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{{Infobox Album |
{{pp-semi|small=yes}}
| Name = Unsigned & Still Major: Da Album Before da Album
{{Two other uses|the mathematical constant|the Greek letter|pi (letter)}}
| Type = Studio
<!--
| Artist = [[Soulja Boy]] & [[Arab]]
| Cover = Unsigned and Still Major.jpg
| Released = [[February 19]], [[2007 in music|2007]]
| Recorded = 2006
| Genre = [[Pop rap]], [[dance pop]], [[snap music|snap]], [[crunk]]
| Length =
| Label = [[Stacks on Deck Entertainment|Stacks on Deck]]
| Reviews =
| Last album =
| This album = '''''Unsigned & Still Major: Da Album Before da Album'''''<br />(2007)
| Next album = ''[[souljaboytellem.com]]''<br />(2007)
}}


'''''Unsigned & Still Major: Da Album Before da Album''''', or simply "'''''Unsigned & Still Major'''''" as it is usually called, is the first studio album by [[United States|American]] rapper [[Soulja Boy]]. It was released on [[February 19]], [[2007]]. (see [[2007 in music]]) The title most likely refers to ''[[souljaboytellem.com]]'', his second studio album which was released later that year.The album sold over 1,000 copies.
IMPORTANT NOTICE: Please note that Wikipedia is not a database to store the 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.


==Tracklisting==
This has been established by a very clear consensus and any editor adding lists of digits of pi is liable to be blocked from editing without further warning.
#"I Got Me Some Bapes"
#"Stop Then Snap"
#"[[Crank Dat (Soulja Boy)|Crank Dat Dance]]"
#"Booty Meat"
#"Soulja Boy Ain't Got No Money"
#"I Got Me Some Bapes <small>(Remix)</small>"
#"Crank Dat Jumprope"
#"Look @ Me"
#"Bring Dat Beat Back"
#"Give Me a High Five"
#"I'm the Hustle Man"
#"Wuzhannanan"
#"Dominican Papi"


== YouTube Singles ==
-->
All music videos for songs shown here were made indepentantly using a camera and an editing program, and posted on [[YouTube]].
[[Image:Pi-unrolled-720.gif|thumb|360px|right|When a circle's diameter is 1, its circumference is π.]]
{| class="infobox" style ="width: 370px;"
| colspan="2" align="center" | [[List of numbers]] – [[Irrational number]]s <br> [[Apéry's constant|&zeta;(3)]] – [[Square root of 2|√2]] – [[Square root of 3|√3]] – [[Square root of 5|√5]] – [[Golden ratio|&phi;]] – [[Feigenbaum constants|&alpha;]] – [[E (mathematical constant)|e]] – [[Pi|&pi;]] – [[Feigenbaum constants|&delta;]]
|-
|[[Binary numeral system|Binary]]
| 11.00100100001111110110…
|-
| [[Decimal]]
| 3.14159265358979323846…
|-
| [[Hexadecimal]]
| 3.243F6A8885A308D31319…
|-
| [[Continued fraction]]
| <math>3 + \cfrac{1}{7 + \cfrac{1}{15 + \cfrac{1}{1 + \cfrac{1}{292 + \ddots}}}}</math><br><small>Note that this continued fraction is not periodic.</small>
|}


'''Shootout (Now Known As Let Me Get Em')''' (Song not shown on album)
'''Pi''' or '''π''' is a [[mathematical constant]] which represents the ratio of any [[circle]]'s circumference to its diameter in [[Euclidean geometry]], which is the same as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.14159. Pi is one of the most important mathematical constants: many formulae from mathematics, [[science]], and [[engineering]] involve π.<ref>{{cite book | title = An Introduction to the History of Mathematics | author = Howard Whitley Eves | year = 1969 | publisher = Holt, Rinehart & Winston | url = http://books.google.com/books?id=LIsuAAAAIAAJ&q=%22important+numbers+in+mathematics%22&dq=%22important+numbers+in+mathematics%22&pgis=1 }}</ref>
*An Indepently made music video was created by Soulja Boy for his youtube account and posted on June 27, 2006. In the music video, soulja boy is seen talking on his cell phone in a car, dancing on roofs and ground, and has closeups on his feet while dancing.<ref>http://www.youtube.com/watch?v=4MBoTTHHzrg</ref>


'''I Got Me Some Bapes (Now Known as Bapes)'''
Pi is an [[irrational number]], which means that it cannot be expressed as a [[fraction]] ''m''/''n'', where ''m'' and ''n'' are [[integer]]s. Consequently its [[decimal representation]] never ends or repeats. Beyond being [[irrational number|irrational]], it is a [[transcendental number]], which means that no finite sequence of algebraic operations on [[integer]]s (powers, roots, sums, etc.) could ever produce it. Throughout the history of mathematics, much effort has been made to determine π more accurately and understand its nature; fascination with the number has even carried over into culture at large.
* I Got Me Some Bapes is also another independant music video made by soulja boy but more popular. The music video was released to youtube on February 1st, 2007. In the beginning of the music video shows the famous scene of where Soulja Boy says: "Arab I just got back from the mall man! Guess what I got?", Arab: "What did you do soulja boy?", Soulja Boy Man I Got Me Some bathin' apes! In the music video, soulja boy is partnered by arab beside a car with many bape products on it and shows amny other clips from his other two videos about him seeing ugly bathin' ape shoes in the mall and one where he finally gets his first pair.
Soulja Boy is also seen destroying other shoes with arab that aren't bathin' apes. Soulja boy is also shown in the mall with friends snap dancing. The end of the music video shows soulja boy riding in to his lawn. <ref>http://www.youtube.com/watch?v=KEn3_kMAIrI</ref> <gallery>
<!-- Image with unknown copyright status removed: Image:arabigotmesomebapes.jpg|Arab in the independant music video. -->
</gallery>


{{hip-hop-album-stub}}
The Greek letter π, often spelled out ''pi'' in text, was adopted for the number from the Greek word for ''perimeter'' "περίμετρος", probably by [[William Jones (mathematician)|William Jones]] in 1706, and popularized by [[Leonhard Euler]] some years later. The constant is occasionally also referred to as the '''circular constant''', '''[[Archimedes]]' constant''' (not to be confused with an [[Archimedes number]]), or '''[[Ludolph van Ceulen|Ludolph]]'s number'''.
{{Soulja Boy}}

==Fundamentals==
=== The letter π ===
[[Image:Pi-symbol.svg|thumb|140px|right|Lower-case ''π'' is used for the constant.]]
{{main|pi (letter)}}
The name of the [[pi (letter)|Greek letter π]] is ''pi'', and this spelling is used in [[typography|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 --> In Greek, the name of this letter is [[Help:IPA|pronounced]] {{IPA|/pi/}}.

