'''Pi''' , or '''[[Archimedes]]' constant''', is the name for most important number in all of [[mathematics]] and engineering. It is defined as the [[ratio]] of the [[circumference]] of a [[circle]] to its [[diameter]], and represented by the Greek letter '''<big><math>\pi</math></big>'''. It was first used with its current meaning in 1706 by a [[Welsh]] mathematician, William Jones,<ref name="hist">{{cite web| url = http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Pi_through_the_ages.html | title = A history or Pi| accessdate = 2012-02-11}}</ref> who selected '''<big><math>\pi</math></big>''' because it is the first letter of the [[Greek]] word for [[perimeter]] (''περίμετρος''), ''i.e.'', the circumference of a circle is its perimeter. A [[Swiss]] mathematician, [[Leonhard Euler]], then popularized the notation in 1737 as he did for other symbols.<ref>https:/big/www.historytoday.com/archive/feature/man-who-invented-pi</ref>, The [[infinite]] length of ostensibly [[random]] decimal points in <math>\pi</math> lead to the assumption that pi is a [[normal number]] (which corresponds is demonstrated by computer calculations of <math>\pi</math> to trillions of digits): it contains every possible finite sequence of digits, and therefore, every birthday, even the English letter equivalent of every book ever written. Simply stated, '''p.''[[Essay:pi contains pi|Pi contains pi]]' '''', beginning an infinite number of digits into it (which avoids the periodicity problem).
<big><math>\pi</math></big> is also the symbol used referenced in mathematics for the ratio between [[Bible]], literature, and even as the diameter title of a circle and its circumference, and which appears in many other placesmovie. It is central to the most famous identity in all of mathematics: [[Euler's Formula|Euler's Identity]].
The value of '''<big><math>\pi</math></big> ''' is approximately 3.1416, or 22/7. The exact value cannot be expressed as a [[fraction]] or as a [[decimal]] number, regardless of how many [[digit]]s are used. Johann Heinrich Lambert proved this in 1761 by showing that '''<big><math>\pi</math></big>''' is an [[irrational number]]; this , which means that it cannot can't be expressed as a fraction, .<ref>{{cite web| url = http://turnbull.mcs.st-and (therefore) cannot .ac.uk/~history/Biographies/Lambert.html| title = Biography of Johann Heinrich Lambert| accessdate = 2012-02-12}}</ref> In 1882, Carl Louis Ferdinand von Lindeman proved that '''<big><math>\pi</math></big>''' is also a [[transcendental number]], which means that it can't be expressed exactly as a decimal no matter how many decimal places it is carried out the solution toany simple [[equation]]. <ref>{{cite web| url = http://turnbull.mcs.st-and.ac.uk/~history/Biographies/Lindemann.html| title = Biography of Carl Louis Ferdinand von Lindemann| accessdate = 2012-02-12}}</ref>
The value of ==History==Mathematicians have worked for centuries to calculate '''<big><math>\pi</math></big> is approximately 3.14159 in ''' to more and more decimalplaces. This value is precise enough for almost all ordinary purposes; it canTo some extent, for examplethe progress of mathematics, or at least of computation, can be used to calculate gauged by the circumference of progress in the Earth with an error number of only 350 feetdigits to which '''<big><math>\pi</math></big>''' has been calculated.
For rough purposesSome ancients expressed '''<big><math>\pi</math></big>''' by using fractional approximations. The Rhind, or Ahmes Papyrus (''c.'' 1650 B.C.)<ref>http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html</ref><ref>https://www.math.tamu.edu/~don.allen/history/egypt/node3.html</ref> has shown that the fraction 22[[ancient Egypt]]ians had determined the value for '''<big><math>\pi</7 (= math></big>''' to be 3.142851605.The [[Babylonia]]n value from the same era was 3 1/8 = 3.125.) is sometimes used. 355<ref>Boyer, ''A History of Mathematics'', 2nd edition</113 is accurate ref>, coming to six placeswithin 1 percent accuracy for both<ref>https://www.maa.org/press/periodicals/convergence/mathematical-treasure-old-babylonian-area-calculation</ref>.
