Arthur Rubin (talk | contribs) Revert #Diameter vs. radis section. Something may need to be said about that, but I can't find anything in the immediate suggestion that I would use. |
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[[Image:PiCM200.svg|right|thumb|150px|right|Lower-case ''π'' (the lower case letter is usually used for the constant)]] |
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I would like to move this page so as to avoid confusion with the Eastern Front in World War I. Is "Eastern Front (WW2)" acceptable? [[User:Oberiko|Oberiko]] 14:53, 21 Mar 2004 (UTC) |
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<!-- IMPORTANT NOTICE: Please note that Wikipedia is not a database to store millions of digits of π; please refrain from adding those to Wikipedia, as it could cause technical problems (and it makes the page unreadable or at least unattractive in the opinion of most readers). Instead, you could add links in the "External links" section, to other web sites containing information regarding digits of π. -->{{Two other uses|the mathematical constant|the letter in Greek|pi (letter)}} |
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{{wrongtitle|title=π}} |
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[[Image:Pi-unrolled.gif|frame|right|'''The circumference of a circle is ''π'' times its diameter.''' A straight line may be marked off in equal measures, each the diameter of a given circle. This may be done legitimately with the classical construction methods of [[Euclidean geometry]], using [[compass and straightedge]]]] |
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The [[mathematical constant]] '''π''' is a [[real number]], approximately equal to 3.14159, which is the [[ratio]] of a [[circle]]'s [[circumference]] to its [[diameter]] in [[Euclidean geometry]], and has many uses in [[mathematics]], [[physics]], and [[engineering]]. It is also known as '''[[Archimedes]]' constant''' (not to be confused with [[Archimedes number]]) and as '''[[Ludolph van Ceulen|Ludolph]]'s number'''. |
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:I just got an edit conflict saving my naming rationale... Of the various names, "Eastern Front" is by far the most common in English sources, and although there have been many eastern fronts, this is by far the biggest of all and could be said to "own" the term. Also, of the dozen-odd links to the term from existing WP articles, every one expected to link to this one. So the usual thing to do is to let this one be "Eastern Front" by itself, and add a note in italics to the top linking to [[Eastern Front (World War I)]]. See [[Thomas Mann]] for something similar that I did just yesterday. |
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==The letter ''π''== |
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Why does there need to be a separate article at all? There is nothing in this article which does not already appear in the main WW2 articles. [[User:Adam Carr|Adam]] 22:31, 21 Mar 2004 (UTC) |
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The name of the [[Pi (letter)|Greek letter ''π'']] is ''pi'', and this spelling is used in typographical contexts where the Greek letter is not available or where its usage could be problematic. When referring to this constant, the symbol ''π'' is always pronounced like "pie" in [[English language|English]], the conventional ''English'' pronunciation of the letter.<!--only state this fact, try not to justify here: see Talk page --> |
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:Hey, I can't spend every minute of the day on Wikipedia! :-) This article is the right place for the more detailed description of the campaigns that would be excessive detail for the already-too-long WWII article. Somewhat of a placeholder right now, but destined to expand - surely the largest and bitterest struggle of the war deserves more than three paragraphs, eh? [[User:Stan Shebs|Stan]] 06:20, 22 Mar 2004 (UTC) |
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The constant is named ''π'' because it is the first letter of the [[Greek language|Greek]] words "''περιφέρεια''" ([[periphery]]) and "''περίμετρον''" ([[perimeter]]). The Swiss mathematician [[Leonhard Euler]] proposed that this number be given a particular name and suggested the use of ''π''. |
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== Russia's attack on Finland == |
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==Definition== |
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I restored the reference to [[preemptive attack]]. First of all since I think the truth has to be honored. Secondly, since I guess Kahkonen's wording[http://en.wikipedia.org/w/wiki.phtml?title=Eastern_Front_(WWII)&diff=3522229&oldid=3521905] pretty soon could be exchanged for the usual less truthful stuff from Soviet history books (i.e. "Finland attacked SSSR"), which is somewhat less likely if a hint to German troops in northern Finland is kept in the text. Finally, this wording might be seen as a NPOV-balanced compromise version. ;-) /[[User:Tuomas|Tuomas]] 09:10, 10 May 2004 (UTC) |
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[[Image:Circle Area.svg|right|thumb|Area of the circle = ''π'' × area of the shaded square]] |
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In [[Euclidean geometry|Euclidean plane geometry]], ''π'' is defined either as the [[ratio]] of a [[circle]]'s [[circumference]] to its [[diameter]], or as the ratio of a circle's [[area]] to the area of a square whose side is the radius. The constant ''π'' may be defined in many other ways, for example as the smallest positive ''x'' for which [[trigonometric function|sin]](''x'') = 0. The formulæ below illustrate other (equivalent) definitions. |
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==Numerical value== |
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:Ok, thats better :-) [[User:Kahkonen|Kahkonen]] 10:15 10 May 2004 (UTC) |
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<!-- IMPORTANT NOTICE: Please note that Wikipedia is not a database to store millions of digits of π; please refrain from adding those to Wikipedia, as it could cause technical problems (and it makes the page unreadable or at least unattractive in the opinion of most readers). Instead, you could add links in the "External links" section, to other web sites containing information regarding digits of π.--> |
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The numerical value of ''π'' [[truncate]]d to 50 [[decimal|decimal places]] is: |
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:<!--Please discuss any changes to this on the Talk page.-->3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 |
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So how do we add in the Soviet Campaign in the East meaning Asia during 1945? [[User:Tomtom|Tomtom]] |
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''See [[#External links|the links below]] and those at sequence [[oeis:A00796|A00796]] in [[On-Line Encyclopedia of Integer Sequences|OEIS]] for more digits.'' |
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:I actually think that should be a seperate campaign (even though it is a small one), but I don't know what name to give it. [[User:Oberiko|Oberiko]] 14:02, 9 Jul 2004 (UTC) |
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With the 50 digits given here, the circumference of any circle that would fit in the observable universe (ignoring the [[curvature of space]]) could be computed with an error less than the size of a proton.<ref name = "universe"> {{cite journal |
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== 'Counteroffensive' Section == |
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|author = [[David H. Bailey|Bailey, David H.]], [[Peter Borwein|Borwein, Peter B.]], and [[Jonathan Borwein|Borwein, Jonathan M.]] |
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| year = 1997 | month = January |
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| title = The Quest for Pi |
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| journal = Mathematical Intelligencer |
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| valume = 19 | issue = 1 | pages = 50-57 |
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| url = http://crd.lbl.gov/~dhbailey/dhbpapers/pi-quest.pdf |
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}} |
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</ref> Nevertheless, the exact value of ''π'' has an infinite [[decimal expansion]]: its decimal expansion never ends and does not [[Recurring decimal|repeat]], since ''π'' is an [[irrational number]] (and indeed, a [[transcendental number]]). This infinite sequence of digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing more digits and investigating the number's properties. Despite much analytical work, and [[supercomputer]] calculations that have determined over 1 [[Orders of magnitude (numbers)#1012|trillion]] digits of ''π'', no simple pattern in the digits has ever been found. Digits of ''π'' are available on many web pages, and there is [[software for calculating π|software for calculating ''π'']] to billions of digits on any [[personal computer]]. ''See'' [[history of numerical approximations of π|history of numerical approximations of ''π'']]. |
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=== Calculating ''π'' === |
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Much of this section must be moved to [[Battle of Kursk]] and replaced by summary here. It looks like recent editors were not aware of the separate article on the topic. [[User:Mikkalai|Mikkalai]] 23:24, 8 Aug 2004 (UTC) |
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The formulae often given for calculating the digits of <math>\pi</math> have desirable mathematical properties, but are often hard to understand without a background in trigonometry and calculus. Nevertheless, it is possible to compute <math>\pi</math> using techniques involving only algebra and geometry. |
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== Great Patriotic War == |
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For example, one common classroom activity for experimentally measuring the value of <math>\pi</math> involves drawing a large circle on graph paper, then measuring its approximate area by counting the number of cells inside the circle. Since the area of the circle is known to be |
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I believe that the [[Great Patriotic War]] deserves a short article of its own. It should explain specifically: |
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# History of the term |
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# the starting date controversy |
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# Differences between the terms [[World War II]], [[Eastern Front (WWII)|the Eastern Front of World War II]] and the [[Great Patriotic War]] |
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Note that, although the term is quite popular even in the West, it is not equal to WWII or Eastern Front. [[User:Halibutt|[[User:Halibutt|Halibu]][[User Talk:Halibutt|tt]]]] 23:30, Sep 15, 2004 (UTC) |
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:<math> a = \pi r^2,\,\!</math> |
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:Are there significant differences between the Great Patriotic War and the Eastern Front? So far as I know, the main change would be the inclusion of the Finnish War(s), which are already listed as being considered part of the Eastern Front by many. [[User:Oberiko|Oberiko]] 13:21, 16 Sep 2004 (UTC) |
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<math>\pi</math> can be derived using algebra: |
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::My ''feeling'' (if such are of any value at all) is that this article ought to be re-named (and focused) on the [[Great Patriotic War]], while an article on the [[Eastern Front (WWII)]] could be fairly much shorter. In my opinion, it's yet another expression of Americo-centrism or Western-centrism to call the article on the Great Patriotic War for the Eastern Front, '''but''' I see that it may be easy to misinterpret me here. I am not in favor of parallell articles in principle. I would not at all like a situation were virtually the same scope was covered by two articles, one with the "eastern name" and one with the "western name". /[[User:Tuomas|Tuomas]] 00:09, 18 Sep 2004 (UTC) |
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:<math> \pi = a/r^2.\,\!</math> |
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:::This '''is''' the English and not the Russian WP, so we're expected to title according to what's most familiar to English readers. Calling it Americo-centrism seems a little like an attempt to lean it towards the Soviet POV - to some extent the Soviets brought it on themselves by embedding POV into the name, not unlike the Confederates calling it the "[[War of Northern Aggression]]", or the Bushies calling it "[[Operation Iraqi Freedom]]", and you'll notice that both of those link to more-neutral titles (at least as I write this :-) ) [[User:Stan Shebs|Stan]] 05:40, 18 Sep 2004 (UTC) |
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This process works mathematically as well as experimentally. If a circle with radius ''r'' is drawn with its center at the point (0,0), any point whose distance from the origin is less than ''r'' will fall inside the circle. The [[pythagorean theorem]] gives the distance from any point (''x'',''y'') to the center: |
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::::Our policy is that articles names should be that which '''english''' speakers most commonly call it. This is BY FAR most commonly called the Eastern front, so the article belongs here. [[User:Raul654|→Raul654]] 06:46, Sep 18, 2004 (UTC) |
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:<math>d=\sqrt{x^2+y^2}.</math> |
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:::::Last point from me on the issue, I don't really think the Fin's much care for the title Great Patriot War anyway. [[User:Oberiko|Oberiko]] |
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Mathematical "graph paper" is formed by imagining a 1x1 square centered around each point (''x'',''y''), where ''x'' and ''y'' are [[integers]] between ''-r'' and ''r''. Squares whose center resides inside the circle can then be counted by testing whether, for each point (''x'',''y''), |
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::::::Just avoid to include the Winter War of November 1939 – March 1940.<br>--[[User:Ruhrjung|Ruhrjung]] 08:30, 2004 Sep 21 (UTC) |
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:<math>\sqrt{x^2+y^2} < r.</math></center> |
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The total number of points satisfying that condition thus approximates the area of the circle, which then can be used to calculate an approximation of <math>\pi</math>. Mathematically, this formula can be written: |
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Due to the move to the current name of '''Eastern Front (WWII)''', the following sentence is wrong: |
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:''The war began as Operation Barbarossa on 22 June 1941 4:00 am, when Germany invaded the Soviet Union; and ended on 8 May 1945 when Germany surrendered following the Battle of Berlin. |
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:<math>\pi \approx \frac{1}{r^2} \sum_{x=-r}^{r} \; \sum_{y=-r}^{r} \Big(1\hbox{ if }\sqrt{x^2+y^2} < r,\; 0\hbox{ otherwise}\Big).</math> |
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The World War II did not begin on 22 June 1941. For that matter the war did end after the "Battle for Berlin" but it did not end '''because''' of that battle. (BTW for the Soviets the war finished on the 9th not the 8th). It finished because Hitler was dead and Grand Admiral Karl Dönitz the new Führer of the Third Reich was willing to surrender as the Eastern and Western Fronts had met (of which there is no metion in this article). [[User:Philip Baird Shearer|Philip Baird Shearer]] 23:27, 29 Sep 2004 (UTC) |
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In other words, begin by choosing a value for ''r''. Consider all points (''x'',''y'') in which both ''x'' and ''y'' are integers between ''-r'' and ''r''. Starting at 0, add 1 for each point whose distance to the origin (0,0) is less than ''r''. Divide the sum, representing the area of a circle of radius ''r'', by ''r''<sup>2</sup> to find the approximation of <math>\pi</math>. Closer approximations can be produced by using larger values of ''r''. |
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==Poland== |
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For example, if ''r'' is set to 2, then the points (-2,-2), (-2,-1), (-2,0), (-2,1), (-2,2), (-1,-2), (-1,-1), (-1,0), (-1,1), (-1,2), (0,-2), (0,-1), (0,0), (0,1), (0,2), (1,-2), (1,-1), (1,0), (1,1), (1,2), (2,-2), (2,-1), (2,0), (2,1), (2,2) are considered. The 9 points (-1,-1), (-1,0), (-1,1), (0,-1), (0,0), (0,1), (1,-1), (1,0), (1,1) are found to be inside the circle, so the approximate area is 9, and π is calculated to be approximately 2.25. Results for larger values of ''r'' are shown in the table below: |
