- Numerical approximations of π
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 πfor other aspects of the evolution of our knowledge about mathematical properties of π.
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
Ahmeswrote the oldest known text to give an approximate value for π. The Rhind Mathematical Papyrusdates 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 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.
Indian astronomer Yajnavalkyagave 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,
Archimedesproved 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 the arctangent function leads to a simple modern proof that indeed frac|3|1|7 exceeds π.) Later, in the second century AD, Ptolemyusing 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 Huiin 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.
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):
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). See Proof that π is irrationalfor an elementary 20th-century proof.
14th century, the Indian mathematician and astronomer Madhava of Sangamagrama, founder of the Kerala 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
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
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 9 sexagesimaldigits. ["Al-Kashi", author: Adolf P. Youschkevitch, chief editor: Boris A. Rosenfeld, p. 256] This figure is equivalent to 16 decimal digits as
which equates to
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 Vegain 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, when William Rutherfordcalculated 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.
Gauss-Legendre algorithmis used for calculating digits of π.
In 1910, the Indian mathematician
Srinivasa Ramanujanfound several rapidly converging infinite series of π, including
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
πwere done with the help of calculatorsor 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, an expansion of π to 100,265 decimal placesRp|78 was computed by Maryland mathematician Dr.
Daniel Shanksand 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 brotherscorrectly computed π to over a billion decimal places on the supercomputer IBM 3090using the following variation of Ramanujan's infinite series of π:
Yasumasa Kanadaand his team at the University of Tokyocorrectly 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. [ [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.
It is often claimed that the
Biblestates 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 Jerusalemas having a diameter of 10 cubits and a circumference of 30 cubits. Rabbi Nehemiahexplained this in his "Mishnat ha-Middot" (the earliest known Hebrewtext 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
Hebrewstended 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
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
For fast calculations, one may use formulæ such as Machin's:
together with the
Taylor seriesexpansion of the function arctan("x"). This formula is most easily verified using polar coordinatesof complex numbers, starting with
Another example is::
Formulæ of this kind are known as "
Other classical formulae
Other formulæ that have been used to compute estimates of π include:: :Liu Hui
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:
::David Chudnovsky and
Many other expressions for π were developed and published by the incredibly intuitive Indian mathematician
Srinivasa Ramanujan. He worked with mathematician G. H. Hardyin England for a number of years.
Extremely long decimal expansions of π are typically computed with iterative formulae like the
Gauss–Legendre algorithmand Borwein's algorithm. The Salamin–Brent algorithmwhich 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 and:where , the sequence 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) by Yasumasa Kanadaof Tokyo Universitystands at 1,241,100,000,000 digits, which were computed in September 2002 on a 64-node Hitachi supercomputerwith 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits). The following Machin-like formulæ were used for this:
::K. Takano (1982).
: (F. C. W. Störmer (1896)).
These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers. (Normality of π will always depend on the infinite string of digits on the end, not on any finite computation.)
Formulae for binary digits
David H. Bailey, Peter Borweinand Simon Plouffepublished a paper (Bailey, 1997) on a new formula for π as an infinite series:
This formula permits one to easily compute the "k"th binary or
hexadecimaldigit 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 various programming languages. The PiHexproject computed 64-bits around the quadrillionth bit of π (which turns out to be 0).
"π" and a fractal
A feature of the
Mandelbrot setrecently 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
where the complex number
where is the non-zero precision required, and counts the number "N" of iterations until
i.e. stop when an iteration hits the circle radius 2 in the complex plane. The result is simply the approximation
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).
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
sexagesimaldigit 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:::
*accurate to 9 places::: : This is from
Ramanujan, who claimed the goddess Namagiriappeared to him in a dream and told him the true value of π.Fact|date=February 2007
*Another approximation by Ramanujan is the following:::
* accurate to 4 digits:::
* accurate to 3 digits::: [A nested radical approximation for pi. [http://www.mschneider.cc/papers/pi.pdf] ]
* accurate to 3 digits::: :
Karl Popperconjectured that Platoknew this expression, that he believed it to be exactly π, and that this is responsible for some of Plato's confidence in the omnicompetenceof mathematical geometry — and Plato's repeated discussion of special right triangles that are either isoscelesor halves of equilateraltriangles.
* accurate to 7 decimal places:::
* accurate to 10 decimal places:::
continued fractionrepresentation 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:
List of formulae involving π
* 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.
title = The Crest of the Peacock: Non-European Roots of Mathematics
edition=New ed., London : Penguin
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