A Real Number denoted which is defined as the ratio of a Circle's Circumference to its
Diameter ,
(1) 
(2) 
The Simple Continued Fraction for , which gives the ``best'' approximation of a given order, is [3, 7, 15, 1, 292, 1, 1,
1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...] (Sloane's A001203). The very large term 292 means that the Convergent
(3) 
Gosper has computed 17,001,303 terms of 's Continued Fraction (Gosper 1977, Ball and Coxeter 1987), although the computer on which the numbers are stored may no longer be functional (Gosper, pers. comm., 1998). According to Gosper, a typical Continued Fraction term carries only slightly more significance than a decimal Digit. The sequence of increasing terms in the Continued Fraction is 3, 7, 15, 292, 436, 20776, ... (Sloane's A033089), occurring at positions 1, 2, 3, 5, 308, 432, ... (Sloane's A033090). In the first 26,491 terms of the Continued Fraction (counting 3 as the 0th), the only fiveDigit terms are 20,776 (the 431st), 19,055 (15,543rd), and 19,308 (23,398th) (Beeler et al. 1972, Item 140). The first 6Digit term is 528,210 (the 267,314th), and the first 8Digit term is 12,996,958 (453,294th). The term having the largest known value is the whopping 9Digit 87,878,3625 (the 11,504,931st term).
The Simple Continued Fraction for does not show any obvious patterns, but clear patterns do emerge in the
beautiful nonsimple Continued Fractions
(4) 
(5) 
crops up in all sorts of unexpected places in mathematics besides Circles and Spheres. For example, it occurs in the normalization of the Gaussian Distribution, in the distribution of Primes, in the construction of numbers which are very close to Integers (the Ramanujan Constant), and in the probability that a pin dropped on a set of Parallel lines intersects a line (Buffon's Needle Problem). Pi also appears as the average ratio of the actual length and the direct distance between source and mouth in a meandering river (Støllum 1996, Singh 1997).
A brief history of Notation for pi is given by Castellanos (1988). is sometimes known as Ludolph's Constant
after Ludolph van Ceulen (15391610), a Dutch calculator. The symbol was first used by William Jones in 1706, and
subsequently adopted by Euler. In Measurement of a Circle, Archimedes (ca. 225
BC ) obtained the first rigorous approximation by Inscribing and
Circumscribing gons on a Circle using the Archimedes Algorithm. Using
(a 96gon), Archimedes obtained
(6) 
The Bible contains two references (I Kings 7:23 and Chronicles 4:2) which give a value of 3 for . It should be mentioned, however, that both instances refer to a value obtained from physical measurements and, as such, are probably well within the bounds of experimental uncertainty. I Kings 7:23 states, ``Also he made a molten sea of ten Cubits from brim to brim, round in compass, and five cubits in height thereof; and a line thirty cubits did compass it round about.'' This implies . The Babylonians gave an estimate of as . The Egyptians did better still, obtaining in the Rhind papyrus, and 22/7 elsewhere. The Chinese geometers, however, did best of all, rigorously deriving to 6 decimal places.
A method similar to Archimedes' can be used to estimate by starting with an gon and then
relating the Area of subsequent gons. Let be the Angle from the center of one of the
Polygon's segments,
(7) 
(8) 
(9) 
(10) 
(11) 
(12)  
(13) 
The Area and Circumference of the Unit Circle are given by
(14)  
(15) 
(16)  
(17) 
(18)  
(19) 
is known to be Irrational (Lambert 1761, Legendre 1794) and even Transcendental (Lindemann 1882). Incidentally, Lindemann's proof of the transcendence of also proved that the Geometric Problem of Antiquity known as Circle Squaring is impossible. A simplified, but still difficult, version of Lindemann's proof is given by Klein (1955).