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This article is about the ''' Soviet-German War''' or '''German-Soviet War'''. There are other campaings in the Eastern Front of WWII. I think either this artcle needs expanding to include a mention of them with links to them or this article should be moved to another name like '''Soviet-German War''' and a replacement article with a brief overview of each campaing/war be put into its place. |
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<center> |
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The primary campaing that should be mentioned as part of the Eastern front which is not mentioned at the moment is the '''Polish September Campaign''' of 1939. |
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{| class="wikitable" style="text-align:center" |
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But there should also be a mention of German counter-partisan operations in Yugoslavia, {Fall Weiss, (1942), Operation Weiss II, Operation Weiss III, and Operation Schwarz all which involved up to half a dozen divisions of German soldiers}, as well as the partisan operaions in Yugoslavia. [[User:Philip Baird Shearer|Philip Baird Shearer]] 23:27, 29 Sep 2004 (UTC) |
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|- |
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! r !! area !! approximation of π |
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|- |
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| 3 || 25 || 2.777778 |
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|- |
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| 4 || 45 || 2.8125 |
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|- |
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| 5 || 69 || 2.76 |
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|- |
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| 10 || 305 || 3.05 |
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|- |
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| 20 || 1245 || 3.1125 |
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|- |
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| 100 || 31397 || 3.1397 |
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|- |
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| 1000 || 3141521 || 3.141521 |
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|} |
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</center> |
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Similarly, the more complex approximations of π given below involve repeated calculations of some sort, yielding closer and closer approximations with increasing numbers of calculations. |
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:As has been stated on [[Category talk:World War II campaigns and theatres]] neither the [[Polish Defense War of 1939]] nor the fights in Yugoslavia and Greece are usually referred to as Eastern Front. That's why they were left alone, as separate sub-categories of the WWII operations. [[User:Halibutt|[[User:Halibutt|Halibu]][[User Talk:Halibutt|tt]]]] 23:43, Sep 29, 2004 (UTC) |
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You seem to be using a circular argument here. In that discussion you said |
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:''However, take a look at '''Eastern Front''' and '''Eastern Front (WWII)''' and check what are these articles about. |
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It reminds me of the old war song that went "We're here, because we're here, because we're here". I am suggesting changing this article to include all the campaings on the eastern front of World War II, between all the contestents not a subset as it is at the moment. |
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== Properties == |
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I would agree that the '''pre-Barbarossa''' operations in the Balkans including '''Operation Margarethe''' are not part of the Eastern Front but that the 1942-44 partisan war is. Or are you saying that none of the partisan operations which took place behind the Eastern front should be mentioned in an article on the Eastern Front? [[User:Philip Baird Shearer|Philip Baird Shearer]] 02:18, 30 Sep 2004 (UTC) |
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The constant ''π'' is an [[irrational number]]; that is, it cannot be written as the ratio of two [[integer]]s. This was proven in [[1761]] by [[Johann Heinrich Lambert]]. |
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Furthermore, ''π'' is also [[transcendental number|transcendental]], as was proven by [[Ferdinand von Lindemann]] in [[1882]]. This means that there is no [[polynomial]] with [[rational number|rational]] coefficients of which ''π'' is a root. An important consequence of the transcendence of ''π'' is the fact that it is not [[constructible number|constructible]]. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to [[squaring the circle|square the circle]]: that is, it is impossible to construct, using [[compass and straightedge]] alone, a square whose area is equal to the area of a given circle. |
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::No, the partisan warfare both in Poland and the USSR could be treated as part of the Eastern Front (although technically speaking most of these actions took place far away from the front itself, that's what the partisan warfare is all about..). However, neither the Balkans 1941 nor Poland 1939 seem like a part of Eastern Front to me. [[User:Halibutt|[[User:Halibutt|Halibu]][[User Talk:Halibutt|tt]]]] 03:55, Sep 30, 2004 (UTC) |
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== History == |
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The understanding of what's the true scope for this and other articles is for natural reasons inviting to confusion. The concepts and perceptions are of course different in different languages. The Great Patriotic War and the Eastern European Front is seen from a certain distance by the English speaking world - that's unavoidable - and it would probably be in vain to try to create distinctions that are not recognized by native English speakers anyway. |
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{{main|History of π}} |
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===Use of the symbol ''π''=== |
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My proposal is that an article on the ''front'' concentrates on the actual front. Other events connected to the Great Patriotic War can be covered by articles of their own, that can be referred to from many articles — but a brief reference to another article is much less prone to cause discussion than the inclusion or exclusion of the actual account. /[[User:Johan Magnus|Johan Magnus]] 08:39, 30 Sep 2004 (UTC) |
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Often [[William Jones (mathematician)|William Jones]]' book ''A New Introduction to Mathematics'' from [[1706]] is cited as the first text where the [[Pi (letter)|Greek letter ''π'']] was used for this constant, but this notation became particularly popular after [[Leonhard Euler]] adopted it some years later, (''[[cf]]''. [[History of π|History of ''π'']]). |
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===Early approximations=== |
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==Additional reason why the Nazis lost that should be mentioned== |
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::''Main article: [[History of numerical approximations of π|History of numerical approximations of ''π'']].'' |
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* An important reasons why the Nazis lost was the unnecessary alienation of the civilian population that was hostile to the Soviets by German arrogance and Nazi racial policy, treating the Slavic popuation as [[Untermensch]]en. The Nazis could have profited from the civilian hostilty to communism and Ukranian nationalism. (See also [[Lebensraum]], [[Commissar_order]], [[Einsatzgruppen]], and [http://www.ess.uwe.ac.uk/genocide/USSR5.htm Fueher decree (do not read , top secret)]). The Soviet POWs were not sufficiently feeded, which the Soviet soldiers got to know in the course of time. In other words, the Germans left the Soviets no choice. Hitler objected to slavic [[Division_%28military%29|division]]s because of his opinions on the race of slavic people until the very last moment, see [[Vlasov army]]. [[User:Andries|Andries]] 12:13, 23 Oct 2004 (UTC) |
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The value of ''π'' has been known in some form since antiquity. As early as the 19th century BC, |
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== Historiography == |
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[[Babylonian mathematics|Babylonian mathematicians]] were using ''π'' = 25/8, which is within 0.5% of the true value. |
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The [[Egyptian mathematics|Egyptian]] scribe [[Ahmes]] wrote the oldest known text to give an approximate value for ''π'', citing a [[Middle Kingdom of Egypt|Middle Kingdom]] [[papyrus]], corresponding to a value of 256 divided by 81 or 3.160. |
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I think that the section Historiography needs attention. |
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It is sometimes claimed that the [[Bible]] states that ''π'' = 3, based on a passage in [[1 Kings]] 7:23 giving measurements for a round basin as having a 10 [[cubit]] diameter and a 30 cubit circumference. [[Rabbi Nehemiah]] explained this by the diameter being from outside to outside while the circumference was the ''inner'' brim; but it may suffice that the measurements are given in round numbers. Also, the basin may not have been exactly circular. |
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:''As V-E Day came, Allied forces in Western Europe [not including Italy] consisted of 4 ½ million men, including 9 armies (5 of them American—one of which, the Fifteenth, saw action only at the last), 23 corps, 91 divisions (61 of them American), 6 tactical air commands (4 American), and 2 strategic air forces (1 American). The Allies had 28,000 combat aircraft, of which 14,845 were American, and they had brought into Western Europe more than 970,000 vehicles and 18 million tons of supplies. At the same time they were achieving final victory in Italy with 18 divisions (7 of them American).'' [http://www.army.mil/cmh-pg/books/AMH/AMH-22.htm] |
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[[Image:Archimedes pi.png|thumb|300px|Principle of Archimedes' method to approximate ''π''.]] |
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--[[User:Philip Baird Shearer|Philip Baird Shearer]] 13:58, 5 Nov 2004 (UTC) |
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[[Archimedes|Archimedes of Syracuse]] discovered, by considering the perimeters of 96-sided polygons inscribing a circle and inscribed by it, that ''π'' is between 223/71 and 22/7. The average of these two values is roughly 3.1419. |
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== Proposed background section == |
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The [[Chinese mathematics|Chinese mathematician]] [[Liu Hui]] computed ''π'' to 3.141014 (good to three decimal places) in AD [[263]] and suggested that 3.14 was a good approximation. |
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The following was contributed by [http://en.wikipedia.org/w/wiki.phtml?title=Special:Contributions&target=161.142.96.218 161.142.96.218]. I think the proposal would need considerable editing to fit into the article, and I do not have the required time right now. Reading the contribution, I wonder if it maybe was rather thought of as an introduction to the article on [[Operation Barbarossa]]. In any case, the text would need serious work with regard to [[Wikipedia:Neutral point of view]]. |
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The [[Indian mathematics|Indian mathematician]] and astronomer [[Aryabhata]] in the [[5th century]] gave the approximation ''π'' = 62832/20000 = 3.1416, correct when rounded off to four decimal places. |
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:''The decision to go eastwards had been made partly because of Britain's surprisingly stubborn resistance in the Western theatre, and partly because the Nazi government harboured (credible) suspicions of the Kremlin's ominous military actions since 1939. Hitler had long considered invading the U.S.S.R., even before he had launched his concentrated campaign against Britain. He had a morbid fear and rabid hatred of Communism and vowed to quash its primary champion once and for all.'' |
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The Chinese mathematician and astronomer [[Zu Chongzhi]] computed ''π'' to be between 3.1415926 and 3.1415927 and gave two approximations of ''π'', 355/113 and 22/7, in the [[5th century]]. |
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:''Furthermore, when the Germans were busy overrunning France in June 1940, Stalin had already seized the opportunity to gobble up the Baltic States (old German territory which Hitler had hoped to re-annex into his Reich). Later on, again without notifying Hitler, the Soviet government made demands on Romania to hand over Bessarabia. This brought the Soviets closer to the precious oilfields vital to feed the German war machine.'' |
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The Indian mathematician and astronomer [[Madhava of Sangamagrama]] in the [[14th century]] computed the value of ''π'' after transforming the [[power series]] expansion of ''π'' /4 into the form |
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:''Hitler waged war on the Soviet Union against the grave misgivings of his generals, but his ego refused to listen. When the High Command argued that it would mean a two-front war, he retorted that they could not risk defeating Britain first as long as Soviet Russia remained a threat at the "back door".'' |
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:''π'' = √12 (1 - 1/(33) + 1/(532) - 1/(733) + ... |
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:''Thus the grand scheme was carried out on an early June morning in 1941. The two-front war, which Hitler himself originally condemned, but later brought upon himself, had begun — with fatal consequences for his Thousand-Year Reich.'' |
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--[[User:Ruhrjung|Ruhrjung]] 08:26, 2004 Nov 8 (UTC) |
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and using the first 21 terms of this series to compute a rational approximation of ''π'' correct to 11 decimal places as 3.14159265359. By adding a remainder term to the original power series of ''π'' /4, he was able to compute ''π'' to an accuracy of 13 decimal places. |
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* The decision analysis is wrong. Britian's resisance was not a reason for going East. |
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* "Nazi government harboured (credible) suspicions of the Kremlin's ominous military actions since 1939" LOL |
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* The "gobble up the Baltic States" was agreed in treaty with the Germans! |
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* Don't know about the Rumania bit. |
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* 2 fronts. Did he say that? |
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* Last paragraph would do without the first sentence. [[User:Philip Baird Shearer|Philip Baird Shearer]] 18:17, 8 Nov 2004 (UTC) |
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The [[Persian Empire|Persian]] astronomer [[Ghyath ad-din Jamshid Kashani]] (1350-1439) correctly computed ''π'' to 9 digits in the base of 60, which is equivalent to 16 decimal digits as: |
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== Anon proposals == |
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:2''π'' = 6.2831853071795865 |
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I moved the following from my talk page, I believe it belongs here. [[User:Halibutt|[[User:Halibutt|Halibu]][[User Talk:Halibutt|tt]]]] |
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By [[1610]], the German mathematician [[Ludolph van Ceulen]] had finished computing the first 35 decimal places of ''π''. It is said that he was so proud of this accomplishment that he had them inscribed on his [[tomb stone|tombstone]]. |
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::No, how could I forget Poland. I don't think I did, but the Polish campaign was hardly part of the same war as what's covered by the article on the Eastern Front, i.e. what the Russians call the Great Patriotic War. |