The [[constant]] is named "π" because "π" is the first letter of the [[Greek language|Greek]] words περιφέρεια (periphery) and περίμετρος (perimeter), probably referring to its use in the formula to find the circumference, or perimeter, of a circle.<ref name="adm">{{cite web|url=http://mathforum.org/dr.math/faq/faq.pi.html|title=About Pi|work=Ask Dr. Math FAQ|accessdate=2007-10-29}}</ref> π is [[Unicode]] [[character (computing)|character]] U+03C0 ("[[Greek alphabet|Greek small letter pi]]").<ref>{{cite web|url=http://www.w3.org/TR/MathML2/bycodes.html|title=Characters Ordered by Unicode|publisher=[[World Wide Web Consortium|W3C]]|accessdate=2007-10-25}}</ref>

===Definition===
[[Image:Pi eq C over d.svg|thumb|left|Circumference = π × diameter]]
In [[Euclidean geometry|Euclidean plane geometry]], π is defined as the [[ratio]] of a [[circle]]'s [[circumference]] to its [[diameter]]:<ref name="adm"/>

:<math> \pi = \frac{c}{d}. </math>

Note that the ratio <sup>''c''</sup>/<sub>''d''</sub> does not depend on the size of the circle. For example, if a circle has twice the diameter ''d'' of another circle it will also have twice the circumference ''c'', preserving the ratio <sup>''c''</sup>/<sub>''d''</sub>. This fact is a consequence of the [[similarity (geometry)|similarity]] of all circles.

[[Image:Circle Area.svg|right|thumb|Area of the circle = π × area of the shaded square]]
Alternatively π can be also defined as the ratio of a circle's [[area]] (A) to the area of a square whose side is equal to the [[radius]]:<ref name="adm"/><ref>{{cite web|url=http://www.wku.edu/~tom.richmond/Pir2.html|title=Area of a Circle|first=Bettina|last=Richmond|publisher=[[Western Kentucky University]]|date=[[1999-01-12]]|accessdate=2007-11-04}}</ref>

:<math> \pi = \frac{A}{r^2}. </math>

The constant π may be defined in other ways that avoid the concepts of [[arc (geometry)|arc]] length and area, for example, as twice the smallest positive ''x'' for which [[trigonometric function|cos]](''x'')&nbsp;=&nbsp;0.<ref>{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |id=ISBN 0-07-054235-X | pages = 183}}</ref> The formulas below illustrate other (equivalent) definitions.

===Irrationality and transcendence===
{{main|Proof that π is irrational}}
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]].<ref name="adm"/> In the 20th century, proofs were found that require no prerequisite knowledge beyond integral calculus. One of those, due to [[Ivan M. Niven|Ivan Niven]], is widely known.<ref>{{cite journal|title=A simple proof that &pi; is irrational|first=Ivan|last=Niven|authorlink=Ivan Niven|journal=[[Bulletin of the American Mathematical Society]]|volume=53|number=6|pages=509|year=1947|url=http://www.ams.org/bull/1947-53-06/S0002-9904-1947-08821-2/S0002-9904-1947-08821-2.pdf|format=[[Portable Document Format|PDF]]|accessdate=2007-11-04|doi=10.1090/S0002-9904-1947-08821-2}}</ref><ref>{{cite web|first=Helmut|last=Richter|url=http://www.lrz-muenchen.de/~hr/numb/pi-irr.html|title=Pi Is Irrational|date=[[1999-07-28]]|publisher=Leibniz Rechenzentrum|accessdate=2007-11-04}}</ref> A somewhat earlier similar proof is by [[Mary Cartwright]].<ref>{{cite book|first=Harold|last=Jeffreys|authorlink=Harold Jeffreys|title=Scientific Inference|edition=3rd|publisher=[[Cambridge University Press]]|year=1973}}</ref>

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 (mathematics)|root]].<ref name="ttop">{{cite web|first=Steve|last=Mayer|url=http://dialspace.dial.pipex.com/town/way/po28/maths/docs/pi.html|title=The Transcendence of &pi;|accessdate=2007-11-04}}</ref> 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 constructions|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.<ref>{{cite web|url=http://www.cut-the-knot.org/impossible/sq_circle.shtml|title=Squaring the Circle|publisher=[[cut-the-knot]]|accessdate=2007-11-04}}</ref>

===Numerical value===
{{seealso|numerical approximations of π}}
<!-- 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.-->
The numerical value of π [[truncation|truncated]] to 2,500th [[decimal|decimal places]] is:<ref>{{cite web|url=http://www.research.att.com/~njas/sequences/A000796|title=A000796: Decimal expansion of Pi|publisher=[[On-Line Encyclopedia of Integer Sequences]]|accessdate=2007-11-04}}</ref>

:<!--Please discuss any changes to this on the Talk page.--> Pi = Pi = 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679
8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196
4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273
7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094
3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912
9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132
0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235
4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859
5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303
5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989
3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151
5574857242 4541506959 5082953311 6861727855 8890750983 8175463746 4939319255 0604009277 0167113900 9848824012
8583616035 6370766010 4710181942 9555961989 4676783744 9448255379 7747268471 0404753464 6208046684 2590694912
9331367702 8989152104 7521620569 6602405803 8150193511 2533824300 3558764024 7496473263 9141992726 0426992279
6782354781 6360093417 2164121992 4586315030 2861829745 5570674983 8505494588 5869269956 9092721079 7509302955
3211653449 8720275596 0236480665 4991198818 3479775356 6369807426 5425278625 5181841757 4672890977 7727938000
8164706001 6145249192 1732172147 7235014144 1973568548 1613611573 5255213347 5741849468 4385233239 0739414333
4547762416 8625189835 6948556209 9219222184 2725502542 5688767179 0494601653 4668049886 2723279178 6085784383
8279679766 8145410095 3883786360 9506800642 2512520511 7392984896 0841284886 2694560424 1965285022 2106611863
0674427862 2039194945 0471237137 8696095636 4371917287 4677646575 7396241389 0865832645 9958133904 7802759009
9465764078 9512694683 9835259570 9825822620 5224894077 2671947826 8482601476 9909026401 3639443745 5305068203
4962524517 4939965143 1429809190 6592509372 2169646151 5709858387 4105978859 5977297549 8930161753 9284681382
6868386894 2774155991 8559252459 5395943104 9972524680 8459872736 4469584865 3836736222 6260991246 0805124388
4390451244 1365497627 8079771569 1435997700 1296160894 4169486855 5848406353 4220722258 2848864815 8456028506
0168427394 5226746767 8895252138 5225499546 6672782398 6456596116 3548862305 7745649803 5593634568 1743241125



:''See [[#External links|the links below]] and those at sequence [[oeis:A000796|A000796]] in [[On-Line Encyclopedia of Integer Sequences|OEIS]] for more digits.''

