Talk:Pi

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Calculating/measuring pi

The Gregory-Leibniz form is indeed beautiful, but sheds no light on pi without an explanation of why it works, and thus shouldn't appear (with or without explanation) so early in the article. I was going to edit the sections, but the comment sent me here. I would have moved the "measuring pi" section up, merging in the para about simple/complicated formulas and then left discussion of all the series to the following sections and the "computing pi" article. Lunkwill 22:08, 2 February 2007 (UTC)

I've added a brief mention of $tan(\pi /4)=1$ to explain the geometrical origins of the series. --Michael C. Price ^talk 10:40, 3 February 2007 (UTC)

I don't think that really helps, except to readers who know the power series for arctan − which are definitely not the people that section is written for. There has been quite some debate about the desirability of having a section early in the article that explains to "the mathematically non-sophisticated reader" that there are definite mathematical procedures for approximating pi, i.e. that it is \emph{not} just an empiric matter of measuring large circles in physical space. I remain fairly unconvinced that we need such a section, but as long as we do have it, please let it do its job without confusing its target audience with mathematical prerequisites that they don't have. (I'll resist the temptation to revert MichaelCPrice for the moment). Henning Makholm 23:02, 3 February 2007 (UTC)

I think the section is needed because otherwise people will think it is just a mysterious empirical thing. I reason I added the tan and sin reference is that the section says it has a geometrical origin but explains no further. As Lunkwill said above, this is not really satisfying. (I didn't write out the power series for arcsin and arctan, so don't see that I added any mathematical complexity to the section, just a few concepts and pointers.) --Michael C. Price ^talk 23:30, 3 February 2007 (UTC)

As it is now, the very target audience of the section will be left out puzzled when you expect them to be able to make the connection from tan(pi/4) to an infinite series. Your text does expect them to make that connection, because it clearly has the form "because of A, B must be true". Any reader who cannot see what A and B have to do with each other will conclude that the matter is too complex or advanced for him to understand, and will give up on learning anything from the section. As far as I can see, the text was perfectly allright for its purpose until you added some advanced claim without explaining further. Henning Makholm 01:25, 5 February 2007 (UTC)

Is it any less baffling when there is no indication of where the series comes from? --Michael C. Price ^talk 12:26, 5 February 2007 (UTC)

Yes, no indication of where the series comes from is vastly less baffling than an indication that will make no sense at all for the intended audience and thus convince the intended reader that he is not the intended reader at all. Henning Makholm 19:15, 5 February 2007 (UTC)

I found the previous lack of explanation baffling. --Michael C. Price ^talk 13:16, 6 February 2007 (UTC)

The current lack of explanation is far more baffling because now there is something that the unitiated will wrongly believe explains anything. Henning Makholm 19:13, 6 February 2007 (UTC)

I think the recent changes to that section have needlessly complicated what used to be a straightforward example for laypeople. It doesn't matter what the derivation of the Gregory-Leibniz form is. There are many laypeople who want to know if there is a simple way of calculating the value of Pi. I have pointed three or four of my friends/colleagues to the article and to that section in the past, and they all appreciated and understood it as previously stated. But as soon as you throw in the part about the right isoceles triangle and tangents, it's no longer a simple statement about a way to calculate Pi. Putting it in there also contradicts the previous sentence, about how calculating Pi doesn't require understanding of trigonometry. When I was in high school tangents were covered in Trigonometry. The idea that the derivation belongs here or is required in this section is overthinking it.

Another point. I dislike the way the Gregory-Leibniz series has been modified. It used to say:

$\pi =4-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}\cdots$

Now it says:

$\pi =4\left(1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-{\frac {1}{11}}\cdots \right)$

The first form is the simplest statement to understand. It may not be as mathematically beautiful, but if it's more understandable it should be what's used there. -- Moondigger 18:59, 5 February 2007 (UTC)

I agree, on the form of the series. Readers will either see no difference, or follow the 4's more easily. Septentrionalis PMAnderson 19:19, 5 February 2007 (UTC)

Hmm, I'd change it back to the original one, but the HTML table-based fractions are really ugly. Is there any way to force it to appear in an image? ~ Keiji (iNVERTED) (Talk | Contribs) 19:48, 5 February 2007 (UTC)

