- Numerical approximations of π
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**numerical approximations of the**. There is a summarizing table atmathematical constant π chronology of computation of π . See alsohistory of π for other aspects of the evolution of our knowledge about mathematical properties of π.**Early history**Believed to be built during the Fourth Dynasty of Egypt's

Old Kingdom , the Great Pyramid was constructed with an approximate ratio of height to circumference of the base of 2π. Each side is 440 cubits long, and the height is believed to have been 280 cubits tall at the time of its construction. This puts the value at approximately 3.142, or 0.04% above the exact value.An Egyptian scribe named

Ahmes wrote the oldest known text to give an approximate value for π. TheRhind Mathematical Papyrus dates from the EgyptianSecond Intermediate Period — though Ahmes stated that he copied a Middle Kingdompapyrus (i.e. from before 1650 BC) — and describes the value in such a way that the result obtained comes out to frac|256|81, which is approximately 3.16, or 0.6% above the exact value.As early as the 19th century BC, Babylonian mathematicians were using π ≈ frac|25|8, which is about 0.5% below the exact value.

The

India n astronomerYajnavalkya gave astronomical calculations in the "Shatapatha Brahmana " (c.9th century BC ) that led to a fractional approximation of π ≈ frac|339|108 (which equals 3.13888…, which is correct to two decimal places when rounded, or 0.09% below the exact value).In the third century BC,

Archimedes proved the sharp inequalities frac|223|71 < π < frac|22|7, by means of regular 96-gons; these values are 0.02% and 0.04% off, respectively. (Differentiating thearctan gent function leads to a simple modern proof that indeed frac|3|1|7 exceeds π.) Later, in the second century AD,Ptolemy using a regular 360-gon obtained a value of 3.141666....Fact|date=June 2008 which is correct to three decimal places.The Chinese mathematician

Liu Hui in 263 AD computed π with to between 3.141024 and 3.142708 with inscribe 96-gon and 192-gon; the average of these two values = 3.141864, less than 0.01% off. However, he suggested that 3.14 was a good enough approximation for practical purpose. Later he obtained a more accurate result "π" = 3927/1250 = 3.1416.**Middle ages**Until 1000, π was known to fewer than 10

decimal digit s only.The Indian mathematician and astronomer,

Aryabhata , in the5th century , gave an accurate approximation for π, and may have realized that π is irrational. He writes, in the second part of the Aryabhatiyam (gaṇitapāda 10):quotation|chaturadhikam śatamaśṭaguṇam dvāśaśṭistathā sahasrāṇām Ayutadvayaviśkambhasyāsanno vrîttapariṇahaḥ.

meaning "Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given."

In other words (4+100)×8 + 62000 is the circumference of a circle with diameter 20000. This provides a value of π ≈ frac|62832|20000 = 3.1416, correct to three decimal places. The commentator

Nilakantha Somayaji , (Kerala school of astronomy and mathematics , 15th century) has argued that the word āsanna (approaching), appearing just before the last word, here means not only that this is an approximation, but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, for the irrationality of pi was proved in Europe only in 1761 (Lambert). SeeProof that π is irrational for an elementary 20th-century proof.The Chinese mathematician and astronomer,

Zu Chongzhi , in the5th century , computed π between 3.1415926 and 3.1415927, which was correct to 7 decimal places. He gave two other approximations of $pi\; approxeq\; frac\{22\}\{7\}$ and $pi\; approxeq\; frac\{355\}\{113\}$.In the

14th century , the Indian mathematician and astronomerMadhava of Sangamagrama , founder of theKerala school of astronomy and mathematics , discovered the infinite series for π, now known as the Madhava-Leibniz series, [*citation|title=Special Functions|last=George E. Andrews, Richard Askey|first=Ranjan Roy|publisher=*] [Cambridge University Press |year=1999|isbn=0521789885|page=58*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*] and gave two methods for computing the value of π. One of these methods is to obtain a rapidly converging series by transforming the original infinite series of π. By doing so, he obtained the infinite series::$pi\; =\; sqrt\{12\}left(1-\{1over\; 3cdot3\}+\{1over5cdot\; 3^2\}-\{1over7cdot\; 3^3\}+cdots\; ight)$

and used the first 21 terms to compute an approximation of π correct to 11 decimal places as 3.14159265359.

The other method he used was to add a remainder term to the original series of π. He used the remainder term

::$frac\{n^2\; +\; 1\}\{4n^3\; +\; 5n\}$

in the infinite series expansion of frac|π|4 to improve the approximation of π to 13 decimal places of accuracy when "n" = 75.

