Numerical approximations of π

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Best-known estimates of the value of π over the centuries

This page is about the history of numerical approximations of the mathematical constant π. There is a summarizing table at chronology of computation of π. See also history of π for other aspects of the evolution of our knowledge about mathematical properties of π.

1 Early history
2 Middle ages
3 16th to 19th centuries
4 20th century
5 Less accurate approximations
- 5.1 Biblical value
- 5.2 The Indiana bill
6 Development of efficient formula
7 See also
8 Notes
9 References

[edit] Early history

An Egyptian scribe named Ahmes wrote the oldest known text to give an approximate value for π. The Rhind Mathematical Papyrus dates from the Egyptian Second Intermediate Period — though Ahmes stated that he copied a Middle Kingdom papyrus (i.e. from before 1650 BC) — and describes the value in such a way that the result obtained comes out to 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 ≈ ²⁵⁄₈, which is about 0.5% below the exact value.

The Indian astronomer Yajnavalkya gave astronomical calculations in the Shatapatha Brahmana (c. 9th century BC) that led to a fractional approximation of π ≈ 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 ²²³⁄₇₁ < π < ²²⁄₇, by means of regular 96-gons; these values are 0.02% and 0.04% off, respectively. (Differentiating the arctangent function leads to a simple modern proof that indeed 3+¹⁄₇ exceeds π.) Later, in the second century AD, Ptolemy using a regular 360-gon obtained a value of 3.141666....^{[citation needed]} 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 96gon and 192gon; the average of this two value =3.141864, less then 0.01% off. However, he suggested that 3.14 was a good enough approximation for practical purpose. Later he obtained a more accurate result $\pi={3927 \over 1250}=3.1416$ .

[edit] Middle ages

Until 1000, π was known to fewer than 10 decimal digits only.

The Indian mathematician and astronomer Aryabhata in the 5th 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):

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 ≈ ⁶²⁸³²⁄₂₀₀₀₀ = 3.1416, correct to three decimal places. The commentator Nilakantha Somayaji, (Kerala School, 15th c.) 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). See Proof that π is irrational for an elementary 20th-century proof.

The Chinese mathematician and astronomer Zu Chongzhi in the 5th 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 astronomer Madhava of Sangamagrama 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

$\pi = \sqrt{12}\left(1-{1\over 3\cdot3}+{1\over5\cdot 3^2}-{1\over7\cdot 3^3}+\cdots\right)$

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 ^π⁄₄ to improve the approximation of π to 13 decimal places of accuracy when n = 75.

The Persian Muslim mathematician and astronomer Ghyath ad-din Jamshid Kashani (1380–1429) correctly computed 2π to 9 sexagesimal digits.^[1] This figure is equivalent to 16 decimal digits as

$\ 2\pi = 6.2831853071795865$

which equates to

$\ \pi = 3.14159265358979325.$

He achieved this level of accuracy by calculating the perimeter of a regular polygon with 3 × 2×10¹⁸ sides.^{[citation needed]}

[edit] 16th to 19th centuries

The German mathematician Ludolph van Ceulen (circa 1600) computed the first 32 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 [2] and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct. Vega improved John 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.

[edit] 20th century

In 1910, the Indian mathematician Srinivasa Ramanujan found several rapidly converging infinite series of π, including

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

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 of calculators or computers.

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, the first expansion of π to 1,000,000 decimal places was computed by Maryland mathematician Dr. Daniel Shanks and his team at the United States Naval Research Laboratory (N.R.L.) in 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 Europe).

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.^[2]

In 1989, the Chudnovsky brothers correctly computed π to over a billion decimal places on the supercomputer 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 the University of Tokyo correctly computed π to over 200 billion decimal places on the supercomputer HITACHI 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.^[3]

[edit] Less accurate approximations

Some approximations which have been given for π are notable in that they were less precise than previously known values.

[edit] 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 the Temple in Jerusalem as having a diameter of 10 cubits and a circumference of 30 cubits. Rabbi Nehemiah explained this in his Mishnat ha-Middot (the earliest known Hebrew text on geometry, 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. ^[4]

[edit] 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 = ¹⁶⁄₅ = 3.2.

[edit] Development of efficient formula

[edit] Machin-like formulae

For fast calculations, one may use formulæ such as Machin's:

$\frac{\pi}{4} = 4 \arctan\frac{1}{5} - \arctan\frac{1}{239}$

together with the Taylor series expansion of the function arctan(x). This formula is most easily verified using polar coordinates of complex numbers, starting with

$(5+i)^4\cdot(-239+i)=-114244-114244i.$

Another example is:

$\frac{\pi}{4} = \arctan\frac{1}{2} + \arctan\frac{1}{3}$

Formulæ of this kind are known as Machin-like formulae.