Okay, I've tried to please both of you; the gregory-leibniz is there so people can see that there are simple formulae, and I mentioned the algorithm from the computing pi article that uses pythagoras for inside-outside tests so that people get the notion of why a simple formula can yield an irrational number. Lunkwill 20:56, 6 February 2007 (UTC)

I like the section much better now. Consensus above seems to be that the series is probably easier to grasp for unsophisticated readers if we distribute the 4 into each fraction; I have done so. I also tried to switch to slashed fractions; does that fix the rendering problem Inverted alludes to? (On the other hand, I wonder how many readers will now get the precedence wrong and think it's some kind of continued fraction ...) Henning Makholm 23:56, 6 February 2007 (UTC)

I put the brackets back in, but leaving the 4 on each fraction. That makes it appear in an image, and it shouldn't look too odd even without a coefficient in front of the brackets. ~ Keiji (iNVERTED) (Talk | Contribs) 07:12, 7 February 2007 (UTC)

Remove semi-protection

As per semi-protection policy this is no longer the 'only reasonable option left to deal with vandalism.' It's been semi-protected since 14th January. I'm putting a request in to remove this here. The18thDoctor 13:05, 6 February 2007 (UTC)

Recommendations

Just a few suggestions I have for this article -- some major, others minor.

1. What is the source for the assertion that "The constant is named "π" because it is the first letter of the Greek words περιφέρεια 'periphery' and περίμετρος 'perimeter', i.e. 'circumference'"?
2. Do we really need to know what Unicode character is used to represent the greek letter pi? That material would be more appropriate for a different article.
3. "The exact value of pi has an infinite decimal expansion" needs to be reworded. The value of pi does not have an infinite decimal expansion -- the number has an infinite decimal expansion. At least that's what I'd say.
4. The second method for approximating pi should include the term "lattice point" somewhere.
5. The paragraph on the Biblical "approximation" of pi needs to be reworded (and sourced). It's almost unintelligible.
6. "For many purposes, 3.14 or ²²/₇ is close enough" should probably not mention 22/7, since it fits better in the context of the next sentence, which discusses fractional approximations rather than decimal approximations.
7. Most of the formulae involving pi need to be diked out and inserted into either Computing pi or List of formulae involving pi.
   • The section Continued fractions exists in Computing Pi, should we remove it here? Deathbob 17:34, 24 February 2007 (UTC)

It is not clear that that section is relevant at computing pi. Are anyone seriously using the systematic continued fractions to approximate pi? I would have pegged them as merely "interesting properties". –Henning Makholm 19:49, 24 February 2007 (UTC)

1. The statement on the Indiana State Senate really, really needs to be correct. As it stands it is unclear whether it is correct.
   • See Indiana Pi Bill and the links there named. Septentrionalis PMAnderson 02:45, 14 February 2007 (UTC)
     • I meant that the statement didn't actually say what the bill did. It did not attempt to define a new value of pi, as many people have stated incorrectly. It proclaimed that a certain circle-squarer had succeeded, and that the state of Indiana had the exclusive rights to his discovery. I think the article should make this a little more clear. --N Shar 06:15, 15 February 2007 (UTC)
2. Lots of citations are needed

N Shar 02:24, 14 February 2007 (UTC)

Significant digits in engineering calculations

The article says, practical science and engineering will rarely require more than 100 digits. Can anybody cite anything to support the claim that anywhere near that many digits are ever used in engineering calculations? The moon is about 10^9 feet away from the Earth. If NASA worked the Apollo calculations to 10 digits, they would have hit the moon within one foot of their target, which is clearly far higher precision than was needed. If you told me they used 12 or even 15 digits, I would believe you. But 100? That's absurd. -- RoySmith (talk) 00:55, 25 February 2007 (UTC)

No claim that "anywhere near that many digits are ever used" is made in the article, and therefore there is no need to support such a hypothetical claim. The sentence you quote asserts that the required number of digits for almost everything is less than 101. Last time I checked, 12 and 15 were both indeed less than 101. –Henning Makholm 01:09, 25 February 2007 (UTC)

