WHAT IS PI?
The mathematical constant pi represents the ratio of the circumference of a circle to its diameter. It is the most famous ratio in mathematics both on Earth and probably for any advanced civilization in the universe. The number pi, like other fundamental constants of mathematics such as e = 2.718..., is a transcendental number. The digits of pi and e never end, nor has anyone detected an orderly pattern in their arrangement. Humans know the value of pi to over a trillion digits.
HISTORY OF PI A little known verse of the Bible reads
And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. (I Kings 7, 23)
The same verse can be found in II Chronicles 4, 2. It occurs in a list of specifications for the great temple of Solomon, built around 950 BC and its interest here is that it gives π = 3. Not a very accurate value of course and not even very accurate in its day, for the Egyptian and Mesopotamian values of 25/8 = 3.125 and √10 = 3.162 have been traced to much earlier dates: though in defence of Solomon's craftsmen it should be noted that the item being described seems to have been a very large brass casting, where a high degree of geometrical precision is neither possible nor necessary. There are some interpretations of this which lead to a much better value.
The fact that the ratio of the circumference to the diameter of a circle is constant has been known for so long that it is quite untraceable. The earliest values of π including the 'Biblical' value of 3, were almost certainly found by measurement. In the Egyptian Rhind Papyrus, which is dated about 1650 BC, there is good evidence for 4 
(8/9)2 = 3.16 as a value for π.
(8/9)2 = 3.16 as a value for π. The first theoretical calculation seems to have been carried out by Archimedes of Syracuse (287-212 BC). He obtained the approximation
223/71 < π < 22/7.
Before giving an indication of his proof, notice that very considerable sophistication involved in the use of inequalities here. Archimedes knew, what so many people to this day do not, that π does not equal 22/7, and made no claim to have discovered the exact value. If we take his best estimate as the average of his two bounds we obtain 3.1418, an error of about 0.0002.
Here is Archimedes' argument.
Consider a circle of radius 1, in which we inscribe a regular polygon of 3 2n-1 sides, with semiperimeter bn, and superscribe a regular polygon of 3 2n-1 sides, with semiperimeter an.
2n-1 sides, with semiperimeter bn, and superscribe a regular polygon of 3 2n-1 sides, with semiperimeter an. 
The diagram for the case n = 2 is on the right.
The effect of this procedure is to define an increasing sequence
b1 , b2 , b3 , ...
and a decreasing sequence
a1 , a2 , a3 , ...
such that both sequences have limit π.
Using trigonometrical notation, we see that the two semiperimeters are given by
an = K tan(π/K), bn = K sin(π/K),
where K = 3 2n-1. Equally, we have
2n-1. Equally, we have an+1 = 2K tan(π/2K), bn+1 = 2K sin(π/2K),
and it is not a difficult exercise in trigonometry to show that
(1/an + 1/bn) = 2/an+1 . . . (1)an+1bn = (bn+1)2 . . . (2)
Archimedes, starting from a1 = 3 tan(π/3) = 3√3 and b1 = 3 sin(π/3) = 3√3/2, calculated a2 using (1), then b2 using (2), then a3 using (1), then b3 using (2), and so on until he had calculated a6 and b6. His conclusion was that
b6 < π < a6 .
It is important to realise that the use of trigonometry here is unhistorical: Archimedes did not have the advantage of an algebraic and trigonometrical notation and had to derive (1) and (2) by purely geometrical means. Moreover he did not even have the advantage of our decimal notation for numbers, so that the calculation of a6 and b6 from (1) and (2) was by no means a trivial task. So it was a pretty stupendous feat both of imagination and of calculation and the wonder is not that he stopped with polygons of 96 sides, but that he went so far.
For of course there is no reason in principle why one should not go on. Various people did, including:
Ptolemy (c. 150 AD) 3.1416
Zu Chongzhi (430-501 AD) 355/113
al-Khwarizmi (c. 800 ) 3.1416
al-Kashi (c. 1430) 14 places
Viète (1540-1603) 9 places
Roomen (1561-1615) 17 places
Van Ceulen (c. 1600) 35 places
Except for Zu Chongzhi, about whom next to nothing is known and who is very unlikely to have known about Archimedes' work, there was no theoretical progress involved in these improvements, only greater stamina in calculation. Notice how the lead, in this as in all scientific matters, passed from Europe to the East for the millennium 400 to 1400 AD.