The Persian mathematician and astronomer,

Ghyath ad-din Jamshid Kashani (1380–1429), correctly computed 2π to 9sexagesimal digits. [*"Al-Kashi", author: Adolf P. Youschkevitch, chief editor: Boris A. Rosenfeld, p. 256*] This figure is equivalent to 16 decimal digits as::$2pi\; =\; 6.2831853071795865$

which equates to

::$pi\; =\; 3.14159265358979325.$

He achieved this level of accuracy by calculating the perimeter of a regular polygon with 3 × 2e|18 sides.Fact|date=February 2007

**16th to 19th centuries**The German mathematician

Ludolph van Ceulen ("circa" 1600) computed the first 35 decimal places of π. He was so proud of this accomplishment that he had them inscribed on his tombstone.The Slovene mathematician

Jurij Vega in 1789 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, whenWilliam Rutherford calculated 208 decimal places of which the first 152 were correct. Vega improvedJohn Machin 's formula from 1706 and his method is still mentioned today.The English amateur mathematician

William Shanks , a man of independent means, spent over 20 years calculating π to 707 decimal places (accomplished in 1873). His routine was as follows: he would calculate new digits all morning; and then he would spend all afternoon checking his morning's work. His work was made possible by the recent invention of the logarithm and its tables by Napier and Briggs. This was the longest expansion of π until the advent of the electronic digital computer a century later.The

Gauss-Legendre algorithm is used for calculating digits of π.**20th century**In 1910, the Indian mathematician

Srinivasa Ramanujan found several rapidly converging infinite series of π, including:$frac\{1\}\{pi\}\; =\; frac\{2sqrt\{2\{9801\}\; sum^infty\_\{k=0\}\; frac\{(4k)!(1103+26390k)\}\{(k!)^4\; 396^\{4k$

which computes a further 8 decimal places of π with each term in the series. His series are now the basis for the fastest algorithms currently used to calculate π.

From the mid-20th century onwards, all calculations of

π were done with the help ofcalculators orcomputers .In 1944, D. F. Ferguson, with the aid of a mechanical

desk calculator , found that William Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were incorrect.In the early years of the computer, an expansion of π to 100,265 decimal placesRp|78 was computed by Maryland mathematician Dr.

Daniel Shanks and his team at the United States Naval Research Laboratory (N.R.L.) in Washington, D.C. (Dr. Shanks's son Oliver Shanks, also a mathematician, states that there is no connection to William Shanks, and the family's roots are in Central EuropeFact|date=June 2008).In 1961, 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 N.R.L.citation|first1=D.|last1=Shanks|first2=J. W.|last2=Wrench, Jr.|title=Calculation of pi to 100,000 decimals|journal=Mathematics of Computation|volume=16|year=1962|pages=76-99.] Rp|80–99 The authors outlined what would be needed to calculate π to 1,000,000 decimal places and concluded that the task was beyond that day's technology, but would be possible in 5 to 7 years. Rp|78

In 1989, the

Chudnovsky brothers correctly computed π to over a billion decimal places on thesupercomputer IBM 3090 using the following variation of Ramanujan's infinite series of π::$frac\{1\}\{pi\}\; =\; 12\; sum^infty\_\{k=0\}\; frac\{(-1)^k\; (6k)!\; (13591409\; +\; 545140134k)\}\{(3k)!(k!)^3\; 640320^\{3k\; +\; 3/2.$

In 1999,

Yasumasa Kanada and his team at theUniversity of Tokyo correctly computed π to over 200 billion decimal places on the supercomputerHITACHI SR8000/MPP (128 nodes) using another variation of Ramanujan's infinite series of π. In October 2005 they claimed to have calculated it to 1.24 trillion places. [*[*]*http://www.super-computing.org/pi_current.html Announcement*] at the Kanada lab web site.**Less accurate approximations**Some approximations which have been given for π are notable in that they were less precise than previously known values.

**Biblical value**It is often claimed that the

Bible states that π is exactly 3, based on a passage in 1 Kings 7:23 (ca. 971-852 BCE) and 2 Chronicles 4:2 giving measurements for the round basin located in front of theTemple in Jerusalem as having a diameter of 10cubit s and a circumference of 30 cubits.Rabbi Nehemiah explained this in his "Mishnat ha-Middot" (the earliest knownHebrew text ongeometry , ca. 150 CE) by saying that the diameter was measured from the "outside" of the brim while the circumference was measured along the "inner" rim. The stated dimensions would be exact if measured this way on a brim about four inches wide.This is disputed, however, and other explanations have been offered, including that the measurements are given in round numbers (as the

Hebrews tended to round off measurements to whole numbers), that cubits were not exact units, or that the basin may not have been exactly circular, or that the brim was wider than the bowl itself. Many reconstructions of the basin show a wider brim extending outward from the bowl itself by several inches. [*[*]*http://mathforum.org/library/drmath/view/52573.html Math Forum - Ask Dr. Math*]**The Indiana bill**The "