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In [[1789]], the Slovene mathematician [[Jurij Vega]] improved [[John Machin]]'s formula from [[1706]] and calculated the first 140 decimal places for ''π'' of which the first 126 were correct [http://www.southernct.edu/~sandifer/Ed/History/Preprints/Talks/Jurij%20Vega/Vega%20math%20script.pdf] and held the world record for 52 years until [[1841]], when [[William Rutherford]] calculated 208 decimal places of which the first 152 were correct. |
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::But wouldn't it be fair to tell that the Soviet Union actually didn't succeed in their attempted invasion of Finland? After all, they managed to take eastern Poland and the Baltic republics, but only some tenth of Finland, which was far less then they had aspired to. /M.L. |
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The English amateur mathematician [[William Shanks]], a man of independent means, spent over 20 years calculating ''π'' to 707 decimal places (accomplished in [[1873]]). In [[1944]], D. F. Ferguson found that Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were fallacious. By 1947, Ferguson had recalculated pi to 808 decimal places (with the aid of a mechanical desk calculator). |
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:That's the same confusion I had with this article. Currently it's about the Eastern Front of World War II. As such, Polish Defence War of 1939 was a part of the same war, but not the same front. If we renamed the article to [[Great Patriotic War]] that would be much easier, but so far saying that the war started in 1941 even if several sentences above it is stated that "the war" refers to [[World War II]], would be misleading. |
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== Numerical approximations == |
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: As to whether Soviet Union succeed in Finland or not is a matter of dispute, I believe the completely neutral statement I proposed is much better. Especially that from other perspective, the Soviet Union managed to crush all Finnish resistance by March 1940 and forced the nation into submission - which is equally true. I'd say let's stick to simple, NPOV sentences. Also, what they aspired to is not that clear either. [[User:Halibutt|[[User:Halibutt|Halibu]][[User Talk:Halibutt|tt]]]] 08:36, Nov 9, 2004 (UTC) |
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{{main|History of numerical approximations of π}} |
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Due to the transcendental nature of ''π'', there are no closed form expressions for the number in terms of algebraic numbers and functions. Roughly speaking, this means that any formula which uses simple math operations to calculate ''π'' must go on forever. This is why formulæ for calculating ''π'' are often written with a "..." to indicate that in order to reach ''π'' exactly, an infinite number of additional terms would have to follow the terms given. |
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Consequently, numerical calculations must use [[approximation]]s of ''π''. For many purposes, 3.14 or [[Proof that 22 over 7 exceeds π|22/7]] is close enough, although engineers often use 3.1416 (5 [[significant figures]]) or 3.14159 (6 significant figures) for more accuracy. The approximations 22/7 and 355/113, with 3 and 7 significant figures respectively, are obtained from the simple [[continued fraction]] expansion of ''π''. The approximation 355/113 (3.1415929…) is the best one that may be expressed with a three-digit numerator and denominator. |
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[[Image:Map of Finnish areas ceded to Soviet Union 1940.png|right|frame|most of what was ceded in 1940 was lost at the negotiation table, not in battle]] |
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::Halibutt, you must have been reading Soviet history books. No, it's established beyond doubt that the Soviet Union attempted to conquer all of Finland (including [[Åland]] of course), but (in accordance with the Molotov-Ribbentrop Pact) halt at the Swedish border — at least temporarily. I consider attempts to falsify the history on this point as pro-Soviet propaganda, that there is absolutely no excuse for now, after the collapse of Communism in Europe. |
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The earliest numerical approximation of ''π'' is almost certainly the value {{num|3}}. In cases where little precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the fact that it is the ratio of the [[perimeter]] of an [[inscribe]]d [[regular polygon|regular]] [[hexagon]] to the [[diameter]] of the [[circle]]. |
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::It's not a "neutral" statement to give the impression that an invasion was somehow realized when the truth is that the operation was aborted after over four months of heavy fighting. And it makes a brutally confusing impression when the article somewhat later can't avoid to mention that the Finns fought back a second time, in July 1941. The Soviet Union had most definitely not ''"managed to crush all Finnish resistance",'' a wording that would seriously offend not so few Finns. [[Taipale]] was still in Finnish hands, some miles from the old border, and [[Viipuri]] was approached, but far from conquered, by the Red Army. |
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All further improvements to the above mentioned "historical" approximations were done with the help of [[computer]]s. |
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::I've expressed support for renaming this article before. /[[User:Tuomas|Tuomas]] 10:29, 9 Nov 2004 (UTC) |
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== Formulæ == |
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:::You got my words wrong, Tuomas. I had no intention to offend any of the heroic defenders of Finland. I personally also resent that the French general staff did not approve to send the Polish Carpathian Rifle Brigade to aid the Finns and sent it to Narvik instead. This however does not change the fact that by the beginning of spring the war was lost for Finland. No serious help arrived, much of the defence lines along the borders was lost and there wasn't much chance for continuation of organised resistance. The Finns would most surely conduct armed operations for months if not years, but the battle-hardened Finnish troops would not be able to sustain technical and numerical supperiority of the enemy for much longer. That is why Mannerheim asked for peace negotiations. |
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===Geometry=== |
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The constant ''π'' appears in many formulæ in [[geometry]] involving [[circle]]s and [[sphere]]s. |
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{| class="wikitable" |
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:::The Winter War was not won by the Finns since it couldn't have been won. The heroic defence against overwhelming odds can be considered both a moral and tactical victory. The Moscow Peace was also a great success of Finnish diplomacy (contrary to what might seem from the harsh peace terms). However, on a larger scale claiming that the Finns won the war is an overstatement. It was a draw at best, if not a defeat. |
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!Geometrical shape |
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!Formula |
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|- |
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|[[Circumference]] of circle of [[radius]] ''r'' and [[diameter]] ''d'' |
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|<math>C = 2 \pi r = \pi d \,\!</math> |
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|- |
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|[[area (geometry)|Area]] of circle of radius ''r'' |
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|<math>A = \pi r^2 = \frac{1}{4} \pi d^2 \,\!</math> |
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|- |
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|Area of [[ellipse]] with semiaxes ''a'' and ''b'' |
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|<math>A = \pi a b \,\!</math> |
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|- |
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|[[Volume]] of sphere of radius ''r'' and diameter ''d'' |
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|<math>V = \frac{4}{3} \pi r^3 = \frac{1}{6} \pi d^3 \,\!</math> |
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|- |
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|[[Surface area]] of sphere of radius ''r'' and diameter ''d'' |
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|<math>A = 4 \pi r^2 = \pi d^2 \,\!</math> |
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|- |
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|Volume of [[cylinder (geometry)|cylinder]] of height ''h'' and radius ''r'' |
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|<math>V = \pi r^2 h \,\!</math> |
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|- |
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|Surface area of cylinder of height ''h'' and radius ''r'' |
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|<math>A = 2 (\pi r^2) + ( 2 \pi r)h = 2 \pi r (r+h) \,\!</math> |
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|- |
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|Volume of [[cone (geometry)|cone]] of height ''h'' and radius ''r'' |
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|<math>V = \frac{1}{3} \pi r^2 h \,\!</math> |
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|- |
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|Surface area of cone of height ''h'' and radius ''r'' |
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|<math>A = \pi r \sqrt{r^2 + h^2} + \pi r^2 = \pi r (r + \sqrt{r^2 + h^2}) \,\!</math> |
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|} |
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(All of these are a consequence of the first one, as the area of a circle can be written as |
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:::Anyway, back to the topic. The invasion did not fail since Soviet Union effectively did invade Finnish land. The war was fought entirely on Finnish soil and in the effect it was Finland who lost territory. Perhaps the Soviets wished to gain more from this conflict and that's what I personally believe. However, I have yet to see a document proving that the Red Army did not successfuly invade Finland and was repulsed on the borders. |
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''A'' = ∫(2''πr'') d''r'' ("sum of [[annulus (mathematics)|annuli]] of infinitesimal width"), and others concern a surface or [[solid of revolution]].) |
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Also, the [[angle]] measure of 180° ([[Degree (angle)|degrees]]) is equal to ''π'' [[radian]]s. |
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:::Finally, stating that there was a Soviet invasion of Finland is not more POV than stating that there was a failed invasion of Finland. I did read Soviet history books too (fascinating lecture, really), but I can think for myself. [[User:Halibutt|[[User:Halibutt|Halibu]][[User Talk:Halibutt|tt]]]] 11:35, Nov 9, 2004 (UTC) |
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===Analysis=== |
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::::Successful invasions there are many of in history. [[Iraq]] and [[Poland]] comes easily to mind. That doesn't say that the following occupation is a similar success. But in the case of Finland, the Soviet union had to (temporarily!) give up its plans for an invasion when they after four months had conquered about five percents of the territory they aimed at. They did so, well knowing that the occupation would be as problematic as the fightings had been. Equally much (approximately) they gained in the peace negotiations, and some more after the peace, but you can not call this a realized invasion of Finland. It was an ''attempted and aborted'' (if you find the word ''failed'' too judgemental) invasion. Already in November 1940, Stalin had to be halted by Hitler not to finish the invasion. By then, noone else could. /[[User:Tuomas|Tuomas]] 12:51, 9 Nov 2004 (UTC) |
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Many formulæ in [[Mathematical analysis|analysis]] contain ''π'', including [[infinite series]] (and [[infinite product]]) representations, [[integral]]s, and so-called [[List of mathematical functions|special functions]]. |
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*The area of the [[unit disc]]: |
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When the Soviet Union invaded the Baltic states etc it was, attempting to reinstate the old Russian Empire Boarders which the Soviets had been forced to conceded when they signed [[Brest-Litovsk Peace Treaty]] with the Germans when Russia finished participating in WWI. In the [[aftermath of World War I]] if the Tsar had still been on the throne a pound for a penny Russia would have had all of its pre-war territory returned to it whether or not Finland had dragged back kicking and screaming. This did not happened because the victors of WWI detested and feared the Soviet regime. The [[Molotov-Ribbentrop Pact#Franco-British negotiations with the Soviet Union|Franco-British negotiations with the Soviet Union]] in the late 1930s stalled in part because France and Britain would not recognise the right of the Soviet Union to interfere against "a change of policy favourable to an aggressor" in old Russian Empire areas given up in 1917, which included Finland. When invading Finland why would the Soviets not have taken the whole country if the Finns had not put up such a good resistance? I think that Stalin would have annexed the country and not put in a nominally in independent government, (like he did in Eastern Europe in 1945,) because as far as Stalin and many Soviets were concerned, Finland was in the same category as Estonia etc. To say that the Soviets won and the Finns lost is not true. The cost for the Soviets was not worth the price they would have to pay for the rest of the country, so they setteled for more than the Finns were willing to give, which is why the Finns went for round two. |
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::<math>2\int_{-1}^1 \sqrt{1-x^2}\,dx = \pi</math> |
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*Half the [[circumference]] of the [[unit circle]]: |
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I do not think that the the [[Winter War]] should be included in the Eastern Front because it is not normally considered to be part of WWII as it is not clear on who's side the two warring parties were fighting on (apart from their own). |
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::<math>\int_{-1}^1\frac{dx}{\sqrt{1-x^2}} = \pi</math> |
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:Maybe they didn't fight on anyone's side. Well, a Finnish view would be that the Finns fought on the side of the West, defending Scandinavia, as has been the Finns' role since Christianization some 900 years ago. :-)) /[[User:Tuomas|Tuomas]] 12:51, 9 Nov 2004 (UTC) |
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But I would include the [[Continuation War]] because as Churchill said "If Hitler invaded hell I would make at least a favourable reference to the devil in the House of Commons." The Finns had thrown in their lot with Hitler and could not expect any further moral or material support from the Western Allies. |
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:The Finns were in dire need of protection against a 50 times as big neighboring aggressor; and none except the Germans had help to offer. It was, according to contemporary views, the lesser of two evils. A pity that the rest of the West joined the Communists. But of course it has to be mentioned as a front of the Great Patriotic War. /[[User:Tuomas|Tuomas]] 12:51, 9 Nov 2004 (UTC) |
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*[[François Viète]], 1593 ([[Viète formula|proof]]): |
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I agree that this article should be renamed to "Great Patriotic War", "Russo-German War", "Nazi-Soviet War" or something similar. I think War (and not theatre or campaign) is an appropriate as part of the name because just as the [[Peninsular War]] was part of the [[Napoleonic Wars]] it was by and large self contained and different from the other campaigns and fronts. This front article should be a brief overview (smilar to the current time line) of all actions on the Eastern Front from '''1939''' until 1945 with links into detailed articles of all the wars and campaigns on the Eastern Front. If this was done then it would be possible to mention the Winter War in a paragraph giving context, between the Polish campaign of 1939 and the start of the Great Patriotic War. It would allow a starting paragraph about the German 2 Front dilemma, and the Western problems with Stalin's demands for the same thing (part of the reason for the strategic bombing campaigns which tied down 100 of thousands of German troops and thousands of guns). At the moment the way this article is structured there is no easy way to include this material because it is an article about the Great Patriotic War not the Eastern Front. --[[User:Philip Baird Shearer|Philip Baird Shearer]] 12:13, 9 Nov 2004 (UTC) |