Al-Khwarizmi lived in Baghdad, and incidentally gave his name to 'algorithm', while the words al jabr in the title of one of his books gave us the word 'algebra'. Al-Kashi lived still further east, in Samarkand, while Zu Chongzhi, one need hardly add, lived in China.
The European Renaissance brought about in due course a whole new mathematical world. Among the first effects of this reawakening was the emergence of mathematical formulae for π. One of the earliest was that of Wallis (1616-1703)
2/π = (1.3.3.5.5.7. ...)/(2.2.4.4.6.6. ...)
and one of the best-known is
π/4 = 1 - 1/3 + 1/5 - 1/7 + ....
This formula is sometimes attributed to Leibniz (1646-1716) but is seems to have been first discovered by James Gregory (1638- 1675).
These are both dramatic and astonishing formulae, for the expressions on the right are completely arithmetical in character, while π arises in the first instance from geometry. They show the surprising results that infinite processes can achieve and point the way to the wonderful richness of modern mathematics.
From the point of view of the calculation of π, however, neither is of any use at all. In Gregory's series, for example, to get 4 decimal places correct we require the error to be less than 0.00005 = 1/20000, and so we need about 10000 terms of the series. However, Gregory also showed the more general result
tan-1 x = x - x3/3 + x5/5 - ... (-1 ≤ x ≤ 1) . . . (3)
from which the first series results if we put x = 1. So using the fact that
tan-1(1/√3) = π/6 we getπ/6 = (1/√3)(1 - 1/(3.3) + 1/(5.3.3) - 1/(7.3.3.3) + ...
which converges much more quickly. The 10th term is 1/(19 39√3), which is less than 0.00005, and so we have at least 4 places correct after just 9 terms.
39√3), which is less than 0.00005, and so we have at least 4 places correct after just 9 terms. An even better idea is to take the formula
π/4 = tan-1(1/2) + tan-1(1/3) . . . (4)
and then calculate the two series obtained by putting first 1/2 and the 1/3 into (3).
Clearly we shall get very rapid convergence indeed if we can find a formula something like
π/4 = tan-1(1/a) + tan-1(1/b)
with a and b large. In 1706 Machin found such a formula:
π/4 = 4 tan-1(1/5) - tan-1(1/239) . . . (5)
Actually this is not at all hard to prove, if you know how to prove (4) then there is no real extra difficulty about (5), except that the arithmetic is worse. Thinking it up in the first place is, of course, quite another matter.
With a formula like this available the only difficulty in computing π is the sheer boredom of continuing the calculation. Needless to say, a few people were silly enough to devote vast amounts of time and effort to this tedious and wholly useless pursuit. One of them, an Englishman named Shanks, used Machin's formula to calculate π to 707 places, publishing the results of many years of labour in 1873. Shanks has achieved immortality for a very curious reason which we shall explain in a moment.
Here is a summary of how the improvement went:
Here is a summary of how the improvement went:
1699: Sharp used Gregory's result to get 71 correct digits
1701: Machin used an improvement to get 100 digits and the following used his methods:
1719: de Lagny found 112 correct digits
1789: Vega got 126 places and in 1794 got 136
1841: Rutherford calculated 152 digits and in 1853 got 440
1873: Shanks calculated 707 places of which 527 were correct
A more detailed Chronology is available.
Shanks knew that π was irrational since this had been proved in 1761 by Lambert. Shortly after Shanks' calculation it was shown by Lindemann that π is transcendental, that is, π is not the solution of any polynomial equation with integer coefficients. In fact this result of Lindemann showed that 'squaring the circle' is impossible. The transcendentality of π implies that there is no ruler and compass construction to construct a square equal in area to a given circle.