The word rarely isn't nearly strong enough. Something is rare when 5% of the instances do it that way. I would say, practical science and engineering will rarely require more than 6 digits -- RoySmith (talk) 03:01, 25 February 2007 (UTC)

You might as well say that practical science and engineering only uses the first Million digits of Pi, or that the Earth is bigger than a proton. It may be true, but it's not useful. Paladinwannabe2 18:17, 27 February 2007 (UTC)

I'd agree with Roy that the statement does suggest that there are computations which require 50-100 digits. However, I appreciate it won't be easy to reformulate it in an acceptable (i.e., attributable) way. That won't stop me from giving it a try though. How about

While the value of pi has been computed to billions of digits, practical science and engineering computations typically use only 16 digits. It is hard to envisage a situation where more than a hundred digits will be needed for this purpose.

The 16 digits refers to standard double precision. I think it's fair to say that is the precision typically used, though perhaps we do officially need a reference to back this up. -- Jitse Niesen (talk) 03:29, 25 February 2007 (UTC)

You could say something like, The roughly 16 decimal digits of precision supplied by the ubiquitous IEEE 754 floating point format is more than sufficient for almost all practical scientific and engineering calculations -- RoySmith (talk) 04:20, 25 February 2007 (UTC)

I agree, this is very silly. 5 digits of Pi gets us within 0.0001 inches at my work, and it looks like we got men to the moon on 5 digits as well, but I can't find an authoritative source yet. I'm going to change it, since I can always supply my own reference in the form of mathematics. Paladinwannabe2 17:18, 27 February 2007 (UTC)

Hardly an attributable source, but there's a scene in Apollo 13 where the guys in mission control are working calculations with slide rules, which are typically good to 3 digits. -- RoySmith (talk) 17:27, 27 February 2007 (UTC)

I changed the example to something sensible. Millimeters are something we can see, we know the earth is huge, this will hopefully get the point across better. Plus, I can provide my own numbers for anyone to see and double-check.

(Using 6,378.137km as the Earth's radius (r) at the equator, 2*(r*3.1415926536 - r*pi) ~= 0.130mm.)Paladinwannabe2 18:03, 27 February 2007 (UTC)

I am unhappy that the "circle the size of the galaxy" example has disappeared. It gave a nice direct demonstration that a realtively small number of digits is sufficient for anything anybody in their wildest dreams could imagine computing. The "six digits for an Earth-sized circle" does not impress this as vividly. Non-matematician readers who don't deeply grok how accuracy in a positional system improves exponentially with the number of digits might end up with a fuzzy expectation that to calculate a circle that encompasses a thousand earths, one would need six thousand digits – and the point that computing millions of digits is practically useless would not be driven home as efficiently. –Henning Makholm 21:05, 27 February 2007 (UTC)

Is there an error here?

Under the section Analysis, the second example begins with the sentence: Half the circumference of the unit circle: but the formula states it's equal to Pi; which is the whole circumference, not half of it! What am I missing here? -- Daniel B. Sedory 67.150.121.211 21:57, 25 February 2007 (UTC)

The unit circle has radius 1. Therefore its diameter is 2, and its total circumference is $2\pi$. –Henning Makholm 23:17, 25 February 2007 (UTC)

"Guillaume Pelletier" trigonometric limits

Why has the equations : $\lim _{P\to \infty }{\pi }=P\left[{\frac {\sin \left({\frac {360}{2P}}\right)}{\sin \left({90-{\frac {360}{2P}}}\right)}}\right]$ and $\lim _{P\to \infty }{\pi }=P\cos \left({90-{\frac {360}{2P}}}\right)$ been removed? The reason provided is not clear. The formulae do not require the value of pi to be inserted. The only value required is that of P which is the number of sides on a polygon. The formulae are effective in large values of P, as the perimeter of the polygon approaches that of the circumference of the circle. The approach of the formulae is not that of supreme practicality, but of functionality, simplicity and concise interpretation: replace P by an extremely high numerical value and you have an extremely accurate pi value. Even higher values of P return even more accurate values of pi — Preceding unsigned comment added by Epgui (talk • contribs) 2007-02-25T23:37:32 (UTC)