In hexadecimal [[Archimedes]] of Syracuse (base287-16212 BC) notation, carried out "the first theoretical calculation" of '''<big><math>\pi</math></big> is approximately 3''',<ref>[http://veling.243fnl/anne/templars/Pi_through_the_ages.html Pi through the ages]</ref> using regular polygons with a total of 96 sides, within and circumscribing a circle, and in about 225 B.C. he came up with a formula between the folowing numbers: <center><math>3 \dfrac {1} {7} < \pi < 3 \dfrac {10} {71}</math></center>
This is ten times better than the Egyptian and Babylonian values, and it is within 0.04 percent of the correct value. The German-Dutch mathematician Ludolph Van Ceulen used Archimedes's formula to calculate '''<big><math>\pi</math></big>''' to 32 digits in 1615. He was so proud of his achievement that he had the digits engraved on his tombstone<ref>https://www.ams.org/publicoutreach/math-history/hap-6-pi.pdf</ref>. Towards the end of the 17th century mathematical analysis had new methods of making calculations, and '''<big><math>\pi</math></big>''' was no exception. Brilliant mathematicians had made individual calculations for '''<big><math>\pi</math></big>''' in a variety of ways. In 1593 [[François Viète]] (1540-1603) wrote the expression: <center><math>\pi =2 \times \dfrac 2 {\sqrt 2} \times \dfrac 2 {\sqrt {2 + \sqrt 2} } \times \dfrac 2 {\sqrt {2 + \sqrt {2 + \sqrt 2} } } \times \dfrac 2 {\sqrt {2 + \sqrt {2 + \sqrt {2 + \sqrt 2 } } } } \times \cdots</math></center> [[John Wallis]] (1616-1703) made his expression 1655, proving that:<center><math>\frac{\pi}{2} =\frac21\cdot\frac23\cdot\frac43\cdot\frac45\cdot\frac65\cdot\frac67\cdots</math></center> [[William Brouncker]] (1620-1684) discovered the continued fraction in 1658:<center><math>\frac \pi 4 = \cfrac{1}{1+\cfrac{1^2}{2+\cfrac{3^2}{2+\cfrac{5^2}{2+\cfrac{7^2}{2+\cfrac{9^2}{2+\ddots}}}}}}</math></center> During the plague of 1665-1666, [[Isaac Newton]] was confined to the English village of Woolsthorpe. There he calculated '''<big><math>\pi</math></big>''' to 16 digits. <center><math>\pi = \dfrac {3 \sqrt 3} 4 + 24 \left({\dfrac 1 {12} - \dfrac 1 {5 \times 2^5} - \dfrac 1 {28 \times 2^7} - \dfrac 1 {72 \times 2^9} - \cdots}\right)</math></center> Unlike Van Ceulen, Newton did not consider his calculations to be an achievement. "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time," he wrote.<ref>Beckmann, Petr (2015-01-06). ''A Historyof Pi'' (Kindle Locations 2249-2250). St. Martin's Press. Kindle Edition.</ref> [[Gottfried Leibniz|Gottfried Wilhelm von Leibniz]] (1646-1716) published a series for '''<big><math>\pi</math></big>''' in 1673, coming up with the formulas:<center><math>\dfrac \pi 4 =1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0.78539 \, 81633 \, 9744 \ldots</math><ref>https://oeis.org/A003881</ref></center> <center><math>\displaystyle \pi =4 \sum_{k \mathop \ge 0} \left({-1}\right)^k \frac 1 {2 k + 1}</math></center>To some extentBrouncker's convergents are related to the [[Leibniz formula for pi]]: for instance<center><math>\frac{1}{1+\frac{1^2}{2}} = \frac{2}{3} = 1 - \frac{1}{3}</math>and:<math>\frac{1}{1+\frac{1^2}{2+\frac{3^2}{2}}} = \frac{13}{15} = 1 - \frac{1}{3} + \frac{1}{5}.</math></center> In 1706, John Machin, secretary of [[England]]'s Royal Society, developed a quickly converging formula for '''<big><math>\pi</math></big>''' and used it to calculate the progress first 100 digits. In 1844, the idiot savant Johann Dase of [[Hamburg]] used Machin's formula to calculate 200 digits in less than two months.