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::<math>\frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdot \cdots = \frac2\pi</math> |
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*[[Gottfried Leibniz|Leibniz]]' formula ([[Leibniz formula for pi|proof]]): |
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:I do not agree that the Winter War was fought at something that contemporaries considered an "Eastern Front". News coverage from that time (in English) didn't use that term. I don't know about the Polish campaign, but I wouldn't think so in that case either. The context of the Polish Campaign and the Winter War is rather the Molotov-Ribbentrop Pact. /[[User:Tuomas|Tuomas]] 12:51, 9 Nov 2004 (UTC) |
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::<math>\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}</math> |
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*[[John Wallis|Wallis]] product, 1655 ([[Wallis product|proof]]): |
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::Tuomas, I believe the conflict is because of the words used. Some native speaker should confirm this, but as far as I can tell the verb "to invade" does not mean "to invade and conquer" or "to invade successfuly". If so, then the invasion of Finland was factual, not attempted. [[User:Halibutt|[[User:Halibutt|Halibu]][[User Talk:Halibutt|tt]]]] 15:23, Nov 9, 2004 (UTC) |
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::<math> \prod_{n=1}^{\infty} \left ( \frac{n+1}{n} \right )^{(-1)^{n-1}} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots = \frac{\pi}{2} </math> |
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*Faster product (see Sondow, 2005 and [http://home.earthlink.net/~jsondow/ Sondow web page]) |
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:::Or maybe that the invasion of [[Finnish Karelia|(Finnish) Karelia]] was factual? ...however, their aim wasn't Karelia, it wasn't even [[Old Finland]], it was all of Finland - and they were delayed. --[[User:Johan Magnus|Johan Magnus]] 15:51, 9 Nov 2004 (UTC) |
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::<math> \left ( \frac{2}{1} \right )^{1/2} \left (\frac{2^2}{1 \cdot 3} \right )^{1/4} \left (\frac{2^3 \cdot 4}{1 \cdot 3^3} \right )^{1/8} \left (\frac{2^4 \cdot 4^4}{1 \cdot 3^6 \cdot 5} \right )^{1/16} \cdots = \frac{\pi}{2} </math> |
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:where the ''n''th factor is the 2<sup>''n''</sup>th root of the product |
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::<math>\prod_{k=0}^n (k+1)^{(-1)^{k+1}{n \choose k}}.</math> |
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*Symmetric formula (see Sondow, 1997) |
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"attempted Soviet invasion" implies that the invasion did not take place. "attempted Soviet occupation" would be better because it implied an invasion took place and its failure is recognised. |
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::<math> \frac {\displaystyle \prod_{n=1}^{\infty} \left (1 + \frac{1}{4n^2-1} \right )}{\displaystyle\sum_{n=1}^{\infty} \frac {1}{4n^2-1}} = \frac {\displaystyle\left (1 + \frac{1}{3} \right ) \left (1 + \frac{1}{15} \right ) \left (1 + \frac{1}{35} \right ) \cdots} {\displaystyle \frac{1}{3} + \frac{1}{15} + \frac{1}{35} + \cdots} = \pi </math> |
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*[[Bailey-Borwein-Plouffe formula|Bailey-Borwein-Plouffe]] algorithm (See Bailey, 1997 and [http://www.nersc.gov/~dhbailey/ Bailey web page]) |
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:''"...unfinished invasion..."?'' <tt>;-))</tt> |
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::<math>\sum_{k=0}^\infty\frac{1}{16^k}\left(\frac {4}{8k+1} - \frac {2}{8k+4} - \frac {1}{8k+5} - \frac {1}{8k+6}\right) = \pi</math> |
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:Do you '''really''' see ''"invasion"'' as synonym to ''border incident'' and ''border transgression?'' Also native English speakers must differentiate between an attack that is met at the border and fought at the border, and one that reaches beyond the border region. ...then it's only a question of how broad a border region is considered to be. In the case of the [[Karelian Isthmus]] the main defence line did nothing like crumble, although it was ultimately broken; but that was after two and a half months. The day after, Soviet peace conditions were presented, a week later, all hopes for regular troops from Sweden were lost, and only enticing French delayed peace negotiations for ten more days. During this time, however, secondary defense lines were held. The invaders reached the first town at the last day of the war. To me, the invasion was avoided by the border defense, but I'm no native speaker of English, of course! |
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:We must, naturally, follow the lead of wiser people. In this case that of native speakers. Please excuse my scepsis. --[[User:Johan Magnus|Johan Magnus]] 17:30, 9 Nov 2004 (UTC) |
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* [[Pafnuty Chebyshev|Chebyshev]] series [http://www.jstor.org/journals/08916837.html Y. Luke, Math. Tabl. Aids Comp. 11 (1957) 16] |
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::<math>\sum_{k=0}^\infty\frac{(-1)^k(\sqrt{2}-1)^{2k+1}}{2k+1} = \frac{\pi}{8}.</math> |
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::<math>\sum_{k=0}^\infty\frac{(-1)^k(2-\sqrt{3})^{2k+1}}{2k+1}=\frac{\pi}{12}.</math> |
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*An [[integral]] formula from [[calculus]] (see also [[Error function]] and [[Normal distribution]]): |
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It does not really matter what the contempories called it, because the lable given to historic events often become the norm after the events. For example no one who fought in the [[War of the Roses]] called them that, No one who fought in the Great War called it "World War '''One'''" because they did not expect there to be a second one! It is quite a common name to use eg "[http://titles.cambridge.org/catalogue.asp?isbn=0521529387 Britain, Poland and the Eastern Front, 1939]", by Anita J. Prazmowska, February 2004, ISBN 0521529387. I also suspect that it was a name used during the four weeks it existed. Certainly in the Official German relply to the British decleration of war they talk about the Eastern Frontier and "if soldier falls at the '''front'''". --[[User:Philip Baird Shearer|Philip Baird Shearer]] 16:01, 9 Nov 2004 (UTC) |
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::<math>\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}</math> |
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*[[Basel problem]], first solved by [[Leonhard Euler|Euler]] (see also [[Riemann zeta function]]): |
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:Copied from my talk page: ([[User:Halibutt|[[User:Halibutt|Halibu]][[User Talk:Halibutt|tt]]]]) |
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::<math>\zeta(2)= \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6}</math> |
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::I think you must remember that Finns, like Poles, have lived in the shadow of the Soviet power and distorted Allied propaganda. One must also consider that Finns rightfully feel "proven right by history". In the 1930s, it was common to regard Finns, as Poles, as warmonging exaggerated Nationalists, aiming at expanding their country at the expense of their peaceful neighbours. And Soviet propaganda actually had some success to explain the necessity of a border adjustment, that in retrospect only would have served the purpose to make Finland easier to penetrate. Having been told that Stalin was a nice man who surely wouldn't attack unprovoked, and who surely didn't attempt to conquer Finland, distrustful Finns feel that now, after access to Soviet archives was granted, if not before, the rest of the world finally must understand the danger Finland was in. You are pushing wrong buttons, by hinting at Stalin maybe was a nice man who didn't intend to take more than the homes of 412,000 Karelians, Finland's industrial heart and power source, and cause some "minor" disturbance in Finland's food production, since the thesis of "limited" aims (much more limited than so!) was purported by Soviet propaganda both before and after the war. --[[User:Johan Magnus|Johan Magnus]] 16:23, 9 Nov 2004 (UTC) |
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::<math>\zeta(4)= \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \cdots = \frac{\pi^4}{90}</math> |
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::and generally, <math>\zeta(2n)</math> is a rational multiple of <math>\pi^{2n}</math> for positive integer n |
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*[[Gamma function]] evaluated at 1/2: |
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:I'm afraid both you and Tuomas see in my words more than I actually wanted to say. I'm not suggesting that Stalin might've been a nice guy; I couldn't be further from such suggestions. It was not my intention to comment on the nature of the conflict, its reasons and outcomes either. I was merely nit-picking at the phrase used in one of the earlier versions of the article and pointing at the fact that it might actually be misleading. Whether Stalin wanted to take 5% of Finland, 50% of it or 100% is completely irrelevant here. What is important is that he invaded Finland. He did succeed in invading the country but did not succeed in occupying it or forcing it into submission. That's what I wanted to say and nothing more. Sorry for the misunderstanding. [[User:Halibutt|[[User:Halibutt|Halibu]][[User Talk:Halibutt|tt]]]] 16:41, Nov 9, 2004 (UTC) |
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::<math>\Gamma\left({1 \over 2}\right)=\sqrt{\pi}</math> |
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*[[Stirling's approximation]]: |
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::No, no-one believes you to be a follower of Stalin. ...only that you might be too much influenced by similar propaganda. ...and, it makes a considerable difference to me, if the Red Army achieved 5%, 50% or 100% of its objectives. In the first case, which is closest to the truth, I think "failure" is close to an euphemism. :-) --[[User:Johan Magnus|Johan Magnus]] 17:30, 9 Nov 2004 (UTC) |
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::<math>n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n</math> |
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*[[Euler's identity]] (called by [[Richard Feynman]] "the most remarkable formula in mathematics"): |
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::Let's take some other similar incidents for comparison then. The Germans seized approximately 5 to 15% of Soviet territory during the [[Operation Barbarossa]]. Was it an [[invasion]] then or not? It's not a question of your beliefs, it's a question of logic. [[User:Halibutt|[[User:Halibutt|Halibu]][[User Talk:Halibutt|tt]]]] 19:01, Nov 9, 2004 (UTC) |
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::<math>e^{i \pi} + 1 = 0\;</math> |
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*A property of [[Euler's totient function]] (see also [[Farey sequence]]): |
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==Straw Poll on name of the article== |
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::<math>\sum_{k=1}^{n} \phi (k) \sim \frac{3n^2}{\pi^2}</math> |
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*An application of the [[residue theorem]] |
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Since there seems to be some discrepancy about the name, should we hold a vote to determine if it should be changed, and what it could be changed to? [[User:Oberiko|Oberiko]] 14:46, 9 Nov 2004 (UTC) |
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::<math>\oint\frac{dz}{z}=2\pi i ,</math> |
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:where the path of integration is a closed curve around the origin, traversed in the standard anticlockwise direction. |
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===Continued fractions=== |
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<center>Please sign with <nowiki>#~~~~</nowiki></center> |
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Next to its [[continued fraction|simple continued-fraction]] representation [3; 7, 15, 1, 292, 1, 1, …], which displays no discernible pattern, ''π'' has many [[generalized continued fraction|generalized continued-fraction]] representations that are generated by a simple rule, including: |
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<center>Feel free to add more naming proposals</center> |
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<center>Please add comments in the comments section not in the Votes section.</center> |
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:<math> \frac{4}{\pi} = 1 + \cfrac{1}{3 + \cfrac{4}{5 + \cfrac{9}{7 + \cfrac{16}{9 + \cfrac{25}{11 + \cfrac{36}{13 + \cdots}}}}}} </math> |
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===Arguements for change and alternate names=== |
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(Other representations are available at [http://functions.wolfram.com/Constants/Pi/10/ The Wolfram Functions Site].) |
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* If it was renamed then there could be an article linking all the campaigns and wars which took place on the on the Eastern Front (EF) between 1939 and 1945. This could include a link to the Great Patriotic War (GPW) in the first section along the lines of "''For the artical on the main war on the Eastern Front of World War II see [[Great Patriotic War]]''" for those who thing the EF and GPW are the same thing (I suppose that is most Americans, because the US came into the war after the start of the GPW!). This way the GPW could remain clean without including details other events which happened on the Eastern Front before the start of the GPW and during the war like Yugoslavia and the Warsaw uprising. Redirects can exist from alternative names '''German-Russian War''' etc. [[User:Philip Baird Shearer|Philip Baird Shearer]] 16:01, 9 Nov 2004 (UTC) |
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=== |
===Number theory=== |
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Some results from [[number theory]]: |
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#[[User:Halibutt|[[User:Halibutt|Halibu]][[User Talk:Halibutt|tt]]]] 15:14, Nov 9, 2004 (UTC) |
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*The [[probability]] that two [[random]]ly chosen integers are [[coprime]] is 6/''π''<sup>2</sup>. |
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#[[User:Philip Baird Shearer|Philip Baird Shearer]] 16:01, 9 Nov 2004 (UTC) |
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#[[User:Johan Magnus|Johan Magnus]] 16:30, 9 Nov 2004 (UTC) |
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# [[User:Oberiko|Oberiko]] 20:03, 9 Nov 2004 (UTC) |
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*The probability that a randomly chosen integer is [[square-free integer|square-free]] is 6/''π''<sup>2</sup>. |
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*The [[mean|average]] number of ways to write a positive integer as the sum of two [[perfect square]]s (order matters but not sign) is ''π''/4. |
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Here, "probability", "average", and "random" are taken in a limiting sense, e.g. we consider the probability for the set of integers {1, 2, 3,…, ''N''}, and then take the [[limit (mathematics)|limit]] as ''N'' approaches infinity. |
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===Arguements against change / votes to keep as Eastern Front (WWII)=== |
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*The [[Product (mathematics)|product]] of (1 − 1/''p''<sup>2</sup>) over the [[prime number|prime]]s, ''p'', is 6/''π''<sup>2</sup>. |
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The theory of elliptic curves and [[complex multiplication]] derives the approximation |
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===Comments=== |
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: <math>\pi \approx {\ln(640320^3+744)\over\sqrt{163}}</math> |