Very soon after Shanks' calculation a curious statistical freak was noticed by De Morgan, who found that in the last of 707 digits there was a suspicious shortage of 7's. He mentions this in his Budget of Paradoxes of 1872 and a curiosity it remained until 1945 when Ferguson discovered that Shanks had made an error in the 528th place, after which all his digits were wrong. In 1949 a computer was used to calculate π to 2000 places. In this and all subsequent computer expansions the number of 7's does not differ significantly from its expectation, and indeed the sequence of digits has so far passed all statistical tests for randomness.
You can see 2000 places of π.
We should say a little of how the notation π arose. Oughtred in 1647 used the symbol d/π for the ratio of the diameter of a circle to its circumference. David Gregory (1697) used π/r for the ratio of the circumference of a circle to its radius. The first to use π with its present meaning was an Welsh mathematician William Jones in 1706 when he states "3.14159 andc. = π". Euler adopted the symbol in 1737 and it quickly became a standard notation.
You can see 2000 places of π.
We should say a little of how the notation π arose. Oughtred in 1647 used the symbol d/π for the ratio of the diameter of a circle to its circumference. David Gregory (1697) used π/r for the ratio of the circumference of a circle to its radius. The first to use π with its present meaning was an Welsh mathematician William Jones in 1706 when he states "3.14159 andc. = π". Euler adopted the symbol in 1737 and it quickly became a standard notation.
We conclude with one further statistical curiosity about the calculation of π, namely Buffon's needle experiment. If we have a uniform grid of parallel lines, unit distance apart and if we drop a needle of length k < 1 on the grid, the probability that the needle falls across a line is 2k/π. Various people have tried to calculate π by throwing needles. The most remarkable result was that of Lazzerini (1901), who made 34080 tosses and got
π = 355/113 = 3.1415929
which, incidentally, is the value found by Zu Chongzhi. This outcome is suspiciously good, and the game is given away by the strange number 34080 of tosses. Kendall and Moran comment that a good value can be obtained by stopping the experiment at an optimal moment. If you set in advance how many throws there are to be then this is a very inaccurate way of computing π. Kendall and Moran comment that you would do better to cut out a large circle of wood and use a tape measure to find its circumference and diameter.
Still on the theme of phoney experiments, Gridgeman, in a paper which pours scorn on Lazzerini and others, created some amusement by using a needle of carefully chosen length k = 0.7857, throwing it twice, and hitting a line once. His estimate for π was thus given by
2
from which he got the highly creditable value of π = 3.1428. He was not being serious!
It is almost unbelievable that a definition of π was used, at least as an excuse, for a racial attack on the eminent mathematician Edmund Landau in 1934. Landau had defined π in this textbook published in Göttingen in that year by the, now fairly usual, method of saying that π/2 is the value of x between 1 and 2 for which cos x vanishes. This unleashed an academic dispute which was to end in Landau's dismissal from his chair at Göttingen. Bieberbach, an eminent number theorist who disgraced himself by his racist views, explains the reasons for Landau's dismissal:-
Thus the valiant rejection by the Göttingen student body which a great mathematician, Edmund Landau, has experienced is due in the final analysis to the fact that the un-German style of this man in his research and teaching is unbearable to German feelings. A people who have perceived how members of another race are working to impose ideas foreign to its own must refuse teachers of an alien culture.
G H Hardy replied immediately to Bieberbach in a published note about the consequences of this un-German definition of π
There are many of us, many Englishmen and many Germans, who said things during the War which we scarcely meant and are sorry to remember now. Anxiety for one's own position, dread of falling behind the rising torrent of folly, determination at all cost not to be outdone, may be natural if not particularly heroic excuses. Professor Bieberbach's reputation excludes such explanations of his utterances, and I find myself driven to the more uncharitable conclusion that he really believes them true.
Not only in Germany did π present problems. In the USA the value of π gave rise to heated political debate. In the State of Indiana in 1897 the House of Representatives unanimously passed a Bill introducing a new mathematical truth.
Be it enacted by the General Assembly of the State of Indiana: It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square of one side.
(Section I, House Bill No. 246, 1897)
The Senate of Indiana showed a little more sense and postponed indefinitely the adoption of the Act!
Open questions about the number π
- Does each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 each occur infinitely often in π?
