on a vandalized edition of the Yasumasa Kanada article. I've therefore removed this finding from the "History" of pi in the main article. --Fredrik Orderud 19:52, 9 May 2005 (UTC)
calculating records
I don't think that all of the records from 1954 to 1992 were by Wrench and Smith, as the table indicates now.
A History of Pi by Petr Beckmann, page 197, lists:
1954-1955 NORC is programmed to computer 3089 digits
1957 Pegasus computer (London) computes 7480 places
1959 IBM 704 (Paris) computes 16,167 decimal places
1961 Shanks and Wrench improve computer program for pi, use IBM 7090 (NEw York) t compute 100,000 decimal places
1966 IBM 7030 (Paris) computes 250,000 decimal places
1967 CDC 6600 (Paris) computes 500,000 decimal places
I don't want to change the table (so I don't mess it up again), so I would appreciate it if someone made these changes to the table. --Bubba73 00:50, 31 May 2005 (UTC)
- Well, I think I make all of these changes. --Bubba73 03:29, 6 Jun 2005 (UTC)
Computing pi given enough random numbers?
I have heard of a way to compute pi given enough random numbers. I think it was called the "Monte Carlo Value for Pi." Is there a formula one has to plug the random numbers into to get this pi value? I found "http://www.random.org/stats/", which gives the monte carlo pi value for random numbers on their page in real time, and "http://www.fourmilab.ch/random/", which briefly explains it,
"For very large streams (this approximation converges very slowly), the value will approach the correct value of Pi if the sequence is close to random. A 32768 byte file created by radioactive decay yielded:
Monte Carlo value for Pi is 3.139648438 (error 0.06 percent)."
but I want to know an algorithm or a formula to be able to compute pi with the numbers from A Million Random Digits with 100,000 Normal Deviates. If anybody could help me out, I would be really grateful. Thanks, DC
- One way to do it is to simulate Buffon's needle. Use the random numbers to pick a center point of the needle and the angle. Then use trig to see if it croses a line. I wrote a program for this about 15 years ago, but I couldn't find it. --Bubba73 21:10, 2 Jun 2005 (UTC)
- I think I also have an idea. The area of a square with the side length of 2 units is naturally 4 square units. At the same time the area of a circle with unit radius is <math>\pi\times1^2=\pi<math>. Of course, such a circle is able to touch all the four sides of the square. Divide the area of the circle with the area of the square and you get <math>{\pi\over4}<math>. If you denote this to be <math>n<math>, then <math>\pi=4n<math>. But <math>n<math> is computable, using random numbers! Just put the square (with the inscribed circle) onto a coordinate plane, with its center at <math>(0;\,0)<math> and its sides parallel to the axes. Now generate two random numbers <math>x<math> and <math>y<math>, both in the range <math>[-1;\,1]<math>. So they are just the coordinates of a point in the square and probably also in the circle. To check if a point lies actually in the circle compute <math>x^2+y^2<math>, that is the square of its distance from <math>(0;\,0)<math>. If <math>x^2+y^2\leq1<math>, then the point lies in the circle. Now generate a vast number of such points, count the total number of them (let's say, <math>A<math>) and also the number of the points that are situated inside the circle (<math>I<math>, for example). Then <math>n\approx{I\over A}<math>; the equality becomes the more exact the more points you use. Thus <math>\pi=4n\approx{4I\over A}<math>. This way you can avoid all the trigonometry involved in the Buffon's needle simulation. Any better ideas are welcome! (By the way: you can sign your messages using four tildes: ~~~~) — Pt (T) 21:25, 2 Jun 2005 (UTC)
- I just created a quick C++ program to demonstrate this:
#include <cstdio>
#include <cstdlib>
#include <ctime>
int main()
{
int i=0, j;
const int n=100000000;
long double x, y;
srand(time(0));
for (j=n; j--;)
{
x=(2*(long double)rand())/RAND_MAX-1;
y=(2*(long double)rand())/RAND_MAX-1;
if (x*x+y*y<=1)
i++;
}
printf("Using %d random points, I got that pi is about %.100Lg.\n", n, ((long double)4)*i/n);
return 0;
}
- Its output:
Using 100000000 random points, I got that pi is about 3.14186652000000000008010647700729123243945650756359100341796875.
- That's another good way - basically inscribe a circle inside a square and throw darts with your eves closed and see how many are inside the circle compared to the square. --Bubba73 17:43, 3 Jun 2005 (UTC)
pi = 3?
Is is true, or only inaccurate folklore, that at some point maybe 100 years ago, a U.S. state legislated that pi was equal to three? Was there a bill that never got passed? Or is the whole story nonsense? If the story has some truth, does it rate a mention in this article? Dmharvey Missing image
User_dmharvey_sig.png
Image:User_dmharvey_sig.png
Talk 14:58, 6 Jun 2005 (UTC)
- http://www.cs.uu.nl/wais/html/na-dir/sci-math-faq/indianabill.html - Fredrik | talk 15:12, 6 Jun 2005 (UTC)