Your formulae involve sines and cosines of angles that you specify in degrees, such as 90-360/2P. The procedure for computing a definite value for such a sine or cosine has two steps:

1. Convert angle in degrees to radians by multiplying the number by $\pi /180$.
2. Apply the definition for the (co)sine of a radian angle, e.g.:

   $\sin(x)=x-x^{3}/3!+x^{5}/5!-x^{7}/7!+x^{9}/9!-\cdots$

That your formulae converge towards $\pi$ are entirely due to the step 1, where a preexisting value of pi is supposed to exist. You may not be aware that this is what your calculator does internally when you ask it to compute the sine of a degree value, but it is. The limit has nothing to do with trigonometry; in fact you would get exactly pi by throwing out the trig functions and approximating $\cos(\pi /2-\epsilon )=\sin(\epsilon )\approx \epsilon$ and $\sin(\pi /2-\epsilon )\approx 1$ for small $\epsilon$. –Henning Makholm 23:57, 25 February 2007 (UTC)

By the way, you're not the first to present such formulae; I removed almost the same one from history of numerical approximations of pi a few months ago, and initially thought you were the same editor who added that. Further investigation shows this to be unlikely. Sorry for the snappy edit summary; I should have assumed good faith. –Henning Makholm 00:33, 26 February 2007 (UTC)

Thanks a lot for helping out, I did not realise beforehand that trigonometric functions implied the use of pi. —The preceding unsigned comment was added by Epgui (talk • contribs) 00:44, 26 February 2007 (UTC).

You're welcome. In the interest of telling the complete thruth, I should mention that there are formulae that allow, e.g., going from sin(v) to sin(v/2) without knowing pi (you may be able to figure them out yourself by considering some appropriately chosen congruent right triangles). Then if you start with sin(90°)=1 you could in principle compute your limits for P's of the form 2ⁿ without knowing pi already; in essence this is how Archimedes approximated pi, as mentioned briefly in the article. Some of the early trigonometric tables were computed in this way, too. But modern sine computations invariably work in radians. –Henning Makholm 01:25, 26 February 2007 (UTC)

PS - someone pointed out that the limit was not placed correctly in the equation. We did not do limits in school yet, so I would be curious to see the corrected formulae. (Epgui 00:52, 26 February 2007 (UTC))

It may have confused you that the limit of a sequence article alwas has then "lim" sign is on the far left of an equation. What you should know is that "lim" is something that applies to an expression rather to an entire equation. Thus, for example

1. "$P\sin(180^{\circ }/P)$" is an expression that has some value if P is given.
2. "$\lim _{P\to \infty }P\sin(180^{\circ }/P)$" is an expression that has some value even if we don't give it a P, because the lim symbol explains what the P in the inner expression is.
3. "$\pi =\lim _{P\to \infty }P\sin(180^{\circ }/P)$" is an equation that asserts something about the value of the second expression.

Clearer now? –Henning Makholm 01:25, 26 February 2007 (UTC)

Yes, thank you very much :) (Epgui 11:38, 26 February 2007 (UTC))3.14159265358979323

Transcendental, not just irrational

In the article lead, it says "The mathematical constant π is an irrational real number". Shouldn't that be "... transcendental real number"? It is mentioned 'way below, but I think it should be highlighted up front. (If the page were not semi-protected I would just go ahead and change it, but I cannot be bothered to try to remember my wikipedia password for this.) -- 72.130.182.201 03:23, 15 March 2007 (UTC)

Gottfried Leibniz or Madhava of Sangamagrama?

For me, it seems that the articles "History of Pi" and "Pi" is a little unclear about the creator of

$\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots ={\frac {\pi }{4}}$

In "pi" it is just stated that "Leibniz' formula (proof): [the expression]" while in the article "History of Pi" Madhava of Sangamagrama is credited. One might intepret (if only reading the "Pi"-article) Leibniz as the creator. Perhaps this needs to be changed somehow - or am i missing something? Sorry for the bad English. --Bilgrau 22:38, 20 March 2007 (UTC)

Done--Ķĩřβȳ♥ŤįɱéØ 03:27, 15 March 2007 (UTC)