- Brouwer's question: In the decimal expansion of π, is there a place where a thousand consecutive digits are all zero?
- Is π simply normal to base 10? That is does every digit appear equally often in its decimal expansion in an asymptotic sense?
- Is π normal to base 10? That is does every block of digits of a given length appear equally often in its decimal expansion in an asymptotic sense?
- Is π normal ? That is does every block of digits of a given length appear equally often in the expansion in every base in an asymptotic sense? The concept was introduced by Borel in 1909.
- Another normal question! We know that π is not rational so there is no point from which the digits will repeat. However, if π is normal then the first million digits 314159265358979... will occur from some point. Even if π is not normal this might hold! Does it? If so from what point? Note: Up to 200 million the longest to appear is 31415926 and this appears twice.
As a postscript, here is a mnemonic for the decimal expansion of π. Each successive digit is the number of letters in the corresponding word.
How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics. All of thy geometry, Herr Planck, is fairly hard...:3.14159265358979323846264...
CHRONOLOGY OF PI
The table below is a brief chronology of computed numerical values of, or bounds on, the mathematical constant π. See the history of numerical approximations of π for explanations, comments and details concerning some of the calculations mentioned below.
| Date | Who | Value of π (world records in bold)' |
|---|---|---|
| 20th century BC | Egyptian Rhind Mathematical Papyrus and Moscow Mathematical Papyrus | (16/9)2 = 3.160493... |
| 19th century BC | Babylonian mathematicians | 25/8 = 3.125 |
| 9th century BC | Indian Shatapatha Brahmana | 339/108 = 3.138888... |
| 434 BC | Anaxagoras attempted to square the circle with compass and straightedge | |
| c. 250 BC | Archimedes | 223/71 < π < 22/7 (3.140845... < π < 3.142857...) |
| 20 BC | Vitruvius | 25/8 = 3.125 |
| 5 | Liu Xin | 3.154 |
| 130 | Zhang Heng | √10 = 3.162277... |
| 150 | Ptolemy | 377/120 = 3.141666... |
| 250 | Wang Fan | 142/45 = 3.155555... |
| 263 | Liu Hui | 3.141024 |
| 480 | Zu Chongzhi | 3.1415926 < π < 3.1415927 |
| 499 | Aryabhata | 62832/20000 = 3.1416 |
| 640 | Brahmagupta | √10 = 3.162277... |
| 800 | Al Khwarizmi | 3.1416 |
| 1150 | Bhāskara II | 3.14156 |
| 1220 | Fibonacci | 3.141818 |
| All records from 1400 onwards are given as the number of correct decimal places. | ||
| 1400 | Madhava of Sangamagrama discovered the infinite power series expansion of π, now known as the Madhava-Leibniz series | 11 decimal places 13 decimal places |
| 1424 | Jamshīd al-Kāshī | 16 decimal places |
| 1573 | Valentinus Otho (355/113) | 6 decimal places |
| 1593 | François Viète | 9 decimal places |
| 1593 | Adriaen van Roomen | 15 decimal places |
| 1596 | Ludolph van Ceulen | 20 decimal places |
| 1615 | 32 decimal places | |
| 1621 | Willebrord Snell (Snellius), a pupil of Van Ceulen | 35 decimal places |
| 1665 | Isaac Newton | 16 decimal places |
| 1699 | Abraham Sharp | 71 decimal places |
| 1700 | Takakazu Seki | 10 decimal places |
| 1706 | John Machin | 100 decimal places |
| 1706 | William Jones introduced the Greek letter 'π' | |
| 1719 | Thomas Fantet de Lagny calculated 127 decimal places, but not all were correct | 112 decimal places |
| 1722 | Toshikiyo Kamata | 24 decimal places |
| 1722 | Katahiro Takebe | 41 decimal places |
| 1739 | Yoshisuke Matsunaga | 51 decimal places |
| 1748 | Leonhard Euler used the Greek letter 'π' in his book Introductio in Analysin Infinitorum and assured its popularity. | |
| 1761 | Johann Heinrich Lambert proved that π is irrational | |
