Talk:Pi/Archive 2

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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

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.

I took a college geometry class that began with taxi geometry, where the distance between any 2 points is the sum of the vertical distance and the horizontal distance. (You could not move diagonally.) the first assignment was to calculate the value of pi, defined as the ratio of the circumfence of a circle to the diameter. A circle was defined as the locus of points that all had the same distance in taxi geometry from the center point. Pi was 4.

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.

-- Damian Yerrick

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

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)


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)

It sounds vaguely familiar. There's ddefinitely pi bonds in chemistry. -- Tarquin
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

0.1121231234...

is a counter example. However, I have heard this asserted on a (mostly) serious radio program. 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

F = \frac{4 \pi GMm}{4 \pi r^2}

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,

\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k!)(1103+26390k)}{(k!)^4 396^{4k}}

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)

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)
The pattern that is being hinted at is the counting up. Eg '101112' = '10','11','12'. All the other groups comply. Commander Keane 10:56, 22 July 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: [1]

  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)

OK, you're right. thanks -Lethe | Talk

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)
You seem to believe that my comment (which I made over half a year ago) was expressing a desire to remove the pronunciation entirely. It wasn't I was mere suggesting that it be moved to the beginning of the article, and phrased better. Someone has taken care of that (thank you). I suppose if you really want we could include that in koine Greek the name of the letter was pronoucned more or less like "pee" (and that it still is in Modern Greek, and many other languages, but in Classical Greek it may have been more like "pay!" and so on) but I don't see the point. How a mathematical symbol is pronounced in any language other than the one the article is written in seems tangential at best. --Iustinus 28 June 2005 20:54 (UTC)
I'd like to reopen the possibility of briefly including the current Greek pronounciation for the pi for this ratio (i.e., "pee") in this English article. Why? Well, simply put: it is Greek to me. 'Irrelevancy' is a matter of perspective, Macrakis. Does it detract from being informative? No: as the article on the letter itself indicates, the Greek pronounciation differs from the predominant English pronounciation. As well, how would a Greek today pronounce this letter as the ratio? Not as "pie", but as "pee."
If this were not a Greek character in popular English use, I would not be emphatic regarding this. And as there is no clear consensus on this issue (above or otherwise), feedback is encouraged and I reserve the right to include this in the article unless there is a clear consensus otherwise.
I oppose this nonsense nationalism. The demotic pronunciation of π is post-classical and has nothing to do with its pronuncation in any other European language, including English. The original pronunciation is indicated by its early spelling πει, and is not far from the present English pronunciation, although this is a coincidence. Septentrionalis 20:34, 30 September 2005 (UTC)
Your comments are noted, but the pronunciations are not the same in English and Greek. Post- classical pronunciation does not invalidate its common Greek usage currently. Also, please proceed with caution when referring to other comments as 'nonsense,' since it is not clear-cut. E Pluribus Anthony 20:56, 30 September 2005 (UTC)
And now I must go to ... the loo. :) Thanks! E Pluribus Anthony 18:31, 30 September 2005 (UTC)
TY: the word 'position' is wholly appropriate; I am from the Great White North, after all. :) E Pluribus Anthony 21:40, 2 October 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 Lethe | Talk 05:22, May 7, 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)

The problem with having a pi symbol as the title is it is automaticly changed to the uppercase pi symbol which looks like Π. as demonstrated here:[2]

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:

\pi = \frac{3 \; 141 \; 592 \; ...}{10^{1 \; 351 \; 100 \; 000 \; 000} }

--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 \pi\times1^2=\pi. 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 {\pi\over4}. If you denote this to be n, then π = 4n. But n is computable, using random numbers! Just put the square (with the inscribed circle) onto a coordinate plane, with its center at (0;\,0) and its sides parallel to the axes. Now generate two random numbers x and y, both in the range [-1;\,1]. 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 x2 + y2, that is the square of its distance from (0;\,0). If x^2+y^2\leq1, then the point lies in the circle. Now generate a vast number of such points, count the total number of them (let's say, A) and also the number of the points that are situated inside the circle (I, for example). Then n\approx{I\over A}; the equality becomes the more exact the more points you use. Thus \pi=4n\approx{4I\over A}. 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.
Not so bad at all... — Pt (T) 21:45, 2 Jun 2005 (UTC)
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)
This seems to me the same than the above program, except for the hits outside the square, and the fact that the darts will have a much less homogeious distributions and I don't know anybody who would count 100 M darts.
Concerning the C program, one notices at once the 0's in the output and realizes that 10^8 throws can't give more than 8 digits, so the 'double's are exagerated : no need to calculate and display 40 digits! — MFH: Talk 13:09, 24 Jun 2005 (UTC)
Yep, I know. I just didn't think about that when I was writing that program — it was late at night and it was supposed to be just a quick demo of my idea... — Pt (T) 14:11, 24 Jun 2005 (UTC)
I adopted the above C program (sorry, I don't know C++ ;-) to allow for any n given as optional arg:
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
main(int argc, char **argv){
   int n,j,i=0; float x,y,r=RAND_MAX,r2=r*r;
   if( argc<2 || (n=atoi( argv[1] ))<1 ) n=10000000;
   printf("Using %d random points...\n", n);
   srandom(time(0));
   for (j=n; j--;) {
       x=random(); y=random(); if ( x*x+y*y <= r2 ) i++;
   }
   printf("I got that pi is about %19f.\n", i*4.0/n);
}
It runs in half the time than the previous one (6 vs 12 sec on my PC) and gives
(~/C) time ./rand-pi 99999999
Using 99999999 random points...
I got that pi is about 3.141776671417766842.
5.970u 0.010s 0:05.99 99.8%     0+0k 0+0io 88pf+0w

— MFH: Talk 14:17, 24 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 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)
Wow. That's so cool. Dmharvey Image:User_dmharvey_sig.png Talk 15:47, 6 Jun 2005 (UTC)
It is exaggerated and garbled. It was Indiana; and the bill didn't pass. The President of IU (IIRC) arrived to look after his appropriation, found that they were discussing this bill, and talked them into referring it to the Committee on Swamp Lands and forgetting it. The value was not three; the bill described π in two different wrong ways, inconsistent with each other. See Petr [sic] Beckmann; Π.
Dear Pmanderson, please clarify: which bit exactly is exaggerated and garbled? My initial post on this question? or the description given at http://www.cs.uu.nl/wais/html/na-dir/sci-math-faq/indianabill.html? In either case, is my edit to the main article (now apparently moved out to History of Pi) correct and/or appropriate? Thanks. Dmharvey Image:User_dmharvey_sig.png Talk 20:55, 24 Jun 2005 (UTC)
Okay, if you ever have a question about something like this check Snopes.com. They are the definitive source for urban legends. All research is well done. Incidentally this one is an urban legend - false. http://www.snopes.com/religion/pi.htm NitrogenX (Michael Hines) 08:33, Jun 25, 2005 (UTC)
Different case. "Though the claim about the Alabama state legislature is pure nonsense, it is similar to an event that happened more than a century ago. In 1897 the Indiana House of Representatives unanimously passed a measure redefining the area of a circle and the value of pi. (House Bill no. 246, introduced by Rep. Taylor I. Record.) The bill died in the state Senate." - Fredrik | talk 14:59, 25 Jun 2005 (UTC)

64 or 70

Sukh, the 70 digits were correct, it would have been enough to click on the link to the (sequence A000796 in OEIS) link, in order to verify it! (If even the editors don't use the links to sources...)— MFH: Talk 12:57, 24 Jun 2005 (UTC)

Requested move back to Pi

Mike Rosoft placed a request to change the name from Π to Pi on 28 June 2005.

  • Strongly concur. Pi is the proper, spelled name of this quantity in English. Furthermore, an uppercase Π is totally wrong for this mathematical quantity; it is a lowercase π, not uppercase, and the Wikipedia software always capitalizes the first letter of a title so the one here is wrong. OTOH, the word "pi" is properly capitalized in various circumstances, including this one.
Fix the disambiguation link at the top of this article as well, and all of the screwed-up mess with at least three disambiguation pages at Pi (disambiguation) and Pi (letter) and Π (disambiguation) and who knows what else. Gene Nygaard 28 June 2005 21:37 (UTC)
Pi (letter) isn't actually a disabiguation page though its close to one. i've recombined the two disambiguations. Also as well as the uppercasing issue i strongly beleive that titles should be the names of things not the things themselves. Plugwash 28 June 2005 22:11 (UTC)
BTW i made the move back. There was no reason for this to be listed on requested moves as it didn't require an admin to do it. Furthermore it was basically undoing the actions of someone who got carried away with the new features before proper policy is formed on thier use. Plugwash 28 June 2005 22:14 (UTC)

approximations

Could we prune the approximations a little? Especially the one that uses 18 digits to approximate the first 17 digits of Pi isn't exactly amazing. --W(t) 30 June 2005 23:20 (UTC)