<ref>Beckmann</ref> In contrast, William Shanks spent twenty years calculating '''<big><math>\pi</math></big>''' to 707 places, a task he completed in 1873. In 1945, it was discovered that only the first 527 of Shanks' digits were correct. ENIAC, the first electronic [[computer]], took seventy hours to calculate 2,037 digits in 1949. In 2008, the first million digits of '''<big><math>\pi</math></big>''' were published on [[Project Gutenberg]].<ref>Hemphill, Scott, ''[https://www.gutenberg.org/ebooks/50 Pi to 1,000,000 places]''.</ref> In 2014, the anonymous programmer Houkouonchi calculated the first 13.3 trillion digits of '''<big><math>\pi</math></big>''' in 208 days.<ref>Yee, Alexander, "[http://www.numberworld.org/y-cruncher/ y-cruncher - A Multi-Threaded Pi-Program]"</ref> This result has not been published. =='''<big><math>\pi</math></big>''' in mathematics—or at least ==It's impossible to overestimate the importance of computation—can be gauged '''<big><math>\pi</math></big>''' (and ''[[e]]'') for mathematics. These values are tied by [[Euler's Formula|Euler's identity]]: :<math>e^{\pi \imath} +1 = 0</math>. '''<big><math>\pi</math></big>''' may be used to determine the progress area of a circle:<ref>{{cite web| url = http://www.worsleyschool.net/science/files/circle/area.html | title = The Circle Area Formula| accessdate = 2012-02-13}}</ref> :<big><math>Area= \pi r^{2}</math></big> ==Recreation==Memorizing '''<big><math>\pi</math></big>''' is a challenge that appeals to some people. Mnemonics have been devised. Counting the letters in each word of the number phrase "Now I want a drink—alcoholic, of digits course" gives '''<big><math>\pi</math></big>''' to seven places (which is more than enough for all ordinary purposes). Numerous other mnemonics of this kind have been devised; in 1995, Michael Keith wrote one titled ''[http://users.aol.com/s6sj7gt/mikerav.htm Near a Raven]'' which simultaneously parodies [[Edgar Allen Poe]]'s poem ''The Raven'' while encoding '''<big><math>\pi</math></big> has been calculated''' to 740 places. March 14 marks [[Pi Day]], a holiday on which the mathematical constant is celebrated. The date, 3/14, comes from the first three digits of '''<big><math>\pi</math></big>'''. Some people begin their celebration at 1:59 p.m., derived from the following three digits. Pi Approximation Day is a similar holiday, celebrated on 22 July (from the approximation 22/7).<ref>[https://www.usatoday.com/tech/science/mathscience/2007-03-14-pi-day_N.htm USA Today (3/14/2007) - Pi-day]</ref> The value of pi is approximately: {{quotebox|3.14159265358979323846264338327950288419​7169399375​1058209749​4459230781​6406286208​9986280348​2534211706​7982148086​5132823066​4709384460​9550582231​7253594081​2848111745​0284102701​9385211055​5964462294​8954930381​9644288109​7566593344​6128475648​2337867831​6527120190​9145648566​92... }}
In ==Does the Bible, 1 Kings 7:23 contains the passage "And he made a molten sea, ten cubits from the one brim attempt to the other: it was round define pi?==Virtually all about, and his height was five cubits: and a line serious students of thirty cubits did compass it round aboutthe [[Bible]] say no." Making many assumptionsStill, critics frequently claim that the Bible contains an incorrect value for '''<refbig>Assuming that: <math>\pi</math></big>''', and the "sea" question is perfectly circular; the measurements are raised frequently enough to be understood as exact; earn mention in the measurement of the "compass" and across the "brim" are measurements of the same circle, rather, than say, exterior and interior measurements of a wide lip; etc.</ref> this quotation (which predates Greek mathematics) would appear to equate Pi to 3[[Skeptics Annotated Bible]].