Indiana Pi Bill " of 1897, which never passed out of committee, has been claimed to imply a number of different values for π, although the closest it comes to explicitly asserting one is the wording "the ratio of the diameter and circumference is as five-fourths to four", which would make π = frac|16|5 = 3.2.**Development of efficient formula****Machin-like formulae**For fast calculations, one may use formulæ such as Machin's:

: $frac\{pi\}\{4\}\; =\; 4\; arctanfrac\{1\}\{5\}\; -\; arctanfrac\{1\}\{239\}$

together with the

Taylor series expansion of the functionarctan ("x"). This formula is most easily verified usingpolar coordinates ofcomplex number s, starting with:$(5+i)^4cdot(-239+i)=-114244-114244i.$

Another example is:: $frac\{pi\}\{4\}\; =\; arctanfrac\{1\}\{2\}\; +\; arctanfrac\{1\}\{3\}$

Formulæ of this kind are known as "

Machin-like formula e".**Other classical formulae**Other formulæ that have been used to compute estimates of π include:: $egin\{align\}pi\; approxeq\; 768\; sqrt\{2\; -\; sqrt\{2\; +\; sqrt\{2\; +\; sqrt\{2\; +\; sqrt\{2\; +\; sqrt\{2\; +\; sqrt\{2\; +\; sqrt\{2\; +\; sqrt\{2+1\}approxeq\; 3.141590463236763.end\{align\}$:Liu Hui

:$pi\; =\; sqrt\{12\}left(1-\{1over\; 3cdot3\}+\{1over5cdot\; 3^2\}-\{1over7cdot\; 3^3\}+cdots\; ight)$:Madhava.

:$\{pi\}\; =\; 20\; arctanfrac\{1\}\{7\}\; +\; 8\; arctanfrac\{3\}\{79\}$:

Euler .:$frac\{pi\}\{2\}=sum\_\{k=0\}^inftyfrac\{k!\}\{(2k+1)!!\}=1+frac\{1\}\{3\}left(1+frac\{2\}\{5\}left(1+frac\{3\}\{7\}left(1+frac\{4\}\{9\}(1+cdots)\; ight)\; ight)\; ight)$:Newton.

:$frac\{1\}\{pi\}\; =\; frac\{2sqrt\{2\{9801\}\; sum^infty\_\{k=0\}\; frac\{(4k)!(1103+26390k)\}\{(k!)^4\; 396^\{4k$:

Ramanujan .This converges extraordinarily rapidly. Ramanujan's work is the basis for the

Chudnovsky algorithm , the fastest algorithms used, as of the turn of the millennium, to calculate π; it is based on::$frac\{1\}\{pi\}\; =\; 12\; sum^infty\_\{k=0\}\; frac\{(-1)^k\; (6k)!\; (13591409\; +\; 545140134k)\}\{(3k)!(k!)^3\; 640320^\{3k\; +\; 3/2$:David Chudnovsky and

Gregory Chudnovsky .Many other expressions for π were developed and published by the incredibly intuitive Indian mathematician

Srinivasa Ramanujan . He worked with mathematicianG. H. Hardy in England for a number of years.**Modern algorithms**Extremely long decimal expansions of π are typically computed with iterative formulae like the

Gauss–Legendre algorithm andBorwein's algorithm . TheSalamin–Brent algorithm which was invented in 1976 is an example of the former.Borwein's algorithm , found in 1985 by Jonathan andPeter Borwein , converges extremely fast:For $y\_0=sqrt2-1,\; a\_0=6-4sqrt2$ and:$y\_\{k+1\}=(1-f(y\_k))/(1+f(y\_k))\; ~,~\; a\_\{k+1\}\; =\; a\_k(1+y\_\{k+1\})^4\; -\; 2^\{2k+3\}\; y\_\{k+1\}(1+y\_\{k+1\}+y\_\{k+1\}^2)$where $f(y)=(1-y^4)^\{1/4\}$, the sequence $1/a\_k$ converges quartically to π, giving about 100 digits in three steps and over a trillion digits after 20 steps.The first one million digits of π and frac|1|π are available from

Project Gutenberg (see external links below). The current record (December 2002) byYasumasa Kanada ofTokyo University stands at 1,241,100,000,000 digits, which were computed in September 2002 on a 64-node Hitachisupercomputer 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::$frac\{pi\}\{4\}\; =\; 12\; arctanfrac\{1\}\{49\}\; +\; 32\; arctanfrac\{1\}\{57\}\; -\; 5\; arctanfrac\{1\}\{239\}\; +\; 12\; arctanfrac\{1\}\{110443\}$:K. Takano (1982).

: $frac\{pi\}\{4\}\; =\; 44\; arctanfrac\{1\}\{57\}\; +\; 7\; arctanfrac\{1\}\{239\}\; -\; 12\; arctanfrac\{1\}\{682\}\; +\; 24\; arctanfrac\{1\}\{12943\}$ (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.)