The one I just added (\sqrt{2} + \sqrt{3})isn't all that amazing either, since it only approximates the first 3 digits of π, but I thought it was interesting since it was so simple. Some day I would like to find insight into why it is so close, based on the definitions of pi and the definitions of square root. Anyway, no offense taken if you want to remove it to keep the article simple. -Armaced 30 June 2005 23:24 (UTC)
To start with, I think it would be a good idea to distinguish between cute approximations like 355/113 and exact identities that can be used to compute π to arbitrary precision. I'd like to see one section for each category, possibly with detail moved to subpages. - Fredrik | talk 30 June 2005 23:37 (UTC)
It looks to me like all of the "Miscellaneous formulas" are cute approximations of pi, while the exact identities are in the "Analysis" section. Perhaps these sections need to be renamed to reflect such? -Armaced 30 June 2005 23:45 (UTC)
Oops, I had already taken care of the problem. I created the "miscellaneous formulas" section a couple of days ago, but forgot about it. I guess I'm not getting enough sleep, or something ;-) - Fredrik | talk 30 June 2005 23:47 (UTC)
So you had a good idea twice. More power to you… --W(t) 30 June 2005 23:54 (UTC)
I don't think \sqrt{2} + \sqrt{3} is so close because of any deep relationship between roots and π, but because when you have an alphabet consisting of all the different common mathematical operators and the digits, it's not hard to find a short formula that approximates any number to three digits. By most encodings and metrics, \sqrt{2} + \sqrt{3} contains a lot more entropy (information) than 3.14, so it's actually bloating the information, not slimming it down. I'd suggest we only keep those that are significantly shorter than just the list of digits. And separating the sections is a good idea too yes. --W(t) 30 June 2005 23:53 (UTC)
You are right, of course. Let me just explain my train of thought and then I will drop it. You have two squares, one of area 2 and one of area 3. The edges of those squares add (to within a few thousadths of a unit) to half the circumference of a unit circle. You see, when I first noticed the approximation, I pointed it out to a math teacher, who said "of course... it would be." I always wondered what he meant by that. Perhaps he is just really good at arithmetic, or maybe he was faking it. And maybe this is just a coincidence, but I would like to construct the coincidence somehow. -Armaced 1 July 2005 00:10 (UTC)
Hmm, it still seems a little far fetched to be honest, I think you may have to start considering the possibility that your math teacher was a smug git :-P. --W(t) 1 July 2005 00:16 (UTC)
Probably.  :) -Armaced 1 July 2005 01:57 (UTC)

355/113 should be kept. It's not just "cute", it's a continued fraction approximation. What this means is that you're getting a much better approximation than you'd expect to get with a denominator as small as 113. With such a small denominator, you'd only really expect to get two or three digits after the decimal point; here you're actually getting six. Such an approximation doesn't really "compress" anything (after all you still need to specify the numerator), but it's still interesting. Dmharvey Image:User_dmharvey_sig.png Talk 1 July 2005 13:06 (UTC)

Redirects

A while back, I created redirects from common approximations (3.14, 3.141, etc). Someone has deleted them. Why? --Celestianpower talk 12:40, 12 July 2005 (UTC)

Unsigned note

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?

The official world record is 42,195 places.

Newer coments

Memorization of pi's digits seems like a waste of everyone's time. Next "pi day" why not do the students a favor and read some of the more entertaining sections of Petr Beckmann's The History of π aloud? Less entertaining extracts, but pertinent ones, include:"The digits beyond the first few decimal places are of no practical or scientific value. Four decimal places are sufficient for the design of the finest engines; ten decimal places would be sufficient to obtain the circumference of the earth within a fraction of an inch if the earth were a smooth sphere." or "It is an interesting phenomenon that all the digit hunters concentrate on the number π; none ever attempted to find hundreds of decimal places for the square root of 2 or sin 1° or log 2. There seems to be something magical about the number π; I have known several people who memorized π (in their adolescence) to 12 and even 25 decimal digits; none of them memorized say, the square root of 2. There is no mathematical justification for this, for to calculate or memorize π to many decimal places is the same waste of time as doing this for the square root of two. - Nunh-huh 08:50, 25 Jun 2005 (UTC)
When one searches through the web, he can meet information that the world record in memorizing π is unbelievable 42,000 decimal places. -mkh-
"none ever attempted to find hundreds of decimal places for the square root of 2" -- how about 5 million: [3] or maybe more[4] "This algorithm is extremely efficient and may be used to compute pi up to billion's [sic] of digits. "

Bubba73 29 June 2005 02:56 (UTC)