| 1775 | Euler pointed out the possibility that π might be transcendental | |
| 1794 | Jurij Vega calculated 140 decimal places, but not all are correct | 137 decimal places |
| 1794 | Adrien-Marie Legendre showed that π² (and hence π) is irrational, and mentioned the possibility that π might be transcendental. | |
| 1841 | William Rutherford calculated 208 decimal places, but not all were correct | 152 decimal places |
| 1844 | Zacharias Dase and Strassnitzky calculated 205 decimal places, but not all were correct | 200 decimal places |
| 1847 | Thomas Clausen calculated 250 decimal places, but not all were correct | 248 decimal places |
| 1853 | Lehmann | 261 decimal places |
| 1853 | William Rutherford | 440 decimal places |
| 1855 | Richter | 500 decimal places |
| 1874 | William Shanks took 15 years to calculate 707 decimal places but not all were correct (the error was found by D. F. Ferguson in 1946) | 527 decimal places |
| 1882 | Lindemann proved that π is transcendental (the Lindemann-Weierstrass theorem) | |
| 1897 | The U.S. state of Indiana came close to legislating the value of 3.2 (among others) for π. House Bill No. 246 passed unanimously. The bill stalled in the state Senate due to a suggestion of possible commercial motives involving publication of a textbook.[1] | |
| 1910 | Srinivasa Ramanujan finds several rapidly converging infinite series of π, which can compute 8 decimal places of π with each term in the series. Since the 1980s, his series have become the basis for the fastest algorithms currently used by Yasumasa Kanada and the Chudnovsky brothers to compute π. | |
| 1946 | D. F. Ferguson (using a desk calculator) | 620 decimal places |
| 1947 | Ivan Niven gave a very elementary proof that π is irrational | |
| January 1947 | D. F. Ferguson (using a desk calculator) | 710 decimal places |
| September 1947 | D. F. Ferguson (using a desk calculator) | 808 decimal places |
| 1949 | D. F. Ferguson and John Wrench, using a desk calculator | 1,120 decimal places |
| All records from 1949 onwards were calculated with electronic computers. | ||
| 1949 | John W. Wrench, Jr, and L. R. Smith were the first to use an electronic computer (the ENIAC) to calculate π (it took 70 hours) (also attributed to Reitwiesner et al.) [2] | 2,037 decimal places |
| 1953 | Kurt Mahler showed that π is not a Liouville number | |
| 1954 | S. C. Nicholson & J. Jeenel, using the NORC (it took 13 minutes) [3] | 3,092 decimal places |
| 1957 | G. E. Felton, using the Ferranti Pegasus computer (London) [4] | 7,480 decimal places |
| January 1958 | Francois Genuys, using an IBM 704 (1.7 hours) [5] | 10,000 decimal places |
| May 1958 | G. E. Felton, using the Pegasus computer (London) (33 hours) | 10,020 decimal places |
| 1959 | Francois Genuys, using the IBM 704 (Paris) (4.3 hours) [6] | 16,167 decimal places |
| 1961 | IBM 7090 (London) (39 minutes) | 20,000 decimal places |
| 1961 | Daniel Shanks and John Wrench, using the IBM 7090 (New York) (8.7 hours) [7] | 100,265 decimal places |
| 1966 | Jean Guilloud and J. Filliatre, using the IBM 7030 (Paris) (taking 28 hours??) | 250,000 decimal places |
| 1967 | Jean Guilloud and M. Dichampt, using the CDC 6600 (Paris) (28 hours) | 500,000 decimal places |
| 1973 | Jean Guilloud and Martin Bouyer, using the CDC 7600 | 1,001,250 decimal places |
| 1981 | Yasumasa Kanada and Kazunori Miyoshi, FACOM M-200 | 2,000,036 decimal places |
| 1981 | Jean Guilloud, Not known | 2,000,050 decimal places |
| 1982 | Yoshiaki Tamura, MELCOM 900II | 2,097,144 decimal places |
| 1982 | Yasumasa Kanada, Yoshiaki Tamura, HITAC M-280H | 4,194,288 decimal places |
| 1982 | Yasumasa Kanada, Yoshiaki Tamura, HITAC M-280H | 8,388,576 decimal places |