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see:[[Dispute_resolution#Conduct a survey]] "''Note that informal [[straw poll]]s can be held at any time if there are enough participants in the discussion, but publicizing the survey can get more of the community involved and increase the weight given to the results.''" |
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which is valid to about 30 digits. |
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===Dynamical systems and ergodic theory=== |
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:What are the justifications for the options in this poll? [[User:Gdr|Gdr]] 19:02, 2004 Nov 9 (UTC) |
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Consider the [[recurrence relation]] |
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:<math>x_{i+1} = 4 x_i (1 - x_i) \,</math> |
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Then for [[almost everywhere|almost every]] initial value ''x''<sub>0</sub> in the [[unit interval]] [0,1], |
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:<math> \lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^{n} \sqrt{x_i} = \frac{2}{\pi} </math> |
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This recurrence relation is the [[logistic map]] with parameter ''r'' = 4, known from [[dynamical system]]s theory. See also: [[ergodic theory]]. |
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===Physics=== |
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I've never really cared for the Soviet-centered name "Great Patriotic War", but it is by far the most common and well known. Also, we need to distinguish between this event (Which widely known as the war against Germany and Finland) and the smaller Soviet Campaign in East Asia. [[User:Oberiko|Oberiko]] 20:03, 9 Nov 2004 (UTC) |
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The number ''π'' appears routinely in equations describing fundamental principles of the universe, due in no small part to its relationship to the nature of the circle and, correspondingly, spherical coordinate systems. |
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*The [[cosmological constant]]: |
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:So what are the pros and cons of the two options? [[User:Gdr|Gdr]] 20:24, 2004 Nov 9 (UTC) |
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:<math>\Lambda = {{8\pi G} \over {3c^2}} \rho</math> |
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*[[Uncertainty principle|Heisenberg's uncertainty principle]]: |
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:<math> \Delta x \Delta p \ge \frac{h}{4\pi} </math> |
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*[[Einstein's field equation]] of [[general relativity]]: |
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:<math> R_{ik} - {g_{ik} R \over 2} + \Lambda g_{ik} = {8 \pi G \over c^4} T_{ik} </math> |
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*[[Coulomb's law]] for the [[electric force]]: |
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:<math> F = \frac{\left|q_1q_2\right|}{4 \pi \epsilon_0 r^2}</math> |
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*[[Permeability (electromagnetism)|Magnetic permeability of free space]]: |
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:<math> \mu_0 = 4 \pi \cdot 10^{-7}\,\mathrm{N/A^2}\,</math> |
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===Probability and statistics=== |
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::Good point, I'll add a sub-category for that. [[User:Oberiko|Oberiko]] 20:35, 9 Nov 2004 (UTC) |
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In [[probability]] and [[statistics]], there are many [[probability distribution|distributions]] whose formulæ contain ''π'', including: |
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*[[probability density function]] (pdf) for the [[normal distribution]] with [[mean]] μ and [[standard deviation]] σ: |
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:<math>f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-(x-\mu )^2/(2\sigma^2)}</math> |
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:::The "arguments" section above doesn't explain why a change of name is necessary and what the relative points of the two proposals are. In particular, why is "Eastern Front" bad? Does "Great Patriotic War" imply any POV on the conflict? (Does "Eastern Front"?) Will there be edit wars over the name? What is the war called in Germany? I have read the discussion above and I am still in the dark on these points. [[User:Gdr|Gdr]] 22:10, 2004 Nov 9 (UTC) |
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*pdf for the (standard) [[Cauchy distribution]]: |
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:<math>f(x) = \frac{1}{\pi (1 + x^2)}</math> |
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Note that since <math>\int_{-\infty}^{\infty} f(x)\,dx = 1</math>, for any pdf ''f''(''x''), the above formulæ can be used to produce other integral formulae for ''π''. |
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A semi-interesting empirical approximation of ''π'' is based on [[Buffon's needle]] problem. Consider dropping a needle of length ''L'' repeatedly on a surface containing parallel lines drawn ''S'' units apart (with ''S'' > ''L''). If the needle is dropped ''n'' times and ''x'' of those times it comes to rest crossing a line (''x'' > 0), then one may approximate ''π'' using: |
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:<math>\pi \approx \frac{2nL}{xS}</math> |
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[As a practical matter, this approximation is poor and [[rate of convergence|converges]] very slowly.] |
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Another approximation of ''π'' is to [http://www.statisticool.com/pi.htm throw points randomly] into a quarter of a circle with radius 1 that is inscribed in a square of length 1. ''π'', the area of a unit circle, is then approximated as 4*(points in the quarter circle) / (total points). |
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===Efficient methods=== |
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In the early years of the computer, the first expansion of ''π'' to 100,000 decimal places was computed by Maryland mathematician Dr. [[Daniel Shanks]] and his team at the United States Naval Research Laboratory (N.R.L.) in 1961. Dr. Shanks's son [[Oliver Shanks]], also a mathematician, states that there is no family connection to [[William Shanks]], and in fact, his family's roots are in Central Europe.{{citation needed}} |
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Daniel Shanks and his team used two different power series for calculating the digital of ''π''. For one it was known that any error would produce a value slightly high, and for the other, it was known that any error would produce a value slightly low. And hence, as long as the two series produced the same digits, there was a very high confidence that they were correct. The first 100,000 digits of ''π'' were published by the [[US Naval Research Laboratory]] |
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None of the formulæ given above can serve as an efficient way of approximating ''π''. For fast calculations, one may use a formula such as [[John Machin|Machin's]]: |
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: <math>\frac{\pi}{4} = 4 \arctan\frac{1}{5} - \arctan\frac{1}{239} </math> |
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together with the [[Taylor series]] expansion of the function [[arctan]](''x''). This formula is most easily verified using [[polar coordinates]] of [[complex number]]s, starting with |
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:<math>(5+i)^4\cdot(-239+i)=-114244-114244i.</math> |
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Formulæ of this kind are known as ''[[Machin-like formula]]e''. |
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Many other expressions for ''π'' were developed and published by the incredibly intuitive Indian mathematician [[Srinivasa Ramanujan]]. He worked with mathematician [[Godfrey Harold Hardy]] in England for a number of years. |
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Extremely long decimal expansions of ''π'' are typically computed with the [[Gauss-Legendre algorithm]] and [[Borwein's algorithm]]; the [[Salamin-Brent algorithm]] which was invented in [[1976]] has also been used. |
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The first one million digits of ''π'' and 1/''π'' are available from [[Project Gutenberg]] (see external links below). |
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The current record (December [[2002]]) by [[Yasumasa Kanada]] of [[Tokyo University]] stands at 1,241,100,000,000 digits, which were computed in September [[2002]] on a 64-node [[Hitachi, Ltd.|Hitachi]] [[supercomputer]] with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits). The following Machin-like formulæ were used for this: |
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:<math> \frac{\pi}{4} = 12 \arctan\frac{1}{49} + 32 \arctan\frac{1}{57} - 5 \arctan\frac{1}{239} + 12 \arctan\frac{1}{110443}</math> |
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:K. Takano ([[1982]]). |
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: <math> \frac{\pi}{4} = 44 \arctan\frac{1}{57} + 7 \arctan\frac{1}{239} - 12 \arctan\frac{1}{682} + 24 \arctan\frac{1}{12943}</math> |
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:F. C. W. Störmer ([[1896]]). |
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These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers. ([[#Open questions|Normality]] of ''π'' will always depend on the infinite string of digits on the end, not on any finite computation.) |
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In [[1997]], [[David H. Bailey]], [[Peter Borwein]] and [[Simon Plouffe]] published a paper (Bailey, 1997) on a new formula for ''π'' as an [[infinite series]]: |
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: <math>\pi = \sum_{k = 0}^{\infty} \frac{1}{16^k} |
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\left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6}\right)</math> |
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This formula permits one to fairly readily compute the ''k''<sup>th</sup> [[Binary numeral system|binary]] or [[hexadecimal]] digit of ''π'', without |
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having to compute the preceding ''k'' − 1 digits. [http://www.nersc.gov/~dhbailey/ Bailey's website] contains the derivation as well as implementations in various [[programming language]]s. The [[PiHex]] project computed 64-bits around the [[quadrillion]]th bit of ''π'' (which turns out to be 0). |
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[[Fabrice Bellard]] claims to have beaten the efficiency record set by Bailey, Borwein, and Plouffe with his formula to calculate binary digits of ''π'' [http://fabrice.bellard.free.fr/pi/pi_bin/pi_bin.html]: |
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:<math>\pi = \frac{1}{2^6} \sum_{n=0}^{\infty} \frac{{(-1)}^n}{2^{10n}} \left( - \frac{2^5}{4n+1} - \frac{1}{4n+3} + \frac{2^8}{10n+1} - \frac{2^6}{10n+3} - \frac{2^2}{10n+5} - \frac{2^2}{10n+7} + \frac{1}{10n+9} \right)</math> |
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Other formulæ that have been used to compute estimates of ''π'' include: |
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:<math> |
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\frac{\pi}{2}= |
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\sum_{k=0}^\infty\frac{k!}{(2k+1)!!}= |
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1+\frac{1}{3}\left(1+\frac{2}{5}\left(1+\frac{3}{7}\left(1+\frac{4}{9}(1+\cdots)\right)\right)\right) |
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</math> |
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:[[Isaac Newton|Newton]]. |
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:<math> \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}} </math> |
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:[[Ramanujan]]. |
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This converges extraordinarily rapidly. Ramanujan's work is the basis for the fastest algorithms used, as of the turn of the millennium, to calculate ''π''. |
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:<math> \frac{1}{\pi} = 12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}} </math> |
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:[[David Chudnovsky (mathematician)|David Chudnovsky]] and [[Gregory Chudnovsky]]. |
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: <math>{\pi} = 20 \arctan\frac{1}{7} + 8 \arctan\frac{3}{79} </math> |
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:[[Euler]]. |
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===Miscellaneous formulæ=== |
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Using [[radix|base]] 60, ''π'' can be approximated to the equivalent of eight significant figures (in base 10) as: |
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:<math> 3 + \frac{8}{60} + \frac{29}{60^2} + \frac{44}{60^3}</math> |
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In addition, the following expressions approximate ''π'' |
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*accurate to 9 digits: |
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::<math>(63/25)(17+15\sqrt 5)/(7+15\sqrt5)</math> |
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*accurate to 3 digits: |
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::<math>\sqrt{2} + \sqrt{3}</math> |
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:[[Karl Popper]] conjectured that [[Plato]] knew this expression; that he believed it to be exactly ''π''; and that this is responsible for some of Plato's confidence in the [[omnicompetence]] of mathematical geometry — and Plato's repeated discussion of special [[right triangle]]s that are either [[isosceles]] or halves of [[equilateral]] triangles. |
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*The [[continued fraction]] representation of ''π'' can be used to generate successively better rational approximations, which start off: 22/7, 333/106, 355/113…. These approximations are the best possible rational approximations of ''π'' relative to the size of their denominators. |
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== Memorising digits == |
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{{main|Piphilology}} |
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Even long before computers have calculated ''π'', memorizing a ''record'' number of digits became an obsession for some people. The current unofficial world record is 83,431 decimal places, and was set by a Japanese mental health counsellor named [[Akira Haraguchi]], who is currently 59 years of age.[http://news.bbc.co.uk/1/hi/world/asia-pacific/4644103.stm] Before Haraguchi accomplished this on July 2, 2005, the world record was 42,195, which was set by Hiroyuki Goto. |
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There are many ways to memorise ''π'', including the use of '''piems''', which are poems that represent ''π'' in a way such that the length of each word (in letters) represents a digit. Here is an example of a piem: ''How I need a drink, alcoholic in nature'' (or: ''of course'')'', after the heavy lectures involving quantum mechanics.'' Notice how the first word has 3 letters, the second word has 1, the third has 4, the fourth has 1, the fifth has 5, and so on. The ''[[Cadaeic Cadenza]]'' contains the first 3834 digits of ''π'' in this manner. Piems are related to the entire field of humorous yet serious study that involves the use of [[mnemonic technique]]s to remember the digits of ''π'', known as [[piphilology]]. See [[:q:English mnemonics#Pi|Pi mnemonics]] for examples. In other languages there are similar methods of memorisation. However, this method proves inefficient for large memorisations of pi. Other methods include remembering "patterns" in the numbers (for instance, the "year" 1971 appears in the first fifty digits of pi). |
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== Open questions == |
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The most pressing open question about ''π'' is whether it is a [[normal number]] -- 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", and that this is true in ''every'' base, not just base 10. Current knowledge on this point is very weak; e.g., it is not even known which of the digits 0,…,9 occur infinitely often in the decimal expansion of ''π''. |
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Bailey and Crandall showed in [[2000]] that the existence of the above mentioned [[Bailey-Borwein-Plouffe formula]] and similar formulæ imply that the normality in base 2 of ''π'' and various other constants can be reduced to a plausible [[conjecture]] of [[chaos theory]]. See Bailey's above mentioned web site for details. |