While the value of pi has been computed to more than a [[orders of magnitude (numbers)#1012|trillion]] (10<sup>12</sup>) digits,<ref>{{cite web |url=http://www.super-computing.org/pi_current.html |title=Current publicized world record of pi |accessdate=2007-10-14}}</ref> elementary applications, such as calculating the circumference of a circle, will rarely require more than a dozen decimal places. For example, a value truncated to 11 decimal places is accurate enough to calculate the circumference of the earth with a precision of a millimeter, and one truncated to 39 decimal places is sufficient to compute the circumference of any circle that fits in the [[observable universe]] to a precision comparable to the size of a [[hydrogen atom]].<ref>{{cite book |title=Excursions in Calculus |last=Young |first=Robert M. |year=1992 |publisher=Mathematical Association of America (MAA)|location=Washington |isbn=0883853175 |pages=417 | url = http://books.google.com/books?id=iEMmV9RWZ4MC&pg=PA238&dq=intitle:Excursions+intitle:in+intitle:Calculus+39+digits&lr=&as_brr=0&ei=AeLrSNKJOYWQtAPdt5DeDQ&sig=ACfU3U0NSYsF9kVp6om4Zyw3a7F82QCofQ }}</ref><ref>{{cite web |url=http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000067000004000298000001&idtype=cvips&gifs=yes |title=Statistical estimation of pi using random vectors |accessdate=2007-08-12 |format= |work=}}</ref>

Because π is an [[irrational number]], its decimal expansion never ends and does not [[Repeating decimal|repeat]]. 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.<ref>{{MathWorld|urlname=PiDigits|title=Pi Digits}}</ref> 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.<ref>{{cite news|first=Chad|last=Boutin|url=http://www.purdue.edu/UNS/html4ever/2005/050426.Fischbach.pi.html|title=Pi seems a good random number generator - but not always the best|publisher=[[Purdue University]]|date=[[2005-04-26]]|accessdate=2007-11-04}}</ref> Digits of π are available on many web pages, and there is [[software for calculating π]] to billions of digits on any [[personal computer]].

===Calculating π===
{{main|Computing π}}

π can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometry-based approach, due to [[Archimedes]]<ref name="NOVA">{{cite web|first=Rick|last=Groleau|url=http://www.pbs.org/wgbh/nova/archimedes/pi.html|title=Infinite Secrets: Approximating Pi|publisher=NOVA|date=09-2003|accessdate=2007-11-04}}</ref>, is to calculate the [[perimeter]], ''P<sub>n</sub> ,'' of a [[regular polygon]] with ''n'' sides [[circumscribe]]d around a circle with diameter ''d.'' Then

:<math>\pi = \lim_{n \to \infty}\frac{P_{n}}{d}</math>

That is, the more sides the polygon has, the closer the approximation approaches π. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides [[Inscribed figure|inscribed]] inside the circle. Using a polygon with 96 sides, he computed the fractional range: <math>\begin{smallmatrix}3\frac{10}{71}\ <\ \pi\ <\ 3\frac{1}{7}\end{smallmatrix}</math>.<ref>{{cite book
| first=Petr | last=Beckmann
| year=1989
| title=A History of Pi
| publisher=Barnes & Noble Publishing
| isbn=0880294183 }}</ref>

π can also be calculated using purely mathematical methods. Most formulas used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in [[trigonometry]] and [[calculus]]. However, some are quite simple, such as this form of the [[Leibniz formula for pi|Gregory-Leibniz series]]:<ref>{{cite book |first=Pierre |last=Eymard |coauthors=Jean-Pierre Lafon |others=Stephen S. Wilson (translator)|title=The Number &pi;|url=http://books.google.com/books?id=qZcCSskdtwcC&pg=PA53&dq=leibniz+pi&ei=uFsuR5fOAZTY7QLqouDpCQ&sig=k8VlN5VTxcX9a6Ewc71OCGe_5jk |accessdate=2007-11-04 |year=2004 |month=02 |publisher=American Mathematical Society |isbn=0821832468 |pages=53 |chapter=2.6 }}</ref>

:<math>\pi = \frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}\cdots\! </math>.

While that series is easy to write and calculate, it is not immediately obvious why it yields π. In addition, this series converges so slowly that 300 terms are not sufficient to calculate '''π''' correctly to 2 decimal places.<ref>{{cite journal|url=http://www.scm.org.co/Articulos/832.pdf|format=[[PDF]]|title=Even from Gregory-Leibniz series &pi; could be computed: an example of how convergence of series can be accelerated|journal=Lecturas Mathematicas|volume=27|year=2006|pages=21–25|first=Vito|last=Lampret, Spanish|accessdate=2007-11-04}}</ref> However, by computing this series in a somewhat more clever way by taking the midpoints of partial sums, it can be made to converge much faster. Let

<math>\pi_{0,1} = \frac{4}{1}, \pi_{0,2} =\frac{4}{1}-\frac{4}{3}, \pi_{0,3} =\frac{4}{1}-\frac{4}{3}+\frac{4}{5}, \pi_{0,4} =\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}, \cdots\! </math>

and then define

<math>\pi_{i,j} = \frac{\pi_{i-1,j}+\pi_{i-1,j+1}}{2}</math> for all <math>i,j\ge 1</math>

then computing <math>\pi_{10,10}</math> will take similar computation time to computing 150 terms of the original series in a brute force manner, and <math>\pi_{10,10}=3.141592653\cdots</math>, correct to 9 decimal places. This computation is an example of the [[Van Wijngaarden transformation]].<ref>A. van Wijngaarden, in: Cursus: Wetenschappelijk Rekenen B, Process Analyse, Stichting Mathematisch Centrum, (Amsterdam, 1965) pp. 51-60.</ref>

==History==
{{seealso|Chronology of computation of π|Numerical approximations of π}}
The history of π parallels the development of mathematics as a whole.<ref>{{cite book |last=Beckmann |first=Petr |authorlink=Petr Beckmann |title=A History of π |year=1976 |publisher=[[St. Martin's Press|St. Martin's Griffin]] |id=ISBN 0-312-38185-9}}</ref> Some authors divide progress into three periods: the ancient period during which π was studied geometrically, the classical era following the development of calculus in Europe around the 17th century, and the age of digital computers.<ref>{{cite web|url=http://numbers.computation.free.fr/Constants/Pi/pi.html|title=Archimedes' constant &pi;|accessdate=2007-11-04}}</ref>

===Geometrical period===

That the ratio of the circumference to the diameter of a circle is the same for all circles, and that it is slightly more than 3, was known to ancient Egyptian, Babylonian, Indian and Greek geometers. The earliest known approximations date from around 1900 BC; they are 25/8 (Babylonia) and 256/81 (Egypt), both within 1% of the true value.<ref name="adm"/> The Indian text ''[[Shatapatha Brahmana]]'' gives π as 339/108 ≈ 3.139. The [[Hebrew Bible|Tanakh]] appears to suggest, in the Book of [[Book of Kings|Kings]], that π = 3, which is notably worse than other estimates available at the time of writing (600 BC). The interpretation of the passage is disputed,<ref>{{cite web|first=H. Peter|last=Aleff|url=http://www.recoveredscience.com/const303solomonpi.htm|title=Ancient Creation Stories told by the Numbers: Solomon's Pi|publisher=recoveredscience.com|accessdate=2007-10-30}}</ref><ref name="ahop">{{cite web|first=J J|last=O'Connor|coauthors=E F Robertson|url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html|title=A history of Pi|date=2001-08|accessdate=2007-10-30}}</ref> as some believe the ratio of 3:1 is of an exterior circumference to an interior diameter of a thinly walled basin, which could indeed be an accurate ratio, depending on the thickness of the walls (See: [[History_of_numerical_approximations_of_%CF%80#Biblical_value|Biblical value of π]]).

[[Archimedes]] (287-212 BC) was the first to estimate π rigorously. He realized that its magnitude can be bounded from below and above by inscribing circles in [[regular polygon]]s and calculating the outer and inner polygons' respective perimeters:<ref name="ahop"/>

[[Image:Archimedes pi.svg|350px|center|]]
[[Image:Cutcircle2.svg|thumb|right|250px|Liu Hui's Pi algorithm]]
By using the equivalent of 96-sided polygons, he proved that 223/71 &lt; π &lt; 22/7.<ref name="ahop"/> Taking the average of these values yields 3.1419.