**Formulae for binary digits**In 1997,

David H. Bailey ,Peter Borwein andSimon Plouffe published a paper (Bailey, 1997) on a new formula for π as aninfinite series :: $pi\; =\; sum\_\{k\; =\; 0\}^\{infty\}\; frac\{1\}\{16^k\}left(\; frac\{4\}\{8k\; +\; 1\}\; -\; frac\{2\}\{8k\; +\; 4\}\; -\; frac\{1\}\{8k\; +\; 5\}\; -\; frac\{1\}\{8k\; +\; 6\}\; ight).$

This formula permits one to easily compute the "k"

^{th}binary orhexadecimal digit of π, without having to compute the preceding "k" − 1 digits. [*http://www.nersc.gov/~dhbailey/ Bailey's website*] contains the derivation as well as implementations in variousprogramming language s. ThePiHex project computed 64-bits around the quadrillionth bit of π (which turns out to be 0).**"π" and a fractal**A feature of the

Mandelbrot set recently reported [*cite web|url=https://home.comcast.net/~davejanelle/mandel.pdf|title=π in the Mandelbrot set*] gives a means to calculate an approximate value of "π" to any chosen accuracy, without seeking the limit of an infinite series. One applies the iteration:$z\; mapsto\; z^2\; +\; c$

where the complex number

:$c\; =\; -0.75\; +\; i\; e$

where $e$ is the non-zero precision required, and counts the number "N" of iterations until

:$(Re(z)^2\; +\; Im(z)^2)\; geqq\; 4\; ,$

i.e. stop when an iteration hits the circle radius 2 in the complex plane. The result is simply the approximation

:$pi\; =\; N\; e\; pm\; e\; ,$

In the graphical view one starts iterating from a point just above the bottom of the "seahorse valley" of the Mandelbrot set at (−0.75, 0).

**Miscellaneous formulæ**Historically, for a long time the base 60 was used for calculations. In this base, π can be approximated to eight (decimal!) significant figures as

:$3\; +\; frac\{8\}\{60\}\; +\; frac\{29\}\{60^2\}\; +\; frac\{44\}\{60^3\}$

(The next

sexagesimal digit is 0, causing truncation here to yield a relatively good approximation.)In addition, the following expressions can be used to estimate π:

* accurate to 9 digits:::$frac\{63\}\{25\}\; imes\; frac\{17\; +\; 15sqrt\{5\{7\; +\; 15sqrt\{5$

*accurate to 9 places::: $sqrt\; [4]\; \{frac\{2143\}\{22$: This is from

Ramanujan , who claimed the goddessNamagiri appeared to him in a dream and told him the true value of π.Fact|date=February 2007*Another approximation by Ramanujan is the following:::$frac\{9\}\{5\}+sqrt\{frac\{9\}\{5$

* accurate to 4 digits:::$sqrt\; [3]\; \{31\}$

* accurate to 3 digits:::$sqrt\{7+sqrt\{6+sqrt\{5\}$ [

*A nested radical approximation for pi. [*]*http://www.mschneider.cc/papers/pi.pdf*]* accurate to 3 digits::: $sqrt\{2\}\; +\; sqrt\{3\}$:

Karl Popper conjectured thatPlato knew this expression, that he believed it to be exactly π, and that this is responsible for some of Plato's confidence in theomnicompetence of mathematical geometry — and Plato's repeated discussion of specialright triangle s that are eitherisosceles or halves ofequilateral triangles.* accurate to 7 decimal places:::$frac\{54648\}\{17395\}$

* accurate to 10 decimal places:::$frac\{833009\}\{265155\}$

* The

continued fraction representation of π can be used to generate successive best rational approximations. These approximations are the best possible rational approximations of π relative to the size of their denominators. Here is a list of the first of these, punctuated at significant steps:

*:$frac\{3\}\{1\},quad\; frac\{13\}\{4\},frac\{16\}\{5\},frac\{19\}\{6\},frac\{22\}\{7\},quad\; frac\{179\}\{57\},frac\{201\}\{64\},frac\{223\}\{71\},frac\{245\}\{78\},frac\{267\}\{85\},frac\{289\}\{92\},frac\{311\}\{99\},frac\{333\}\{106\},quad\; frac\{355\}\{113\},quad\; frac\{52163\}\{16604\}$**ee also***

List of formulae involving π **Notes****References*** cite journal

author = Bailey, David H., Borwein, Peter B., and 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

doi = 10.1090/S0025-5718-97-00856-9

* cite book

author = Joseph, George G.

year =2000

title = The Crest of the Peacock: Non-European Roots of Mathematics

edition=New ed., London : Penguin

id=ISBN 0-14-027778-1

*Wikimedia Foundation.
2010.*

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