| 1983 | Yasumasa Kanada, Yoshiaki Tamura, S. Yoshino, HITAC M-280H | 16,777,206 decimal places |
| October 1983 | Yasumasa Kanada and Yasunori Ushiro, HITAC S-810/20 | 10,013,395 decimal places |
| October 1985 | Bill Gosper, Symbolics 3670 | 17,526,200 decimal places |
| January 1986 | David H. Bailey, CRAY-2 | 29,360,111 decimal places |
| September 1986 | Yasumasa Kanada, Yoshiaki Tamura, HITAC S-810/20 | 33,554,414 decimal places |
| October 1986 | Yasumasa Kanada, Yoshiaki Tamura, HITAC S-810/20 | 67,108,839 decimal places |
| January 1987 | Yasumasa Kanada, Yoshiaki Tamura, Yoshinobu Kubo, NEC SX-2 | 134,214,700 decimal places |
| January 1988 | Yasumasa Kanada and Yoshiaki Tamura, HITAC S-820/80 | 201,326,551 decimal places |
| May 1989 | Gregory V. Chudnovsky & David V. Chudnovsky, CRAY-2 & IBM 3090/VF | 480,000,000 decimal places |
| June 1989 | Gregory V. Chudnovsky & David V. Chudnovsky, IBM 3090 | 535,339,270 decimal places |
| July 1989 | Yasumasa Kanada and Yoshiaki Tamura, HITAC S-820/80 | 536,870,898 decimal places |
| August 1989 | Gregory V. Chudnovsky & David V. Chudnovsky, IBM 3090 | 1,011,196,691 decimal places |
| 19 November 1989 | Yasumasa Kanada and Yoshiaki Tamura, HITAC S-820/80 | 1,073,740,799 decimal places |
| August 1991 | Gregory V. Chudnovsky & David V. Chudnovsky, Home made parallel computer (details unknown, not verified) [8] | 2,260,000,000 decimal places |
| 18 May 1994 | Gregory V. Chudnovsky & David V. Chudnovsky, New home made parallel computer (details unknown, not verified) | 4,044,000,000 decimal places |
| 26 June 1995 | Yasumasa Kanada and Daisuke Takahashi (mathematician), HITAC S-3800/480 (dual CPU) [9] | 3,221,220,000 decimal places |
| 28 August 1995 | Yasumasa Kanada and Daisuke Takahashi (mathematician), HITAC S-3800/480 (dual CPU) [10] | 4,294,960,000 decimal places |
| 11 October 1995 | Yasumasa Kanada and Daisuke Takahashi (mathematician), HITAC S-3800/480 (dual CPU) [11] | 6,442,450,000 decimal places |
| 6 July 1997 | Yasumasa Kanada and Daisuke Takahashi (mathematician), HITACHI SR2201 (1024 CPU) [12] | 51,539,600,000 decimal places |
| 5 April 1999 | Yasumasa Kanada and Daisuke Takahashi (mathematician), HITACHI SR8000 (64 of 128 nodes) [13] | 68,719,470,000 decimal places |
| 20 September 1999 | Yasumasa Kanada and Daisuke Takahashi (mathematician), HITACHI SR8000/MPP (128 nodes) [14] | 206,158,430,000 decimal places |
| 24 November 2002 | Professor Yasumasa Kanada & 9 man team, HITACHI SR8000/MPP (64 nodes), 600 hours, Department of Information Science at the University of Tokyo in Tokyo, Japan [15] | 1,241,100,000,000 decimal places |
| 29 April 2009 | Professor Daisuke Takahashi (mathematician) et al., T2K Open Supercomputer (640 nodes), single node speed is 147.2 gigaflops, 29.09 hours, computer memory is 13.5 terabytes, Gauss–Legendre algorithm, Center for Computational Sciences at the University of Tsukuba in Tsukuba, Japan[16] | 2,576,980,377,524 decimal places |
| All records from Dec 2009 onwards are calculated on home computers with commercially available parts. | ||
| 31 December 2009 | Fabrice Bellard
| 2,699,999,990,000 decimal places |
| 2 August 2010 | Shigeru Kondo[19]
| 5,000,000,000,000 decimal places |
TRANSCENDENTAL NUMBER PI
Transcendental numbers cannot be expressed as the root of any algebraic equation with rational coefficients. This means that pi could not exactly satisfy equations of the type: pi2 = 10, or 9pi4 - 240pi2 + 1492 = 0. These are equations involving simple integers with powers of pi. The numbers pi and e can be expressed as an endless continued fraction or as the limit of an infinite series. The remarkable fraction 355/113 expresses pi accurately to six decimal places.