In the following centuries further development took place in India and China. Around 265, the [[Wei Kingdom]] mathematician [[Liu Hui]] provided a simple and rigorous [[Liu Hui's π algorithm|iterative algorithm]] to calculate π to any degree of accuracy. He himself carried through the calculation to 3072-gon and obtained an approximate value for π of 3.1416.
: <math>
\begin{align}
\pi \approx A_{3072} & {} = 768 \sqrt{2 - \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2+1}}}}}}}}} \\
& {} \approx 3.14159.
\end{align}
</math>

Later, Liu Hui invented a [[Liu Hui's π algorithm#Quick method|quick method of calculating π]] and obtained an approximate value of 3.1416 with only a 96-gon, by taking advantage of the fact that the difference in area of successive polygons forms a geometric series with a factor of 4.

Around 480, the Chinese mathematician [[Zu Chongzhi]] demonstrated that π ≈ 355/113, and showed that 3.1415926 &lt; π &lt; 3.1415927 using Liu Hui's algorithm applied to a 12288-gon. This value would stand as the most accurate approximation of π over the next 900 years.

===Classical period===

Until the [[2nd millennium|second millennium]], π was known to fewer than 10 decimal digits. The next major advancement in the study of π came with the development of [[calculus]], and in particular the discovery of [[Series (mathematics)|infinite series]] which in principle permit calculating π to any desired accuracy by adding sufficiently many terms. Around 1400, [[Madhava of Sangamagrama]] found the first known such series:

:<math>\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\!</math>

This is now known as the [[Leibniz formula for pi|Madhava-Leibniz series]]<ref>{{citation|title=Special Functions|last=George E. Andrews, Richard Askey|first=Ranjan Roy|publisher=[[Cambridge University Press]]|year=1999|isbn=0521789885|page=58}}</ref><ref>{{citation|first=R. C.|last=Gupta|title=On the remainder term in the Madhava-Leibniz's series|journal=Ganita Bharati|volume=14|issue=1-4|year=1992|pages=68-71}}</ref> or Gregory-Leibniz series since it was rediscovered by [[James Gregory (astronomer and mathematician)|James Gregory]] and [[Gottfried Leibniz]] in the 17th century. Unfortunately, the rate of convergence is too slow to calculate many digits in practice; about 4,000 terms must be summed to improve upon Archimedes' estimate. However, by transforming the series into

:<math>\pi = \sqrt{12} \, \left(1-\frac{1}{3 \cdot 3} + \frac{1}{5 \cdot 3^2} - \frac{1}{7 \cdot 3^3} + \cdots\right)\!</math>

[[Madhava of Sangamagrama| Madhava]] was able to calculate π as 3.14159265359, correct to 11 decimal places. The record was beaten in 1424 by the [[Islamic mathematics|Persian mathematician]], [[Jamshīd al-Kāshī]], who determined 16 decimals of π.

The first major European contribution since Archimedes was made by the German mathematician [[Ludolph van Ceulen]] (1540&ndash;1610), who used a geometrical method to compute 35 decimals of π. He was so proud of the calculation, which required the greater part of his life, that he had the digits engraved into his tombstone.<ref>{{cite book | title = Mathematical Tables; Containing the Common, Hyperbolic, and Logistic Logarithms... | author = Charles Hutton | publisher = London: Rivington | year = 1811 | pages = p.13 | url = http://books.google.com/books?id=zDMAAAAAQAAJ&pg=PA13&dq=snell+descartes+date:0-1837&lr=&as_brr=1&ei=rqPgR7yeNqiwtAPDvNEV }}</ref>

Around the same time, the methods of calculus and determination of infinite series and products for geometrical quantities began to emerge in Europe. The first such representation was the [[Viète's formula]],
:<math>\frac2\pi = \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdot \cdots\!</math>

found by [[François Viète]] in 1593. Another famous result is [[Wallis product|Wallis' product]],

:<math>\frac{\pi}{2} = \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\!</math>

written down by [[John Wallis]] in 1655. [[Isaac Newton]] himself derived a series for π and calculated 15 digits, although he later confessed: "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time." <ref>[http://query.nytimes.com/gst/fullpage.html?res=9B0DE0DB143FF93BA35750C0A961948260 The New York Times: Even Mathematicians Can Get Carried Away]</ref>

In 1706 [[John Machin]] was the first to compute 100 decimals of π, using the formula

:<math>\frac{\pi}{4} = 4 \, \arctan \frac{1}{5} - \arctan \frac{1}{239}\!</math>

with

:<math>\arctan \, x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots\!</math>

Formulas of this type, now known as [[Machin-like formula]]s, were used to set several successive records and remained the best known method for calculating π well into the age of computers. A remarkable record was set by the calculating prodigy [[Zacharias Dase]], who in 1844 employed a Machin-like formula to calculate 200 decimals of π in his head. The best value at the end of the 19th century was due to [[William Shanks]], who took 15 years to calculate π with 707 digits, although due to a mistake only the first 527 were correct. (To avoid such errors, modern record calculations of any kind are often performed twice, with two different formulas. If the results are the same, they are likely to be correct.)

Theoretical advances in the 18th century led to insights about π's nature that could not be achieved through numerical calculation alone. [[Johann Heinrich Lambert]] proved the irrationality of π in 1761, and [[Adrien-Marie Legendre]] proved in 1794 that also π<sup>2</sup> is irrational. When [[Leonhard Euler]] in 1735 solved the famous [[Basel problem]] &ndash; finding the exact value of

:<math>\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots\!</math>

which is π<sup>2</sup>/6, he established a deep connection between π and the [[prime number]]s. Both Legendre and Leonhard Euler speculated that π might be [[transcendental number|transcendental]], a fact that was proved in 1882 by [[Ferdinand von Lindemann]].

[[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 in 1737.<ref>{{cite web|url=http://www.famousWelsh.com/cgibin/getmoreinf.cgi?pers_id=737|title=About: William Jones|work=Famous Welsh|accessdate=2007-10-27}}</ref> He wrote:
{{cquote|<nowiki>There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to (16/5 - 4/239) - 1/3(16/5^3 - 4/239^3) +&nbsp;...&nbsp;=&nbsp;3.14159...&nbsp;=&nbsp;&pi;</nowiki><ref name="adm"/>}}
{{seealso|history of mathematical notation}}

===Computation in the computer age===

The advent of digital computers in the 20th century led to an increased rate of new π calculation records. [[John von Neumann]] used [[ENIAC]] to compute 2037 digits of π in 1949, a calculation that took 70 hours. Additional thousands of decimal places were obtained in the following decades, with the million-digit milestone passed in 1973. Progress was not only due to faster hardware, but also new algorithms. One of the most significant developments was the discovery of the [[fast Fourier transform]] (FFT) in the 1960s, which allows computers to perform arithmetic on extremely large numbers quickly.