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It is also unknown whether ''π'' and [[E (mathematical constant)|''e'']] are [[algebraically independent]]. However it is known that at least one of ''πe'' and ''π'' + ''e'' is [[transcendental number|transcendental]] (''q.v.'').<!-- redundant wikilink intentional: specifically relevant to this section--> |
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== Naturality == |
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In [[non-Euclidean geometry]] the sum of the angles of a [[triangle (geometry)|triangle]] may be more or less than ''π'' [[radians]], and the ratio of a circle's circumference to its diameter may also differ from ''π''. This does not change the definition of ''π'', but it does affect many formulæ in which ''π'' appears. So, in particular, ''π'' is not affected by the [[shape of the universe]]; it is not a [[physical constant]] but a mathematical constant defined independently of any physical measurements. Nonetheless, it occurs often in physics. |
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For example, consider [[Coulomb's law]] (SI units) |
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:<math> F = \frac{1}{ 4 \pi \epsilon_0} \frac{\left|q_1 q_2\right|}{r^2} </math>. |
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Here, 4''πr''<sup>2</sup> is just the surface area of sphere of radius ''r''. In this form, it is a convenient way of describing the inverse square relationship of the force at a distance ''r'' from a point source. It would of course be possible to describe this law in other, but less convenient ways, or in some cases more convenient. If [[Planck charge]] is used, it can be written as |
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:<math> F = \frac{q_1 q_2}{r^2} </math> |
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and thus eliminate the need for ''π''. |
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== Fictional references == |
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*In [[Carl Sagan]]'s [[science fiction]] novel ''[[Contact (novel)|Contact]]'', Sagan contemplates the possibility of finding a signature embedded in the [[Positional notation|base-11]] expansion of ''π'' by the creators of the universe. |
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*In [[Greg Bear]]'s science fiction novel ''[[Eon (novel)|Eon]]'', the protagonists measure the amount of space curvature using a device that computes ''π''. Only in completely flat space/time will a circle have a circumference–diameter ratio of 3.14159…. |
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*In [[Terry Pratchett]]'s [[fantasy]] novel ''[[Going Postal]]'', the famous inventor [[Bloody Stupid Johnson]] invents an organ/mail sorter that contains a wheel for which ''π'' is exactly 3. This "New ''π''" starts a chain of events that leads to the failure of the [[Ankh-Morpork]] Post Office (and possibly the destruction of the Universe all in one go.) |
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*In the [[cult film]] ''[[Pi (film)|''π'']]'', the relationship between numbers and nature is analyzed. |
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*In the cartoon series ''[[The Simpsons]]'' (episode [[Bye Bye Nerdie]]), "''π'' is exactly three!" was an announcement made by [[Professor Frink]] on behalf of [[Lisa Simpson]] to gain the attention of a hall full of scientists. <!-- According to the provided references, it is Frink, not Lisa, who made the announcement. Please do not change without discussing at the talk page. --> [http://www.snpp.com/episodes/CABF11], [http://download.lardlad.com/sounds/season12/nerdie20.mp3] |
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*The cartoon series ''[[Futurama]]'' also contains several references to π, such as the use of 'π in 1' oil, and shopping at πkea. |
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*In the ''[[Star Trek]]'' episode "[[Wolf in the Fold]]", when the computer of the ''Enterprise'' is taken over by an evil consciousness, Spock tells the computer to figure ''π'' to the last digit, which incapacitates the entity as all computer resources are devoted to this impossible task. |
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*In the ''[[Doctor Who]]'' episode "[[The Five Doctors]]", the [[First Doctor]] uses ''π'' in an application to get across a deadly chessboard floor in the Dark Tower of [[Rassilon]], after figuring out the statement made by [[The Master (Doctor Who)|The Master]] that the chessboard is; "easy as 'pi'." |
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*The [[science fiction]] novel ''[[Time's Eye]]'', by [[Arthur C. Clarke]] and [[Stephen Baxter]], depicts a world restructured by alien forces. A spherical device is observed whose circumference-to-diameter ratio appears to be an exact integer 3 across all planes. It is the first book in [[The Time Odyssey]] series. |
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*In the [[Leslie Nielsen]] spoof film ''[[Spy Hard]]'', a spy ([[Nicollette Sheridan]]) is referred to as "Agent 3.14." |
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*In the ''[[Jimmy Neutron]]'' episode "Revenge of the Nanobots," Jimmy destroys the nanobots (which were designed to fix all errors, and were destroying mankind due to an excessively high standard of "error-free") by making them correct a test paper which states that ''π'' equals 3. |
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*''[[The Ragged Astronauts]]'', a science-fiction novel by [[Bob Shaw]], describes 2 planets, Land & Overland, which share atmospheres. ''π'' is discovered by the most eminent philosopher of Land to be exactly 3. Subsequent books in the trilogy involve another planet (Farland), and the discovery of changes to the value of ''π''. |
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*In ''[[The Time Warp Trio]]'', Sam shuts down a threatening robot by telling it that his number was ''π''. |
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*In the Look Around You episode about maths, the audience is asked to turn to chapter 3.1415926 of their copybooks. |
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*The main character of the [[bande dessinée]] series ''La vache'' (by Belgian authors [[Johann de Moor]] and [[Stephen Desberg]]) is a secret-agent cow called Pi = 3,1416. [http://www.bedetheque.com/serie-1372-BD-Vache-(La).html] |
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*Episode 7 of the [[anime]] [[Seraphim Call]] is devoted to ''π'' and the relationship between a circle and a square. |
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*In the movie [[The Matrix Reloaded]], 314 seconds is "the length and breadth of the window" which [[Neo]] has to reach the "source" of the matrix. |
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== Trivia == |
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[[Image:Diameter and Pi 1.gif|right|frame|A visual definition of π as the ratio of the circumference of a circle to its diameter]] |
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*[[March 14]] (3/14 in [[United States|U.S.]] date format) marks [[Pi Day]] which is celebrated by many lovers of ''π''. Incidentally, it is also [[Einstein]]'s birthday. |
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*On [[July 22]], [[Pi Approximation Day]] is celebrated (22/7 - in European date format - is a popular approximation of ''π''). |
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*355/113 (~3.1415929) is sometimes jokingly referred to as "not ''π'', but an incredible simulation!" |
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*Singer [[Kate Bush]]'s 2005 album "[[Aerial (album)|Aerial]]" contains a song titled "''π''", in which she sings ''π'' to its 137th decimal place; however, for an unknown reason, she omits the 79th to 100th decimal places.[http://www.telegraph.co.uk/connected/main.jhtml?xml=/connected/2005/12/20/ecdeer20.xml] She was preceded in this achievement by several years by a Swedish indie math lyrics artist under the moniker [[Matthew Matics]], who loses track of the decimals at about the same point in the series. |
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*[[John Harrison]] (1693–1776) (of Longitude fame), devised a [[meantone temperament]] musical tuning system derived from ''π'', now called [[Lucy Tuning]]. |
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*Users of the ''[[A9.com]]'' search engine are eligible for an ''[[amazon.com]]'' program offering discounts of (''π''/2)% on purchases. |
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*The [[Heywood Banks]] song "Eighteen Wheels on a Big Rig" has the singer(s) count pi in the final verse; they reach "eight hundred billion" before going into the chorus. |
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*In [[1932]], [[Stanisław Gołąb]] proved that the ratio of the circumference to the diameter of the [[unit disc]] is always in between 3 and 4; these values are attained if and only if the unit "circle" has the shape of a regular [[hexagon]] resp. a [[parallelogram]]. See [[unit disc]] for details. |
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*[[John Squire]] (of [[The Stone Roses]]) mentions ''π'' in a song written for his second band [[The Seahorses]] called "Something Tells Me". The song was recorded in full by the full band, and appears on the bootleg of the never released second-album recordings. The song ends with the lyrics, "''What's the secret of life? It's 3.14159265, yeah yeah!!''" |
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*[[Hard 'n Phirm]]'s fourth track on [[Horses and Grasses]] is "Pi" (and is preceded by "An Intro", which discusses the topic like an educational television program). Many digits are recited through it, and a [http://keithschofield.com/pi/std.html video] appeared online inspired by it. |
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==See also== |
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*[[List of topics related to pi]] |
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*[[Pi (letter)|Greek letter ''π'']] |
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*[[Calculus]] |
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*[[Geometry]] |
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*[[Trigonometric function]] |
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*[[Pi through experiment]] |
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*[[Lindemann-Weierstrass theorem|Proof that ''π'' is transcendental]] |
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*[[Proof that 22 over 7 exceeds π|A simple proof that 22/7 exceeds ''π'']] |
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*[[Feynman point]] |
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*[[Pi Day]] |
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*[[Lucy Tuning]] |
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*[[Cadaeic Cadenza]] |
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*[[Software for calculating π|Software for calculating ''π'']] on personal computers |
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*[[History of π|History of ''π'']] |
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*[[History of numerical approximations of π|History of numerical approximations of ''π'']] |
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*[[Constant|Mathematical Constants]] |
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**[[E (mathematical constant)|e]] |
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**[[Golden ratio|φ]] |
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==References== |
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=== Footnotes === |
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<div class="references-small"> |
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<references /> |
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</div> |
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=== Additional === |
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<div class="references-small"> |
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*{{cite journal |
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| author = [[David H. Bailey|Bailey, David H.]], [[Peter Borwein|Borwein, Peter B.]], and [[Simon Plouffe|Plouffe, Simon]] |
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| year =1997 | month = April |
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| title = On the Rapid Computation of Various Polylogarithmic Constants |
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| journal = Mathematics of Computation |
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| volume = 66 | issue = 218 | pages = 903–913 |
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| url = http://crd.lbl.gov/~dhbailey/dhbpapers/digits.pdf |
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}} |
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*[http://fabrice.bellard.free.fr/pi/pi_bin/pi_bin.html ''A new formula to compute the n'th binary digit of pi''] by Fabrice Bellard, retrieved March 22, 2006 |
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*[[Petr Beckmann]], ''A History of ''π |
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*[http://home.earthlink.net/~jsondow/ Jonathan Sondow], [http://arXiv.org/abs/math.NT/0401406 "A faster product for pi and a new integral for ln pi/2,"] Amer. Math. Monthly 112 (2005) 729-734. |
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*[http://home.earthlink.net/~jsondow/ Jonathan Sondow], Problem 88, Math Horizons 5 (Sept., 1997) 32, 34. |
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*Borwein, Jonathan M.; Borwein, Peter; and Berggren, Lennart (2004). ''Pi: A Source Book'', Springer. ISBN 0-387-20571-3. |
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</div> |
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== External links == |
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;Digits |
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*[http://www.zenwerx.com/pi.php First 4 Million Digits of ''π''] |
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*[http://www.gutenberg.net/etext/50 Project Gutenberg E-Text containing a million digits of ''π''] |
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*[http://www.angio.net/pi/piquery Search the first 200 million digits of ''π'' for arbitrary strings of numbers] |
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*[http://www.codecodex.com/wiki/index.php?title=Digits_of_pi_calculation Source code for calculating the digits of ''π''] |
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*[http://pi-world-ranking-list.com Pi World ranking list] - List of many people who have memorized large numbers of digits of ''π''. |
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;General |
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*[http://backpi.hopto.org Background Pi] |
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*[http://www.joyofpi.com The Joy of Pi by David Blatner] |
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*[http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Pi_through_the_ages.html J J O'Connor and E F Robertson: ''A history of pi''. Mac Tutor project] |
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*[http://www.lrz-muenchen.de/~hr/numb/pi-irr.html A proof that ''π'' Is Irrational] |
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*[http://mathworld.wolfram.com/PiFormulas.html Lots of formulæ for ''π''] at [[MathWorld]] |
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*[http://planetmath.org/encyclopedia/Pi.html PlanetMath: Pi] |
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*[http://mathforum.org/isaac/problems/pi1.html Finding the value of ''π''] |
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*[http://www.cut-the-knot.org/pythagoras/NatureOfPi.shtml Determination of ''π''] at [[cut-the-knot]] |
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*[http://www.cecm.sfu.ca/~jborwein/pi-slides.pdf The Life of Pi by Jonathan Borwein] |
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*[http://www.bbc.co.uk/radio4/science/5numbers2.shtml BBC Radio Program about ''π''] |
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*[http://www.research.att.com/~njas/sequences/A000796 Decimal expansions of Pi and related links] at the [[On-Line Encyclopedia of Integer Sequences]] |
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*[http://www.super-computing.org/pi-decimal_current.html Statistical Distribution Information on PI] based on 1.2 trillion digits of PI |
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*[http://www.eveandersson.com/pi/ Pi Land] |
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Revision as of 13:20, 31 August 2006
The mathematical constant π is a real number, approximately equal to 3.14159, which is the ratio of a circle's circumference to its diameter in Euclidean geometry, and has many uses in mathematics, physics, and engineering. It is also known as Archimedes' constant (not to be confused with Archimedes number) and as Ludolph's number.