Papyrys The claim is based on a verse in the [[I Kings|first book of AhmesKings]]:{{bible quote|He made the Sea of cast metal, dated c. 1650 B.Ccircular in shape, measuring ten [[cubit]]s from rim to rim and five cubits high. circa 1000 years before Book It took a line of Kingsthirty cubits to measure around it.|book=1_Kings|chap=7|verses=23|version=NIV}} Thus, critics say, shows the Bible claims that ancient Egyptians had the value 3 1/6 = 3.166666667. The Babylonian value from same time of '''<big><math>\pi</math></big> 3 1/8 = 3''' is three, and is therefore in error.125(BoyerHowever, A History there are a number of Mathematicsassumptions in this claim, 2nd Edition).any of which might invalidate the argument if wrong:
In 1873 Abraham Shanks spent twenty years calculating * That the tools and system of measurement available to the builders were sufficiently accurate to distinguish between three and '''<big><math>\pi</math></big> to 707 places'''.* That the phrase translated as "circular in shape" means perfectly circular, but unfortunately made not simply "round" as an ellipse is round.* That the Bible is trying to provide a mistake in his calculation value for '''<big><math>\pi</math></big>''', and only 527 not merely describing the historical event of them were correctbuilding this object. When electronic computers were developed, <ref name="math">{{cite web| url = http://mathforum.org/library/drmath/view/52573.html| title = Discussion re rounding Pi| accessdate = 2012-02-10}}</ref>* That a value is ''wrong'' simply because it depends on imprecisely expressed divisors. The ratio that is '''<big><math>\pi</math></big> was soon calculated to tens ''' can be derived completely accurately for a whole range of thousandsvalues<ref name="math" /> even if the Sea is taken as a perfect circle, millionsnamely when, before rounding to the nearest [[whole number]], the diameter is known to be greater than 9.5 and billions of placesless than about 9. As 708, and the circumference greater than about 29.845 and less than 30.5, it being quite common at the time to round to whole numbers.<ref name="jph">{{cite web| url = http://www.tektonics.org/lp/piwrong.html| title = Is the Bible wrong about pi?| accessdate = 2012-02-10}}</ref> * That both the [[diameter]] and the [[circumference]] are measuring the same edges. Since the sides of 2002any practical vessel have thickness, it is possible that the record diameter is held by Yasumasa Kanada an outside measurement and the circumference is an inside measurement.<ref name="jph" /> Taking the measurements as artisan's instructions, the ''molten sea'' would be cast in a dual mold: an outer mold, measured across its cavity's diameter (equal to the outer diameter of Tokyo University at 1the ''molten sea''),241and an inner mold thrust into the molten material (probably bronze) to make the ''molten sea'' into a hollow bowl,100measured around its circumference (equal to the inner circumference of the ''molten sea''),000with measurement taken using a string with regularly spaced knots,000 digitsthus allowing adjustments during construction. Taking a cubit as 18 inches and a handbreadth (4 inches) for the vessel's thickness (according to 2 Chronicles 4:5), we obtain an inner diameter of 172 inches (10 cubits minus 2 handbreadths for the two rims) and a correct value for pi: about 3.14. <ref> Lindsell, '''The Battle for the Bible''', Zondervan, 1976, pp. 165, 166</ref>* That result was never printed outboth the diameter and the circumference are measuring the same part of the object.The object is also described as having an outward-turned rim. The easiest places to measure the diameter would be across the wider rim, and the easiest place to measure the circumference would be around the body below the rim.<ref>{{cite web| url = http://creationontheweb.com/content/view/1731/| title = Does the Bible say Pi equals 3.0?| accessdate = 2012-02-10}}</ref>
==Recreational use==Memorizing Common sense and a rudimentary knowledge of the Bible should cause one to question whether it sets out to define mathematical concepts. The creation of a "sea of cast metal" by human beings in ancient times, without modern construction tools and measuring equipment, does not require nor could it utilize a precise value for '''<big><math>\pi</math></big> '''. An even more fundamental objection is a challenge that appeals to some people. Mnemonics have been devised. Counting the letters in the phrase "Now I want a drink—alcoholic, of course" gives '''<big><math>\pi</math></big> to seven places (which ''' is more than enough for all ordinary purposes). Numerous other mnemonics of this kind have been devised; in 1995, Michael Keith wrote one entitled [http://users.aol.com/s6sj7gt/mikerav.htm Near a Raven] which simultaneously parodies an [[Edgar Allen Poeirrational number]], and therefore has an infinite number of digits. (A "closed form" of 's poem ''The Raven<big><math>\pi</math></big>''' does exist,but requires mathematical notation that was invented many centuries later.) A decimal expression of ''' while encoding <big><math>\pi</math></big> to 740 places''' could not "fit" in the Bible, or in any other finite text.<ref name="math" />
March 14 marks Pi Day, a holiday on which the mathematical constant is celebrated. The date comes from the first three digits of pi; some people begin their celebration at 1:59 pm, derived from the following three digits.==References==<small><references/></small>
==Notes and referencesSee also==<references*[[Pi Day]]*[http://yacas.sourceforge.net/>Algochapter5.html#c5s5 Calculation of pi with computers]
[[Category:Mathematics]]