In the beginning of the 20th century, the Indian mathematician [[Srinivasa Ramanujan]] found many new formulas for π, some remarkable for their elegance and mathematical depth.<ref name="rad">{{cite web|url=http://numbers.computation.free.fr/Constants/Pi/piramanujan.html|title=The constant &pi;: Ramanujan type formulas|accessdate=2007-11-04}}</ref> Two of his most famous formulas are the series

:<math>\frac{1}{\pi} = \frac{2 \sqrt 2}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!</math>
and
:<math>\frac{426880 \sqrt{10005}}{\pi} = \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 (-640320)^{3k}}\!</math>

which deliver 14 digits per term.<ref name="rad"/> The Chudnovsky brothers used this formula to set several π computing records in the end of the 1980s, including the first calculation of over one billion (1,011,196,691) decimals in 1989. It remains the formula of choice for π calculating software that runs on personal computers, as opposed to the [[supercomputer]]s used to set modern records.

Whereas series typically increase the accuracy with a fixed amount for each added term, there exist iterative algorithms that ''multiply'' the number of correct digits at each step, with the downside that each step generally requires an expensive calculation. A breakthrough was made in 1975, when [[Richard Brent (scientist)|Richard Brent]] and [[Eugene Salamin]] independently discovered the [[Gauss–Legendre algorithm|Brent–Salamin algorithm]], which uses only arithmetic to double the number of correct digits at each step.<ref name="brent">{{Citation | last=Brent | first=Richard | author-link=Richard Brent (scientist) | year=1975 | title=Multiple-precision zero-finding methods and the complexity of elementary function evaluation | periodical=Analytic Computational Complexity | publication-place=New York | publisher=Academic Press | editor-last=Traub | editor-first=J F | pages=151–176 | url=http://wwwmaths.anu.edu.au/~brent/pub/pub028.html | accessdate=2007-09-08}}</ref> The algorithm consists of setting

:<math>a_0 = 1 \quad \quad \quad b_0 = \frac{1}{\sqrt 2} \quad \quad \quad t_0 = \frac{1}{4} \quad \quad \quad p_0 = 1\!</math>

and iterating

:<math>a_{n+1} = \frac{a_n+b_n}{2} \quad \quad \quad b_{n+1} = \sqrt{a_n b_n}\!</math>
:<math>t_{n+1} = t_n - p_n (a_n-a_{n+1})^2 \quad \quad \quad p_{n+1} = 2 p_n\!</math>

until ''a<sub>n</sub>'' and ''b<sub>n</sub>'' are close enough. Then the estimate for π is given by

:<math>\pi \approx \frac{(a_n + b_n)^2}{4 t_n}\!</math>.

Using this scheme, 25 iterations suffice to reach 45 million correct decimals. A similar algorithm that quadruples the accuracy in each step has been found by [[Jonathan Borwein|Jonathan]] and [[Peter Borwein]].<ref>{{cite book|first=Jonathan M|last=Borwein|authorlink=Jonathan Borwein|coauthors=Borwein, Peter, Berggren, Lennart|date=2004|title=Pi: A Source Book|publisher=Springer|isbn=0387205713}}</ref> The methods have been used by [[Yasumasa Kanada]] and team to set most of the π calculation records since 1980, up to a calculation of 206,158,430,000 decimals of π in 1999. The current record is 1,241,100,000,000 decimals, set by Kanada and team in 2002. Although most of Kanada's previous records were set using the Brent-Salamin algorithm, the 2002 calculation made use of two Machin-like formulas that were slower but crucially reduced memory consumption. The calculation was performed on a 64-node Hitachi supercomputer with 1 [[terabyte]] of main memory, capable of carrying out 2 trillion operations per second.

An important recent development was the [[Bailey–Borwein–Plouffe formula]] (BBP formula), discovered by [[Simon Plouffe]] and named after the authors of the paper in which the formula was first published, [[David H. Bailey]], [[Peter Borwein]], and Plouffe.<ref name="bbpf">{{cite journal
| author = [[David H. Bailey|Bailey, David H.]], [[Peter Borwein|Borwein, Peter B.]], and [[Simon Plouffe|Plouffe, Simon]]
| year =1997 | month = April
| title = On the Rapid Computation of Various Polylogarithmic Constants
| journal = Mathematics of Computation
| volume = 66 | issue = 218 | pages = 903–913
| url = http://crd.lbl.gov/~dhbailey/dhbpapers/digits.pdf
| format = [[PDF]]
| doi = 10.1090/S0025-5718-97-00856-9
}}</ref> The formula,

:<math>\pi = \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),</math>

is remarkable because it allows extracting any individual [[hexadecimal]] or [[Binary numeral system|binary]] digit of π without calculating all the preceding ones.<ref name="bbpf"/> Between 1998 and 2000, the [[distributed computing]] project [[PiHex]] used a modification of the BBP formula due to [[Fabrice Bellard]] to compute the [[Orders of magnitude (numbers) #1015|quadrillionth]] (1,000,000,000,000,000:th) bit of π, which turned out to be 0.<ref>{{cite web|url=http://fabrice.bellard.free.fr/pi/pi_bin/pi_bin.html|title=A new formula to compute the n<sup>th</sup> binary digit of pi|first=Fabrice|last=Bellard|authorlink=Fabrice Bellard|accessdate=2007-10-27}}</ref>

===Memorizing digits===
{{main|Piphilology}}
[[Image:PiDigits.svg|right|thumb|300px|right|Recent decades have seen a surge in the record number of digits memorized.]]

Even long before computers have calculated ''π'', memorizing a ''record'' number of digits became an obsession for some people.
In 2006, [[Akira Haraguchi]], a retired Japanese engineer, claimed to have recited 100,000 decimal places.<ref name="japantimes">{{cite news|first=Tomoko|last=Otake|url=http://search.japantimes.co.jp/print/fl20061217x1.html|title=How can anyone remember 100,000 numbers?|work=[[The Japan Times]]|date=[[2006-12-17]]|accessdate=2007-10-27}}</ref> This, however, has yet to be verified by [[Guinness World Records]]. The Guinness-recognized record for remembered digits of ''π'' is 67,890 digits, held by [[Lu Chao]], a 24-year-old graduate student from [[China]].<ref>{{cite web|url=http://www.pi-world-ranking-list.com/news/index.htm|title=Pi World Ranking List|accessdate=2007-10-27}}</ref> It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of ''π'' without an error.<ref>{{cite news|url=http://www.newsgd.com/culture/peopleandlife/200611280032.htm|title=Chinese student breaks Guiness record by reciting 67,890 digits of pi|work=News Guangdong|date=[[2006-11-28]]|accessdate=2007-10-27}}</ref>

There are many ways to memorize ''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.''<ref>{{cite web|first=Jonathan M|last=Borwein|authorlink=Jonathan Borwein|url=http://users.cs.dal.ca/~jborwein/pi-culture.pdf|format=[[PDF]]|title=The Life of Pi: From Archimedes to Eniac and Beyond|publisher=[[Dalhousie University]] Computer Science|date=[[2005-09-25]]|accessdate=2007-10-29}}</ref> 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.<ref>{{cite web|first=Mike|last=Keith|authorlink=Mike Keith (mathematician)|url=http://users.aol.com/s6sj7gt/solution.htm|title=Cadaeic Cadenza: Solution & Commentary|date=1996|accessdate=2007-10-30}}</ref> Piems are related to the entire field of humorous yet serious study that involves the use of [[Mnemonic|mnemonic techniques]] to remember the digits of ''π'', known as [[piphilology]]. See [[:q:English mathematics mnemonics#Pi|Pi mnemonics]] for examples. In other languages there are similar methods of memorization. However, this method proves inefficient for large memorizations of pi. Other methods include remembering patterns in the numbers.<ref>{{cite web|first=Yicong|last=Liu|url=http://silverchips.mbhs.edu/inside.php?sid=3577|title=Oh my, memorizing so many digits of pi.|publisher=Silver Chips Online|date=[[2004-05-19]]|accessdate=2007-11-04}}</ref>