The letter π
The name of the Greek letter π is pi, and this spelling is used in typographical contexts where the Greek letter is not available or where its usage could be problematic. When referring to this constant, the symbol π is always pronounced like "pie" in English, the conventional English pronunciation of the letter.
The constant is named π because it is the first letter of the Greek words "περιφέρεια" (periphery) and "περίμετρον" (perimeter). The Swiss mathematician Leonhard Euler proposed that this number be given a particular name and suggested the use of π.
Definition
In Euclidean plane geometry, π is defined either as the ratio of a circle's circumference to its diameter, or as the ratio of a circle's area to the area of a square whose side is the radius. The constant π may be defined in many other ways, for example as the smallest positive x for which sin(x) = 0. The formulæ below illustrate other (equivalent) definitions.
Numerical value
The numerical value of π truncated to 50 decimal places is:
- 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
See the links below and those at sequence A00796 in OEIS for more digits.
With the 50 digits given here, the circumference of any circle that would fit in the observable universe (ignoring the curvature of space) could be computed with an error less than the size of a proton.[1] Nevertheless, the exact value of π has an infinite decimal expansion: its decimal expansion never ends and does not repeat, since π is an irrational number (and indeed, a transcendental number). This infinite sequence of digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing more digits and investigating the number's properties. Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of π, no simple pattern in the digits has ever been found. Digits of π are available on many web pages, and there is software for calculating π to billions of digits on any personal computer. See history of numerical approximations of π.
Calculating π
The formulae often given for calculating the digits of have desirable mathematical properties, but are often hard to understand without a background in trigonometry and calculus. Nevertheless, it is possible to compute using techniques involving only algebra and geometry.
For example, one common classroom activity for experimentally measuring the value of involves drawing a large circle on graph paper, then measuring its approximate area by counting the number of cells inside the circle. Since the area of the circle is known to be
can be derived using algebra:
This process works mathematically as well as experimentally. If a circle with radius r is drawn with its center at the point (0,0), any point whose distance from the origin is less than r will fall inside the circle. The pythagorean theorem gives the distance from any point (x,y) to the center:
Mathematical "graph paper" is formed by imagining a 1x1 square centered around each point (x,y), where x and y are integers between -r and r. Squares whose center resides inside the circle can then be counted by testing whether, for each point (x,y),
The total number of points satisfying that condition thus approximates the area of the circle, which then can be used to calculate an approximation of . Mathematically, this formula can be written:
In other words, begin by choosing a value for r. Consider all points (x,y) in which both x and y are integers between -r and r. Starting at 0, add 1 for each point whose distance to the origin (0,0) is less than r. Divide the sum, representing the area of a circle of radius r, by r2 to find the approximation of . Closer approximations can be produced by using larger values of r.
For example, if r is set to 2, then the points (-2,-2), (-2,-1), (-2,0), (-2,1), (-2,2), (-1,-2), (-1,-1), (-1,0), (-1,1), (-1,2), (0,-2), (0,-1), (0,0), (0,1), (0,2), (1,-2), (1,-1), (1,0), (1,1), (1,2), (2,-2), (2,-1), (2,0), (2,1), (2,2) are considered. The 9 points (-1,-1), (-1,0), (-1,1), (0,-1), (0,0), (0,1), (1,-1), (1,0), (1,1) are found to be inside the circle, so the approximate area is 9, and π is calculated to be approximately 2.25. Results for larger values of r are shown in the table below:
r | area | approximation of π |
---|---|---|
3 | 25 | 2.777778 |
4 | 45 | 2.8125 |
5 | 69 | 2.76 |
10 | 305 | 3.05 |
20 | 1245 | 3.1125 |
100 | 31397 | 3.1397 |
1000 | 3141521 | 3.141521 |
Similarly, the more complex approximations of π given below involve repeated calculations of some sort, yielding closer and closer approximations with increasing numbers of calculations.
Properties
The constant π is an irrational number; that is, it cannot be written as the ratio of two integers. This was proven in 1761 by Johann Heinrich Lambert.
Furthermore, π is also transcendental, as was proven by Ferdinand von Lindemann in 1882. This means that there is no polynomial with rational coefficients of which π is a root. An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.
History
Use of the symbol π
Often William Jones' book A New Introduction to Mathematics from 1706 is cited as the first text where the Greek letter π was used for this constant, but this notation became particularly popular after Leonhard Euler adopted it some years later, (cf. History of π).
Early approximations
- Main article: History of numerical approximations of π.
The value of π has been known in some form since antiquity. As early as the 19th century BC, Babylonian mathematicians were using π = 25/8, which is within 0.5% of the true value.
The Egyptian scribe Ahmes wrote the oldest known text to give an approximate value for π, citing a Middle Kingdom papyrus, corresponding to a value of 256 divided by 81 or 3.160.
It is sometimes claimed that the Bible states that π = 3, based on a passage in 1 Kings 7:23 giving measurements for a round basin as having a 10 cubit diameter and a 30 cubit circumference. Rabbi Nehemiah explained this by the diameter being from outside to outside while the circumference was the inner brim; but it may suffice that the measurements are given in round numbers. Also, the basin may not have been exactly circular.
Archimedes of Syracuse discovered, by considering the perimeters of 96-sided polygons inscribing a circle and inscribed by it, that π is between 223/71 and 22/7. The average of these two values is roughly 3.1419.
The Chinese mathematician Liu Hui computed π to 3.141014 (good to three decimal places) in AD 263 and suggested that 3.14 was a good approximation.
The Indian mathematician and astronomer Aryabhata in the 5th century gave the approximation π = 62832/20000 = 3.1416, correct when rounded off to four decimal places.
The Chinese mathematician and astronomer Zu Chongzhi computed π to be between 3.1415926 and 3.1415927 and gave two approximations of π, 355/113 and 22/7, in the 5th century.
The Indian mathematician and astronomer Madhava of Sangamagrama in the 14th century computed the value of π after transforming the power series expansion of π /4 into the form
- π = √12 (1 - 1/(33) + 1/(532) - 1/(733) + ...
and using the first 21 terms of this series to compute a rational approximation of π correct to 11 decimal places as 3.14159265359. By adding a remainder term to the original power series of π /4, he was able to compute π to an accuracy of 13 decimal places.
The Persian astronomer Ghyath ad-din Jamshid Kashani (1350-1439) correctly computed π to 9 digits in the base of 60, which is equivalent to 16 decimal digits as:
- 2π = 6.2831853071795865
By 1610, the German mathematician Ludolph van Ceulen had finished computing the first 35 decimal places of π. It is said that he was so proud of this accomplishment that he had them inscribed on his tombstone.
In 1789, the Slovene mathematician Jurij Vega improved John Machin's formula from 1706 and calculated the first 140 decimal places for π of which the first 126 were correct [1] and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct.
The English amateur mathematician William Shanks, a man of independent means, spent over 20 years calculating π to 707 decimal places (accomplished in 1873). In 1944, D. F. Ferguson found that Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were fallacious. By 1947, Ferguson had recalculated pi to 808 decimal places (with the aid of a mechanical desk calculator).
Numerical approximations
Due to the transcendental nature of π, there are no closed form expressions for the number in terms of algebraic numbers and functions. Roughly speaking, this means that any formula which uses simple math operations to calculate π must go on forever. This is why formulæ for calculating π are often written with a "..." to indicate that in order to reach π exactly, an infinite number of additional terms would have to follow the terms given.
Consequently, numerical calculations must use approximations of π. For many purposes, 3.14 or 22/7 is close enough, although engineers often use 3.1416 (5 significant figures) or 3.14159 (6 significant figures) for more accuracy. The approximations 22/7 and 355/113, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction expansion of π. The approximation 355/113 (3.1415929…) is the best one that may be expressed with a three-digit numerator and denominator.
The earliest numerical approximation of π is almost certainly the value 3. In cases where little precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the fact that it is the ratio of the perimeter of an inscribed regular hexagon to the diameter of the circle.
All further improvements to the above mentioned "historical" approximations were done with the help of computers.
Formulæ
Geometry
The constant π appears in many formulæ in geometry involving circles and spheres.
Geometrical shape | Formula |
---|---|
Circumference of circle of radius r and diameter d | |
Area of circle of radius r | |
Area of ellipse with semiaxes a and b | |
Volume of sphere of radius r and diameter d | |
Surface area of sphere of radius r and diameter d | |
Volume of cylinder of height h and radius r | |
Surface area of cylinder of height h and radius r | |
Volume of cone of height h and radius r | |
Surface area of cone of height h and radius r |
(All of these are a consequence of the first one, as the area of a circle can be written as A = ∫(2πr) dr ("sum of annuli of infinitesimal width"), and others concern a surface or solid of revolution.)
Also, the angle measure of 180° (degrees) is equal to π radians.
Analysis
Many formulæ in analysis contain π, including infinite series (and infinite product) representations, integrals, and so-called special functions.
- The area of the unit disc:
- Half the circumference of the unit circle:
- François Viète, 1593 (proof):
- Faster product (see Sondow, 2005 and Sondow web page)
- where the nth factor is the 2nth root of the product
- Symmetric formula (see Sondow, 1997)
- Bailey-Borwein-Plouffe algorithm (See Bailey, 1997 and Bailey web page)
- An integral formula from calculus (see also Error function and Normal distribution):
- Basel problem, first solved by Euler (see also Riemann zeta function):
- and generally, is a rational multiple of for positive integer n
- Gamma function evaluated at 1/2:
- Euler's identity (called by Richard Feynman "the most remarkable formula in mathematics"):
- A property of Euler's totient function (see also Farey sequence):
- An application of the residue theorem
- where the path of integration is a closed curve around the origin, traversed in the standard anticlockwise direction.
Continued fractions
Next to its simple continued-fraction representation [3; 7, 15, 1, 292, 1, 1, …], which displays no discernible pattern, π has many generalized continued-fraction representations that are generated by a simple rule, including:
(Other representations are available at The Wolfram Functions Site.)
Number theory
Some results from number theory:
- The probability that two randomly chosen integers are coprime is 6/π2.
- The probability that a randomly chosen integer is square-free is 6/π2.
- The average number of ways to write a positive integer as the sum of two perfect squares (order matters but not sign) is π/4.
Here, "probability", "average", and "random" are taken in a limiting sense, e.g. we consider the probability for the set of integers {1, 2, 3,…, N}, and then take the limit as N approaches infinity.
The theory of elliptic curves and complex multiplication derives the approximation
which is valid to about 30 digits.
Dynamical systems and ergodic theory
Consider the recurrence relation
Then for almost every initial value x0 in the unit interval [0,1],
This recurrence relation is the logistic map with parameter r = 4, known from dynamical systems theory. See also: ergodic theory.
Physics
The number π appears routinely in equations describing fundamental principles of the universe, due in no small part to its relationship to the nature of the circle and, correspondingly, spherical coordinate systems.
- Coulomb's law for the electric force:
Probability and statistics
In probability and statistics, there are many distributions whose formulæ contain π, including:
- probability density function (pdf) for the normal distribution with mean μ and standard deviation σ:
- pdf for the (standard) Cauchy distribution:
Note that since , for any pdf f(x), the above formulæ can be used to produce other integral formulae for π.
A semi-interesting empirical approximation of π is based on Buffon's needle problem. Consider dropping a needle of length L repeatedly on a surface containing parallel lines drawn S units apart (with S > L). If the needle is dropped n times and x of those times it comes to rest crossing a line (x > 0), then one may approximate π using:
[As a practical matter, this approximation is poor and converges very slowly.]