It is also known that is not a Liouville Number (Mahler 1953). In 1974, M. Mignotte showed that
(20) 
(21) 
digit  
0  9,999  99,959  599,963,005  5,000,012,647 
1  10,137  99,758  600,033,260  4,999,986,263 
2  9,908  100,026  599,999,169  5,000,020,237 
3  10,025  100,229  600,000,243  4,999,914,405 
4  9,971  100,230  599,957,439  5,000,023,598 
5  10,026  100,359  600,017,176  4,999,991,499 
6  10,029  99,548  600,016,588  4,999,928,368 
7  10,025  99,800  600,009,044  5,000,014,860 
8  9,978  99,985  599,987,038  5,000,117,637 
9  9,902  100,106  600,017,038  4,999,990,486 
The digits of are also very uniformly distributed ( ), as shown in the following table.
digit  
0  4,999,969,955 
1  5,000,113,699 
2  4,999,987,893 
3  5,000,040,906 
4  4,999,985,863 
5  4,999,977,583 
6  4,999,990,916 
7  4,999,985,552 
8  4,999,881,183 
9  5,000,066,450 
It is not known if , , or are Irrational. However, it is known that they
cannot satisfy any Polynomial equation of degree with Integer Coefficients of
average size 10
satisfies the Inequality
(22) 
Beginning with any Positive Integer , round up to the nearest multiple of , then up to the nearest multiple
of , and so on, up to the nearest multiple of 1. Let denote the result. Then the ratio
(23) 
(24) 
A particular case of the Wallis Formula gives
(25) 
(26) 
(27) 
(28) 
(29) 
(30)  
(31) 
(32)  
(33) 
(34) 
(35) 
(36) 
(37) 
(38) 

(39) 
(40) 
(41) 
(42) 
(43)  
(44) 
(45) 
(46) 
(47)  
(48)  
(49) 
The best formula for Class Number 2 (largest discriminant ) is
(50) 
(51)  
(52)  
(53) 
(54) 
(55)  
(56)  
(57) 
This gives 50 digits per term. Borwein and Borwein (1993) have developed a general Algorithm for generating such
series for arbitrary Class Number. Bellard gives the exotic formula
(58) 
(59) 
A complete listing of Ramanujan's series for found in his second and third notebooks is given by Berndt (1994, pp. 352354),
(60)  
(61)  
(62)  
(63)  
(64)  
(65)  
(66)  
(67) 
(68)  
(69)  
(70)  
(71)  
(72)  
(73)  
(74)  
(75)  
(76) 
A Spigot Algorithm for is given by Rabinowitz and Wagon (1995). Amazingly, a closed form expression giving a digit extraction algorithm which produces digits of (or ) in base16 was recently discovered by Bailey et al. (Bailey et al. 1995, Adamchik and Wagon 1997),
(77) 
(78) 
(79) 
(80) 
(81) 
Another identity is
(82) 
(83) 
(84) 
(85) 
(86) 
A slew of additional identities due to Ramanujan , Catalan, and Newton are given by Castellanos (1988, pp. 8688), including several involving sums of Fibonacci Numbers.
Gasper quotes the result
(87) 
(88) 
(89) 
(90) 
(91) 
(92) 
(93) 
(94) 
(95) 
may also be computed using iterative Algorithms. A quadratically
converging Algorithm due to Borwein is
(96)  
(97)  
(98) 
(99)  
(100)  
(101) 
(102) 
(103) 
(104) 
(105) 
(106) 
A cubically converging Algorithm which converges to the nearest multiple of to is the simple iteration
(107) 
(108) 
A quartically converging Algorithm is obtained by letting
(109)  
(110) 
(111) 
(112) 
(113) 
(114) 
A quintically converging Algorithm is obtained by letting
(115)  
(116) 
(117) 
(118)  
(119)  
(120) 
(121) 
(122) 
Another Algorithm is due to Woon (1995). Define and
(123) 
(124) 
(125) 
(126) 
(127) 
(128) 
(129) 
Other iterative Algorithms are the Archimedes Algorithm, which was derived by Pfaff in 1800, and the BrentSalamin Formula. Borwein et al. (1989) discuss th order iterative algorithms.
Kochansky's Approximation is the Root of
(130) 
(131) 
(132) 
Some approximations due to Ramanujan
(133)  
(134)  
(135)  
(136)  
(137)  
(138)  
(139)  
(140)  
(141)  
(142)  
(143)  
(144) 
Castellanos (1988) gives a slew of curious formulas:
(145)  
(146)  
(147)  
(148)  
(149)  
(150)  
(151)  
(152)  
(153)  
(154)  
(155)  
(156)  
(157) 
(158) 
(159)  
(160)  
(161)  
(162)  
(163)  
(164)  
(165)  
(166)  
(167)  
(168)  
(169)  
(170) 
Ramanujan (191314) and Olds (1963) give geometric constructions for 355/113. Gardner (1966, pp. 9293) gives a geometric construction for . Dixon (1991) gives constructions for and . Constructions for approximations of are approximations to Circle Squaring (which is itself impossible).