==Advanced properties==
===Numerical approximations===
{{main|History of numerical approximations of π}}
Due to the transcendental nature of ''π'', there are no closed form expressions for the number in terms of algebraic numbers and functions.<ref name="ttop"/> Formulas for calculating ''π'' using elementary arithmetic typically include [[series (mathematics)|series]] or [[Summation#Capital-sigma notation|summation notation]] (such as "..."), which indicates that the formula is really a formula for an infinite sequence of approximations to ''π''.<ref>{{cite web|url=http://mathworld.wolfram.com/PiFormulas.html|title=Pi Formulas|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|publisher=[[MathWorld]]|date=[[2007-09-27]]|accessdate=2007-11-10}}</ref> The more terms included in a calculation, the closer to ''π'' the result will get.

Consequently, numerical calculations must use [[approximation]]s of ''π''. For many purposes, 3.14 or [[Proof that 22/7 exceeds π|<sup>22</sup>/<sub>7</sub>]] is close enough, although engineers often use 3.1416 (5 [[significant figures]]) or 3.14159 (6 significant figures) for more precision. The approximations <sup>22</sup>/<sub>7</sub> and <sup>355</sup>/<sub>113</sub>, with 3 and 7 significant figures respectively, are obtained from the simple [[continued fraction]] expansion of ''π''. The approximation [[Milü|<sup>355</sup>⁄<sub>113</sub>]] (3.1415929…) is the best one that may be expressed with a three-digit or four-digit [[fraction (mathematics)|numerator and denominator]].<ref>{{cite news|language=Chinese|author=韩雪涛|title=数学科普：常识性谬误流传令人忧|publisher=中华读书报|date=[[2001-08-29]]|url=http://www.xys.org/~xys/xys/ebooks/others/science/dajia/shuxuekepu.txt|accessdate=2006-10-06}}</ref><ref>{{cite web|url=http://www.kaidy.com/PiReward.htm|title=Magic of 355 ÷ 113|publisher=Kaidy Educational Resources|accessdate=2007-11-08}}</ref><ref>{{cite web|url=http://numbers.computation.free.fr/Constants/Pi/piApprox.html|title=Collection of approximations for &pi;|publisher=Numbers, constants and computation|first=Xavier|last=Gourdon|coauthors=Pascal Sebah|accessdate=2007-11-08}}</ref>

The earliest numerical approximation of ''π'' is almost certainly the value {{num|3}}.<ref name="ahop"/> 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 figure|inscribed]] [[regular polygon|regular]] [[hexagon]] to the [[diameter]] of the [[circle]].

===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.<ref>{{cite web|url=http://mathworld.wolfram.com/NormalNumber.html|title=Normal Number|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|publisher=[[MathWorld]]|date=[[2005-12-22]]|accessdate=2007-11-10}}</ref> 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''.<ref>{{cite news|url=http://www.lbl.gov/Science-Articles/Archive/pi-random.html|title=Are The Digits of Pi Random? Lab Researcher May Hold The Key|first=Paul|last=Preuss|authorlink=Paul Preuss|publisher=[[Lawrence Berkeley National Laboratory]]|date=[[2001-07-23]]|accessdate=2007-11-10}}</ref>

Bailey and Crandall showed in [[2000]] that the existence of the above mentioned [[Bailey-Borwein-Plouffe formula]] and similar formulas imply that the normality in base 2 of ''π'' and various other constants can be reduced to a plausible [[conjecture]] of [[chaos theory]].<ref>{{cite news|url=http://www.sciencenews.org/articles/20010901/bob9.asp|title=Pi à la Mode: Mathematicians tackle the seeming randomness of pi's digits|first=Ivars|last=Peterson|authorlink=Ivars Peterson|work=Science News Online|date=[[2001-09-01]]|accessdate=2007-11-10}}</ref>

It is also unknown whether ''π'' and [[E (mathematical constant)|''e'']] are [[Algebraic independence|algebraically independent]], although [[Yuri Valentinovich Nesterenko|Yuri Nesterenko]] proved the algebraic independence of {π, [[Gelfond's constant|''e''<sup>&pi;</sup>]], [[Gamma function|&Gamma;]](1/4)} in 1996.<ref>{{cite journal|author=Nesterenko, Yuri V|authorlink=Yuri Valentinovich Nesterenko|title=Modular Functions and Transcendence Problems|journal=[[Comptes rendus de l'Académie des sciences]] Série 1|volume=322|number=10|pages=909–914|year=1996}}</ref> However it is known that at least one of ''πe'' and ''π'' + ''e'' is [[transcendental number|transcendental]] (see [[Lindemann–Weierstrass theorem]]).<!-- redundant wikilink intentional: specifically relevant to this section-->

==Use in mathematics and science==
{{main|List of formulas involving π}}
π is ubiquitous in mathematics, appearing even in places that lack an obvious connection to the circles of Euclidean geometry.<ref>{{cite web|url=http://news.bbc.co.uk/1/hi/world/asia-pacific/4644103.stm|title=Japanese breaks pi memory record|work=[[BBC News]]|date=[[2005-07-02]]|accessdate=2007-10-30}}</ref>

===Geometry and trigonometry===
{{seealso|Area of a disk}}
For any circle with radius ''r'' and diameter ''d'' = 2''r'', the circumference is π''d'' and the area is π''r''<sup>2</sup>. Further, π appears in formulas for areas and volumes of many other geometrical shapes based on circles, such as [[ellipse]]s, [[sphere]]s, [[Cone (geometry)|cone]]s, and [[torus|tori]].<ref>{{cite web|url=http://www.math.psu.edu/courses/maserick/circle/circleapplet.html|title=Area and Circumference of a Circle by Archimedes|publisher=[[Pennsylvania State University|Penn State]]|accessdate=2007-11-08}}</ref> Accordingly, π appears in [[Integral|definite integrals]] that describe circumference, area or volume of shapes generated by circles. In the basic case, half the area of the [[unit disk]] is given by:<ref name="udi">{{cite web|url=http://mathworld.wolfram.com/UnitDiskIntegral.html|title=Unit Disk Integral|publisher=[[MathWorld]]|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|date=[[2006-01-28]]|accessdate=2007-11-08}}</ref>
:<math>\int_{-1}^1 \sqrt{1-x^2}\,dx = \frac{\pi}{2}</math>
and
:<math>\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\,dx = \pi</math>
gives half the circumference of the [[unit circle]].<ref>{{cite web|url=http://www.math.psu.edu/courses/maserick/circle/circleapplet.html|title=Area and Circumference of a Circle by Archimedes|publisher=[[Pennsylvania State University|Penn State]]|accessdate=2007-11-08}}</ref> More complicated shapes can be integrated as [[solid of revolution|solids of revolution]].<ref>{{cite web|url=http://mathworld.wolfram.com/SolidofRevolution.html|title=Solid of Revolution|publisher=[[MathWorld]]|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|date=[[2006-05-04]]|accessdate=2007-11-08}}</ref>

From the unit-circle definition of the [[trigonometric function]]s also follows that the sine and cosine have period 2π. That is, for all ''x'' and integers ''n'', sin(''x'') = sin(''x'' + 2π''n'') and cos(''x'') = cos(''x'' + 2π''n''). Because sin(0) = 0, sin(2π''n'') = 0 for all integers ''n''. Also, the angle measure of 180° is equal to π radians. In other words, 1° = (π/180) radians.