Another approximation of π is to throw points randomly into a quarter of a circle with radius 1 that is inscribed in a square of length 1. π, the area of a unit circle, is then approximated as 4*(points in the quarter circle) / (total points).
Efficient methods
In the early years of the computer, the first expansion of π to 100,000 decimal places was computed by Maryland mathematician Dr. Daniel Shanks and his team at the United States Naval Research Laboratory (N.R.L.) in 1961. Dr. Shanks's son Oliver Shanks, also a mathematician, states that there is no family connection to William Shanks, and in fact, his family's roots are in Central Europe.[citation needed]
Daniel Shanks and his team used two different power series for calculating the digital of π. For one it was known that any error would produce a value slightly high, and for the other, it was known that any error would produce a value slightly low. And hence, as long as the two series produced the same digits, there was a very high confidence that they were correct. The first 100,000 digits of π were published by the US Naval Research Laboratory
None of the formulæ given above can serve as an efficient way of approximating π. For fast calculations, one may use a formula such as Machin's:
together with the Taylor series expansion of the function arctan(x). This formula is most easily verified using polar coordinates of complex numbers, starting with
Formulæ of this kind are known as Machin-like formulae.
Many other expressions for π were developed and published by the incredibly intuitive Indian mathematician Srinivasa Ramanujan. He worked with mathematician Godfrey Harold Hardy in England for a number of years.
Extremely long decimal expansions of π are typically computed with the Gauss-Legendre algorithm and Borwein's algorithm; the Salamin-Brent algorithm which was invented in 1976 has also been used.
The first one million digits of π and 1/π are available from Project Gutenberg (see external links below). The current record (December 2002) by Yasumasa Kanada of Tokyo University stands at 1,241,100,000,000 digits, which were computed in September 2002 on a 64-node Hitachi supercomputer with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits). The following Machin-like formulæ were used for this:
- K. Takano (1982).
- F. C. W. Störmer (1896).
These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers. (Normality of π will always depend on the infinite string of digits on the end, not on any finite computation.)
In 1997, David H. Bailey, Peter Borwein and Simon Plouffe published a paper (Bailey, 1997) on a new formula for π as an infinite series:
This formula permits one to fairly readily compute the kth binary or hexadecimal digit of π, without having to compute the preceding k − 1 digits. Bailey's website contains the derivation as well as implementations in various programming languages. The PiHex project computed 64-bits around the quadrillionth bit of π (which turns out to be 0).
Fabrice Bellard claims to have beaten the efficiency record set by Bailey, Borwein, and Plouffe with his formula to calculate binary digits of π [2]:
Other formulæ that have been used to compute estimates of π include:
This converges extraordinarily rapidly. Ramanujan's work is the basis for the fastest algorithms used, as of the turn of the millennium, to calculate π.
Miscellaneous formulæ
Using base 60, π can be approximated to the equivalent of eight significant figures (in base 10) as:
In addition, the following expressions approximate π
- accurate to 9 digits:
- accurate to 3 digits:
- Karl Popper conjectured that Plato knew this expression; that he believed it to be exactly π; and that this is responsible for some of Plato's confidence in the omnicompetence of mathematical geometry — and Plato's repeated discussion of special right triangles that are either isosceles or halves of equilateral triangles.
- The continued fraction representation of π can be used to generate successively better rational approximations, which start off: 22/7, 333/106, 355/113…. These approximations are the best possible rational approximations of π relative to the size of their denominators.
Memorising digits
Even long before computers have calculated π, memorizing a record number of digits became an obsession for some people. The current unofficial world record is 83,431 decimal places, and was set by a Japanese mental health counsellor named Akira Haraguchi, who is currently 59 years of age.[3] Before Haraguchi accomplished this on July 2, 2005, the world record was 42,195, which was set by Hiroyuki Goto.
There are many ways to memorise π, including the use of piems, which are poems that represent π in a way such that the length of each word (in letters) represents a digit. Here is an example of a piem: How I need a drink, alcoholic in nature (or: of course), after the heavy lectures involving quantum mechanics. Notice how the first word has 3 letters, the second word has 1, the third has 4, the fourth has 1, the fifth has 5, and so on. The Cadaeic Cadenza contains the first 3834 digits of π in this manner. Piems are related to the entire field of humorous yet serious study that involves the use of mnemonic techniques to remember the digits of π, known as piphilology. See Pi mnemonics for examples. In other languages there are similar methods of memorisation. However, this method proves inefficient for large memorisations of pi. Other methods include remembering "patterns" in the numbers (for instance, the "year" 1971 appears in the first fifty digits of pi).
Open questions
The most pressing open question about π is whether it is a normal number -- 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", and that this is true in every base, not just base 10. Current knowledge on this point is very weak; e.g., it is not even known which of the digits 0,…,9 occur infinitely often in the decimal expansion of π.
Bailey and Crandall showed in 2000 that the existence of the above mentioned Bailey-Borwein-Plouffe formula and similar formulæ imply that the normality in base 2 of π and various other constants can be reduced to a plausible conjecture of chaos theory. See Bailey's above mentioned web site for details.
It is also unknown whether π and e are algebraically independent. However it is known that at least one of πe and π + e is transcendental (q.v.).
Naturality
In non-Euclidean geometry the sum of the angles of a triangle may be more or less than π radians, and the ratio of a circle's circumference to its diameter may also differ from π. This does not change the definition of π, but it does affect many formulæ in which π appears. So, in particular, π is not affected by the shape of the universe; it is not a physical constant but a mathematical constant defined independently of any physical measurements. Nonetheless, it occurs often in physics.
For example, consider Coulomb's law (SI units)
- .
Here, 4πr2 is just the surface area of sphere of radius r. In this form, it is a convenient way of describing the inverse square relationship of the force at a distance r from a point source. It would of course be possible to describe this law in other, but less convenient ways, or in some cases more convenient. If Planck charge is used, it can be written as
and thus eliminate the need for π.
Fictional references
- In Carl Sagan's science fiction novel Contact, Sagan contemplates the possibility of finding a signature embedded in the base-11 expansion of π by the creators of the universe.
- In Greg Bear's science fiction novel Eon, the protagonists measure the amount of space curvature using a device that computes π. Only in completely flat space/time will a circle have a circumference–diameter ratio of 3.14159….
- In Terry Pratchett's fantasy novel Going Postal, the famous inventor Bloody Stupid Johnson invents an organ/mail sorter that contains a wheel for which π is exactly 3. This "New π" starts a chain of events that leads to the failure of the Ankh-Morpork Post Office (and possibly the destruction of the Universe all in one go.)
- In the cult film π, the relationship between numbers and nature is analyzed.
- In the cartoon series The Simpsons (episode Bye Bye Nerdie), "π is exactly three!" was an announcement made by Professor Frink on behalf of Lisa Simpson to gain the attention of a hall full of scientists. [4], [5]
- The cartoon series Futurama also contains several references to π, such as the use of 'π in 1' oil, and shopping at πkea.
- In the Star Trek episode "Wolf in the Fold", when the computer of the Enterprise is taken over by an evil consciousness, Spock tells the computer to figure π to the last digit, which incapacitates the entity as all computer resources are devoted to this impossible task.
- In the Doctor Who episode "The Five Doctors", the First Doctor uses π in an application to get across a deadly chessboard floor in the Dark Tower of Rassilon, after figuring out the statement made by The Master that the chessboard is; "easy as 'pi'."
- The science fiction novel Time's Eye, by Arthur C. Clarke and Stephen Baxter, depicts a world restructured by alien forces. A spherical device is observed whose circumference-to-diameter ratio appears to be an exact integer 3 across all planes. It is the first book in The Time Odyssey series.
- In the Leslie Nielsen spoof film Spy Hard, a spy (Nicollette Sheridan) is referred to as "Agent 3.14."
- In the Jimmy Neutron episode "Revenge of the Nanobots," Jimmy destroys the nanobots (which were designed to fix all errors, and were destroying mankind due to an excessively high standard of "error-free") by making them correct a test paper which states that π equals 3.
- The Ragged Astronauts, a science-fiction novel by Bob Shaw, describes 2 planets, Land & Overland, which share atmospheres. π is discovered by the most eminent philosopher of Land to be exactly 3. Subsequent books in the trilogy involve another planet (Farland), and the discovery of changes to the value of π.
- In The Time Warp Trio, Sam shuts down a threatening robot by telling it that his number was π.
- In the Look Around You episode about maths, the audience is asked to turn to chapter 3.1415926 of their copybooks.
- The main character of the bande dessinée series La vache (by Belgian authors Johann de Moor and Stephen Desberg) is a secret-agent cow called Pi = 3,1416. [6]
- Episode 7 of the anime Seraphim Call is devoted to π and the relationship between a circle and a square.
- In the movie The Matrix Reloaded, 314 seconds is "the length and breadth of the window" which Neo has to reach the "source" of the matrix.
Trivia
- March 14 (3/14 in U.S. date format) marks Pi Day which is celebrated by many lovers of π. Incidentally, it is also Einstein's birthday.
- On July 22, Pi Approximation Day is celebrated (22/7 - in European date format - is a popular approximation of π).
- 355/113 (~3.1415929) is sometimes jokingly referred to as "not π, but an incredible simulation!"
- Singer Kate Bush's 2005 album "Aerial" contains a song titled "π", in which she sings π to its 137th decimal place; however, for an unknown reason, she omits the 79th to 100th decimal places.[7] She was preceded in this achievement by several years by a Swedish indie math lyrics artist under the moniker Matthew Matics, who loses track of the decimals at about the same point in the series.
- John Harrison (1693–1776) (of Longitude fame), devised a meantone temperament musical tuning system derived from π, now called Lucy Tuning.
- Users of the A9.com search engine are eligible for an amazon.com program offering discounts of (π/2)% on purchases.
- The Heywood Banks song "Eighteen Wheels on a Big Rig" has the singer(s) count pi in the final verse; they reach "eight hundred billion" before going into the chorus.
- In 1932, Stanisław Gołąb proved that the ratio of the circumference to the diameter of the unit disc is always in between 3 and 4; these values are attained if and only if the unit "circle" has the shape of a regular hexagon resp. a parallelogram. See unit disc for details.
- John Squire (of The Stone Roses) mentions π in a song written for his second band The Seahorses called "Something Tells Me". The song was recorded in full by the full band, and appears on the bootleg of the never released second-album recordings. The song ends with the lyrics, "What's the secret of life? It's 3.14159265, yeah yeah!!"
- Hard 'n Phirm's fourth track on Horses and Grasses is "Pi" (and is preceded by "An Intro", which discusses the topic like an educational television program). Many digits are recited through it, and a video appeared online inspired by it.
See also
- List of topics related to pi
- Greek letter π
- Calculus
- Geometry
- Trigonometric function
- Pi through experiment
- Proof that π is transcendental
- A simple proof that 22/7 exceeds π
- Feynman point
- Pi Day
- Lucy Tuning
- Cadaeic Cadenza
- Software for calculating π on personal computers
- History of π
- History of numerical approximations of π
- Mathematical Constants
References
Footnotes
- ^ Bailey, David H., Borwein, Peter B., and Borwein, Jonathan M. (1997). "The Quest for Pi" (PDF). Mathematical Intelligencer (1): 50–57.
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Additional
- Bailey, David H., Borwein, Peter B., and Plouffe, Simon (1997). "On the Rapid Computation of Various Polylogarithmic Constants" (PDF). Mathematics of Computation. 66 (218): 903–913.
{{cite journal}}
: Unknown parameter|month=
ignored (help)CS1 maint: multiple names: authors list (link) - A new formula to compute the n'th binary digit of pi by Fabrice Bellard, retrieved March 22, 2006
- Petr Beckmann, A History of π
- Jonathan Sondow, "A faster product for pi and a new integral for ln pi/2," Amer. Math. Monthly 112 (2005) 729-734.
- Jonathan Sondow, Problem 88, Math Horizons 5 (Sept., 1997) 32, 34.
- Borwein, Jonathan M.; Borwein, Peter; and Berggren, Lennart (2004). Pi: A Source Book, Springer. ISBN 0-387-20571-3.
External links
- Digits
- First 4 Million Digits of π
- Project Gutenberg E-Text containing a million digits of π
- Search the first 200 million digits of π for arbitrary strings of numbers
- Source code for calculating the digits of π
- Pi World ranking list - List of many people who have memorized large numbers of digits of π.
- General
- Background Pi
- The Joy of Pi by David Blatner
- J J O'Connor and E F Robertson: A history of pi. Mac Tutor project
- A proof that π Is Irrational
- Lots of formulæ for π at MathWorld
- PlanetMath: Pi
- Finding the value of π
- Determination of π at cut-the-knot
- The Life of Pi by Jonathan Borwein
- BBC Radio Program about π
- Decimal expansions of Pi and related links at the On-Line Encyclopedia of Integer Sequences
- Statistical Distribution Information on PI based on 1.2 trillion digits of PI
- Pi Land
- The sung Pi