[edit] Other classical formulae

Other formulæ that have been used to compute estimates of π include:

$\begin{align} \pi \approxeq 768 \sqrt{2 - \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2+1}}}}}}}}} \approxeq 3.141590463236763. \end{align}$

Liu Hui

$\pi = \sqrt{12}\left(1-{1\over 3\cdot3}+{1\over5\cdot 3^2}-{1\over7\cdot 3^3}+\cdots\right)$

Madhava.

${\pi} = 20 \arctan\frac{1}{7} + 8 \arctan\frac{3}{79}$

Euler.

$\frac{\pi}{2}= \sum_{k=0}^\infty\frac{k!}{(2k+1)!!}= 1+\frac{1}{3}\left(1+\frac{2}{5}\left(1+\frac{3}{7}\left(1+\frac{4}{9}(1+\cdots)\right)\right)\right)$

Newton.

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

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 mathematician G. H. Hardy in England for a number of years.

[edit] Modern algorithms

Extremely long decimal expansions of π are typically computed with iterative formulae like the Gauss-Legendre algorithm and Borwein's algorithm. The Salamin-Brent algorithm which was invented in 1976 is an example of the former.

Borwein's algorithm, found in 1985 by Jonathan and Peter Borwein, converges extremely fast: For $y_0=\sqrt2-1,\ a_0=6-4\sqrt2$ 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 ¹⁄_π are available from Project Gutenberg (see external links below). The current record (December 2002) by Yasumasa Kanada of Tokyo University stands at 1,241,100,000,000 digits, which were computed in September 2002 on a 64-node Hitachi supercomputer with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits). The following Machin-like formulæ were used for this:

$\frac{\pi}{4} = 12 \arctan\frac{1}{49} + 32 \arctan\frac{1}{57} - 5 \arctan\frac{1}{239} + 12 \arctan\frac{1}{110443}$

K. Takano (1982).

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

[edit] Formulae for binary digits

Main article: Bailey-Borwein-Plouffe formula

In 1997, David H. Bailey, Peter Borwein and Simon Plouffe published a paper (Bailey, 1997) on a new formula for π as an infinite series:

$\pi = \sum_{k = 0}^{\infty} \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6}\right).$

This formula permits one to easily compute the k^th binary or hexadecimal digit of π, without having to compute the preceding k − 1 digits. Bailey's website contains the derivation as well as implementations in various programming languages. The PiHex project computed 64-bits around the quadrillionth bit of π (which turns out to be 0).

[edit] 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:

$\frac{63}{25} \times \frac{17 + 15\sqrt{5}}{7 + 15\sqrt{5}}$

accurate to 9 places:

$\sqrt[4]{\frac{2143}{22}}$

This is from Ramanujan, who claimed the goddess Namagiri appeared to him in a dream and told him the true value of π.^{[citation needed]}

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:

$\sqrt{7+\sqrt{6+\sqrt{5}}}$ ^[5]

accurate to 3 digits:

$\sqrt{2} + \sqrt{3}$

Karl Popper conjectured that Plato knew this expression, that he believed it to be exactly π, and that this is responsible for some of Plato's confidence in the omnicompetence of mathematical geometry — and Plato's repeated discussion of special right triangles that are either isosceles or halves of equilateral triangles.

The continued fraction representation of π can be used to generate 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:

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

[edit] See also

List of formulae involving π

[edit] Notes

^ Al-Kashi, author: Adolf P. Youschkevitch, chief editor: Boris A. Rosenfeld, p. 256
^ Shank, D. & Wrench, Jr., J. W. (1962), “Calculation of pi to 100,000 decimals”, Mathematics of Computation 16: 76-99 .
^ Announcement at the Kanada lab web site.
^ Math Forum - Ask Dr. Math
^ A nested radical approximation for pi. [1]

[edit] References

Bailey, David H., Borwein, Peter B., and Plouffe, Simon (April 1997). "On the Rapid Computation of Various Polylogarithmic Constants". Mathematics of Computation 66 (218): 903–913. doi:10.1090/S0025-5718-97-00856-9.
Joseph, George G. (2000). The Crest of the Peacock: Non-European Roots of Mathematics, New ed., London : Penguin. ISBN 0-14-027778-1.

Categories: History of mathematics | Pi

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