In modern mathematics, π is often ''defined'' using trigonometric functions, for example as the smallest positive ''x'' for which sin ''x'' = 0, to avoid unnecessary dependence on the subtleties of Euclidean geometry and integration. Equivalently, π can be defined using the [[inverse trigonometric function]]s, for example as π = 2 arccos(0) or π = 4 arctan(1). Expanding inverse trigonometric functions as [[power series]] is the easiest way to derive infinite series for π.

===Higher analysis and number theory===

[[Image:Euler's formula.svg|thumb|250px]]

The frequent appearance of π in [[complex analysis]] can be related to the behavior of the [[exponential function]] of a complex variable, described by [[Euler's formula]]

:<math>e^{i\varphi} = \cos \varphi + i\sin \varphi \!</math>

where ''i'' is the [[imaginary unit]] satisfying ''i''<sup>2</sup> = &minus;1 and ''e'' ≈ 2.71828 is [[E (mathematical constant)|Euler's number]]. This formula implies that imaginary powers of ''e'' describe rotations on the [[unit circle]] in the complex plane; these rotations have a period of 360° = 2π. In particular, the 180° rotation ''φ'' = π results in the remarkable [[Euler's identity]]

:<math>e^{i \pi} = -1.\!</math>

There are ''n'' different ''n''-th [[Root of unity|roots of unity]]
:<math>e^{2 \pi i k/n} \qquad (k = 0, 1, 2, \dots, n - 1).</math>

The [[Gaussian integral]]

:<math>\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}.</math>

A consequence is that the [[gamma function]] of a half-integer is a rational multiple of √π.
<!-- need some prose here on the zeta function and primes -->

===Physics===
Although not a [[physical constant]], ''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 system]]s. Using units such as [[Planck units]] can sometimes eliminate ''π'' from formulae.

*The [[cosmological constant]]:<ref>{{cite web|first=Cole|last=Miller|url=http://www.astro.umd.edu/~miller/teaching/astr422/lecture12.pdf|format=[[PDF]]|title=The Cosmological Constant|publisher=[[University of Maryland, College Park|University of Maryland]]|accessdate=2007-11-08}}</ref>
::<math>\Lambda = {{8\pi G} \over {3c^2}} \rho</math>
*[[Uncertainty principle|Heisenberg's uncertainty principle]], which shows that the uncertainty in the measurement of a particle's position (&Delta;''x'') and [[momentum]] (&Delta;''p'') can not both be arbitrarily small at the same time:<ref>{{cite web|first=James M|last=Imamura|url=http://zebu.uoregon.edu/~imamura/208/jan27/hup.html|title=Heisenberg Uncertainty Principle|publisher=[[University of Oregon]]|date=[[2005-08-17]]|accessdate=2007-11-09}}</ref>
::<math> \Delta x\, \Delta p \ge \frac{h}{4\pi} </math>
*[[Einstein field equations|Einstein's field equation]] of [[general relativity]]:<ref name = ein>{{cite journal| last = Einstein| first = Albert| authorlink = Albert Einstein | title = The Foundation of the General Theory of Relativity| journal = [[Annalen der Physik]] |date=1916| url = http://www.alberteinstein.info/gallery/gtext3.html| format = [[PDF]] | id = | accessdate = 2007-11-09 }}</ref>
::<math> R_{ik} - {g_{ik} R \over 2} + \Lambda g_{ik} = {8 \pi G \over c^4} T_{ik} </math>
*[[Coulomb's law]] for the [[Electric field|electric force]], describing the force between two [[electric charge]]s (''q<sub>1</sub>'' and ''q<sub>2</sub>'') separated by distance ''r'':<ref>
{{cite web|first=C. Rod|last=Nave|url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html#c3|title=Coulomb's Constant|work=[[HyperPhysics]]|publisher=[[Georgia State University]]|date=[[2005-06-28]]|accessdate=2007-11-09}}</ref>
::<math> F = \frac{\left|q_1q_2\right|}{4 \pi \varepsilon_0 r^2}</math>
*[[Magnetic constant|Magnetic permeability of free space]]:<ref>{{cite web |url=http://physics.nist.gov/cgi-bin/cuu/Value?mu0 |title=Magnetic constant |accessdate=2007-11-09 |date=2006 [[Committee on Data for Science and Technology|CODATA]] recommended values |publisher=[[National Institute of Standards and Technology|NIST]] }}</ref>
::<math> \mu_0 = 4 \pi \cdot 10^{-7}\,\mathrm{N/A^2}\,</math>
*[[Kepler's laws of planetary motion#Kepler's third law|Kepler's third law constant]], relating the [[orbital period]] (''P'') and the [[semimajor axis]] (''a'') to the [[mass]]es (''M'' and ''m'') of two co-orbiting bodies:
::<math>\frac{P^2}{a^3}={(2\pi)^2 \over G (M+m)} </math>

===Probability and statistics===
In [[probability]] and [[statistics]], there are many [[probability distribution|distributions]] whose formulas contain ''π'', including:
*the [[probability density function]] for the [[normal distribution]] with [[mean]] μ and [[standard deviation]] σ, due to the [[Gaussian integral]]:<ref>{{cite web|url=http://mathworld.wolfram.com/GaussianIntegral.html|title=Gaussian Integral|publisher=[[MathWorld]]|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|date=[[2004-10-07]]|accessdate=2007-11-08}}</ref>

:<math>f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-(x-\mu )^2/(2\sigma^2)}</math>
*the probability density function for the (standard) [[Cauchy distribution]]:<ref>{{cite web|url=http://mathworld.wolfram.com/CauchyDistribution.html|title=Cauchy Distribution|publisher=[[MathWorld]]|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|date=[[2005-10-11]]|accessdate=2007-11-08}}</ref>

:<math>f(x) = \frac{1}{\pi (1 + x^2)}.</math>

Note that since <math>\int_{-\infty}^{\infty} f(x)\,dx = 1</math> for any probability density function ''f''(''x''), the above formulas can be used to produce other integral formulas for ''π''.<ref>{{cite web|url=http://mathworld.wolfram.com/ProbabilityFunction.html|title=Probability Function|publisher=[[MathWorld]]|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|date=[[2003-07-02]]|accessdate=2007-11-08}}</ref>