A short mnemonic for remembering the first eight Decimal Digits of is ``May I have a large container of coffee?'' giving 3.1415926 (Gardner 1959; Gardner 1966, p. 92; Eves 1990, p. 122, Davis 1993, p. 9). A more substantial mnemonic giving 15 digits (3.14159265358979) is ``How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics,'' originally due to Sir James Jeans (Gardner 1966, p. 92; Castellanos 1988, p. 152; Eves 1990, p. 122; Davis 1993, p. 9; Blatner 1997, p. 112). A slight extension of this adds the phrase ``All of thy geometry, Herr Planck, is fairly hard,'' giving 24 digits in all (3.14159265358979323846264).
An even more extensive rhyming mnemonic giving 31 digits is ``Now I will a rhyme construct, By chosen words the young instruct. Cunningly devised endeavour, Con it and remember ever. Widths in circle here you see, Sketched out in strange obscurity.'' (Note that the British spelling of ``endeavour'' is required here.)
The following stanzas are the first part of a poem written by M. Keith based on Edgar Allen Poe's ``The Raven.'' The entire poem gives 740 digits; the fragment below gives only the first 80 (Blatner 1997, p. 113). Words with ten letters represent the digit 0, and those with 11 or more digits are taken to represent two digits.
Poe, E.: Near a Raven.
Midnights so dreary, tired and weary.
Silently pondering volumes extolling all bynow obsolete lore.
During my rather long napthe weirdest tap!
An ominous vibrating sound disturbing my chamber's antedoor.
`This,' I whispered quietly, `I ignore.'
Perfectly, the intellect remembers: the ghostly fires, a glittering ember.Inflamed by lightning's outbursts, windows cast penumbras upon this floor.
Sorrowful, as one mistreated, unhappy thoughts I heeded:That inimitable lesson in eleganceLenore
Is delighting, exciting... nevermore.
An extensive collection of mnemonics in many languages is maintained by A. P. Hatzipolakis. Other mnemonics in various languages are given by Castellanos (1988) and Blatner (1997, pp. 112118).
In the following, the word ``digit'' refers to decimal digit after the decimal point. J. H. Conway has shown that there is a sequence of fewer than 40 Fractions , , ... with the property that if you start with and repeatedly multiply by the first of the that gives an integral answer, then the next Power of 2 to occur will be the th decimal digit of .
The first occurrence of 0s appear at digits 32, 307, 601, 13390, 17534, .... The sequence 9999998 occurs at decimal 762 (which is sometimes called the Feynman Point). This is the largest value of any seven digits in the first million decimals. The first time the Beast Number 666 appears is decimal 2440. The digits 314159 appear at least six times in the first 10 million decimal places of (Pickover 1995). In the following, ``digit'' means digit of . The sequence 0123456789 occurs beginning at digits 17,387,594,880, 26,852,899,245, 30,243,957,439, 34,549,153,953, 41,952,536,161, and 43,289,964,000. The sequence 9876543210 occurs beginning at digits 21,981,157,633, 29,832,636,867, 39,232,573,648, 42,140,457,481, and 43,065,796,214. The sequence 27182818284 (the digits of e) occur beginning at digit 45,111,908,393. There are also interesting patterns for . 0123456789 occurs at 6,214,876,462, 9876543210 occurs at 15,603,388,145 and 51,507,034,812, and 999999999999 occurs at 12,479,021,132 of .
Scanning the decimal expansion of until all digit numbers have occurred, the last 1, 2, ... digit numbers appearing are 0, 68, 483, 6716, 33394, 569540, ... (Sloane's A032510). These end at digits 32, 606, 8555, 99849, 1369564, 14118312, ... (Sloane's A036903).
See also Almost Integer, Archimedes Algorithm, BrentSalamin Formula, BuffonLaplace Needle Problem, Buffon's Needle Problem, Circle, Dirichlet Beta Function, Dirichlet Eta Function, Dirichlet Lambda Function, e, EulerMascheroni Constant, Gaussian Distribution, Maclaurin Series, Machin's Formula, MachinLike Formulas, Relatively Prime, Riemann Zeta Function, Sphere, Trigonometry
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© 19969 Eric W. Weisstein