[[Buffon's needle]] problem is sometimes quoted as a empirical approximation of ''π'' in "popular mathematics" works. Consider dropping a needle of length ''L'' repeatedly on a surface containing parallel lines drawn ''S'' units apart (with ''S''&nbsp;>&nbsp;''L''). If the needle is dropped ''n'' times and ''x'' of those times it comes to rest crossing a line (''x''&nbsp;>&nbsp;0), then one may approximate ''π'' using the [[Monte Carlo method]]:<ref name="bn">{{cite web|url=http://mathworld.wolfram.com/BuffonsNeedleProblem.html|title=Buffon's Needle Problem|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|publisher=[[MathWorld]]|date=[[2005-12-12]]|accessdate=2007-11-10}}</ref><ref>{{cite web|first=Alex|last=Bogomolny|url=http://www.cut-the-knot.org/ctk/August2001.shtml|title=Math Surprises: An Example|work=[[cut-the-knot]]|date=2001-08|accessdate=2007-10-28}}</ref><ref>{{cite journal|last = Ramaley|first = J. F.|title = Buffon's Noodle Problem|journal = The American Mathematical Monthly|volume = 76|issue = 8|date=Oct 1969|pages = 916–918|doi = 10.2307/2317945}}</ref><ref>{{cite web|url=http://www.datastructures.info/the-monte-carlo-algorithmmethod/|title=The Monte Carlo algorithm/method|work=datastructures|date=[[2007-01-09]]|accessdate=2007-11-07}}</ref>
:<math>\pi \approx \frac{2nL}{xS}.</math>
Though this result is mathematically impeccable, it cannot be used to determine more than very few digits of ''π'' ''by experiment''. Reliably getting just three digits (including the initial "3") right requires millions of throws,<ref name="bn"/> and the number of throws grows [[exponential growth|exponentially]] with the number of digits desired. Furthermore, any error in the measurement of the lengths ''L'' and ''S'' will transfer directly to an error in the approximated ''π''. For example, a difference of a single [[atom]] in the length of a 10-centimeter needle would show up around the 9th digit of the result. In practice, uncertainties in determining whether the needle actually crosses a line when it appears to exactly touch it will limit the attainable accuracy to much less than 9 digits.

==See also==
*[[List of topics related to π]]
*[[Proof that 22/7 exceeds π]]
*[[Feynman point]] &ndash; comprising the 762nd through 767th decimal places of π, consisting of the digit 9 repeated six times.
*[[Indiana Pi Bill]].
*[[Pi Day]].
*[[Software for calculating π]] on personal computers.
*[[Mathematical constant]]s: [[E (mathematical constant)|e]] and [[Golden ratio|φ]]
* [[Statistics Online Computational Resource|SOCR]] resource [http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_BuffonNeedleExperiment#Buffon.27s_needle_experiment_and_estimation_of_the_constant_.CF.80 hands-on activity for estimation of ''π'' using needle-dropping simulation].

== References ==

{{reflist|3}}

==External links==
{{commonscat}}
*[http://www.joyofpi.com The Joy of Pi by David Blatner]
*[http://www.research.att.com/~njas/sequences/A000796 Decimal expansions of Pi and related links] at the [[On-Line Encyclopedia of Integer Sequences]]
*[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]
*[http://mathworld.wolfram.com/PiFormulas.html Lots of formulas for ''π''] at [[MathWorld]]
*[http://planetmath.org/encyclopedia/Pi.html PlanetMath: Pi]
*[http://mathforum.org/isaac/problems/pi1.html Finding the value of ''π'']
*[http://www.cut-the-knot.org/pythagoras/NatureOfPi.shtml Determination of ''π''] at [[cut-the-knot]]
*[http://www.bbc.co.uk/radio4/science/5numbers2.shtml BBC Radio Program about ''π'']
*[http://www.super-computing.org/pi-decimal_current.html Statistical Distribution Information on PI] based on 1.2 trillion digits of PI
*[http://www.joyofpi.com/pi.html The Digits of Pi &mdash; First ten thousand]
*[http://www.zenwerx.com/pi.php First 4 Million Digits of ''π''] - ''Warning'' - Roughly 2 [[megabyte]]s will be transferred.
*[http://www.piday.org/million.php One million digits of pi at piday.org]
*[http://www.gutenberg.net/etext/50 Project Gutenberg E-Text containing a million digits of ''π'']
*[http://www.angio.net/pi/piquery Search the first 200 million digits of ''π'' for arbitrary strings of numbers]
*[http://www.codecodex.com/wiki/index.php?title=Digits_of_pi_calculation Source code for calculating the digits of ''π'']
*[http://www.math.utah.edu/~palais/pi.pdf π is Wrong! An opinion column on why 2π is more useful in mathematics.]
*[http://ja0hxv.calico.jp/pai/estart.html 70 Billion digits of Pi(π) downloads.]
*[http://filebin.ca/nastsa/pi_data.txt The first 16 million digits of Pi] (18 mb .txt file)

[[Category:Pi| ]]
[[Category:Transcendental numbers]]
[[Category:Mathematical constants]]
[[Category:Dimensionless numbers]]

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Revision as of 22:34, 12 October 2008

Untitled

Unsigned & Still Major: Da Album Before da Album, or simply "Unsigned & Still Major" as it is usually called, is the first studio album by American rapper Soulja Boy. It was released on February 19, 2007. (see 2007 in music) The title most likely refers to souljaboytellem.com, his second studio album which was released later that year.The album sold over 1,000 copies.

Tracklisting

  1. "I Got Me Some Bapes"
  2. "Stop Then Snap"
  3. "Crank Dat Dance"
  4. "Booty Meat"
  5. "Soulja Boy Ain't Got No Money"
  6. "I Got Me Some Bapes (Remix)"
  7. "Crank Dat Jumprope"
  8. "Look @ Me"
  9. "Bring Dat Beat Back"
  10. "Give Me a High Five"
  11. "I'm the Hustle Man"
  12. "Wuzhannanan"
  13. "Dominican Papi"

YouTube Singles

All music videos for songs shown here were made indepentantly using a camera and an editing program, and posted on YouTube.

Shootout (Now Known As Let Me Get Em') (Song not shown on album)

  • An Indepently made music video was created by Soulja Boy for his youtube account and posted on June 27, 2006. In the music video, soulja boy is seen talking on his cell phone in a car, dancing on roofs and ground, and has closeups on his feet while dancing.[1]

I Got Me Some Bapes (Now Known as Bapes)

  • I Got Me Some Bapes is also another independant music video made by soulja boy but more popular. The music video was released to youtube on February 1st, 2007. In the beginning of the music video shows the famous scene of where Soulja Boy says: "Arab I just got back from the mall man! Guess what I got?", Arab: "What did you do soulja boy?", Soulja Boy Man I Got Me Some bathin' apes! In the music video, soulja boy is partnered by arab beside a car with many bape products on it and shows amny other clips from his other two videos about him seeing ugly bathin' ape shoes in the mall and one where he finally gets his first pair.

Soulja Boy is also seen destroying other shoes with arab that aren't bathin' apes. Soulja boy is also shown in the mall with friends snap dancing. The end of the music video shows soulja boy riding in to his lawn. [2]

Stub icon

This hip hop album-related article is a stub. You can help Wikipedia by expanding it.

  1. ^ http://www.youtube.com/watch?v=4MBoTTHHzrg
  2. ^ http://www.youtube.com/watch?v=KEn3_kMAIrI
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