Ants and Transcendental Numbers
Imagine a race of talking ants. The ants can compress the infinite digits of pi in an interesting way. For example, let us imagine that the ants can speak by manipulating their crude jaws. The first ant in the long parade of ants screams out the first digit, "3". The next yells the number on its back, a "1". The next yells a "4", and so on. Further imagine that each ant speaks its digit in only half the time of the preceding ant. Each ant has a turn to speak. Only the most recent digit is spoken at any instant. If the first digit of pi requires 30 seconds to speak (due to the ant's cumbersome jaws and little brain), the entire ant colony will speak all the digits of pi in a minute! (Again, this is because the infinite sum 1/2 minute + 1/4 minute + 1/8 minute + ... is equal to 1 minute.) Astoundingly, at the end of the minute, there will be a quick-talking ant that will actually say the "last" digit of pi! The geometer God, upon hearing this last digit, may cry, "That's impossible, because pi has no last digit!"INTERESTING FACTS ABOUT PI
- Pi is the most recognized mathematical constant in the world. Scholars often consider Pi the most important and intriguing number in all of mathematics.e
- In the Star Trek episode “Wolf in the Fold,” Spock foils the evil computer by commanding it to “compute to last digit the value of pi.”d
- Comedian John Evans once quipped: “What do you get if you divide the circumference of a jack-o'-lantern by its diameter? Pumpkin π.”d
- Scientists in Carl Sagan’s novel Contact are able to unravel enough of pi to find hidden messages from the creators of the human race, allowing humans to access deeper levels of universal awareness.d
- The symbol for pi (π) has been used regularly in its mathematical sense only for the past 250 years.c
- During the famed O.J. Simpson trial, there were arguments between defense attorney Robert Blasier and an FBI agent about the actual value of pi, seemingly to reveal flaws in the FBI agent’s intellectual acumen.d
- A Givenchy men’s cologne named Pi is marketed as highlighting the sexual appeal of intelligent and visionary men.h
- We can never truly measure the circumference or the area of a circle because we can never truly know the value of pi. Pi is an irrational number, meaning its digits go on forever in a seemingly random sequence.f
- Darren Aronofsky’s fascinating movie π (Pi: Faith in Chaos) shows how the main character’s attempt to find simple answers about pi (and, by extension, the universe) drives him mad. The film won the Directing Award at the 1988 Sundance Film Festival.g
- In the Greek alphabet, π (piwas) is the sixteenth letter. In the English alphabet, p is also the sixteenth letter.d
- The letter π is the first letter of the Greek word “periphery” and “perimeter.” The symbol π in mathematics represents the ratio of a circle’s circumference to its diameter. In other words, π is the number of times a circle’s diameter will fit around its circumference.a
- Egyptologists and followers of mysticism have been fascinated for centuries by the fact that the Great Pyramid at Giza seems to approximate pi. The vertical height of the pyramid has the same relationship to the perimeter of its base as the radius of a circle has to its circumference.d
- The first 144 digits of pi add up to 666 (which many scholars say is “the mark of the Beast”). And 144 = (6+6) x (6+6).d
- If the circumference of the earth were calculated using π rounded to only the ninth decimal place, an error of no more than one quarter of an inch in 25,000 miles would result.i
- In 1995, Hiroyoki Gotu memorized 42,195 places of pi and is considered the current pi champion. Some scholars speculate that Japanese is better suited than other languages for memorizing sequences of numbers.a
- A mysterious 2008 crop circle in Britain shows a coded image representing the first 10 digits of pi.b
- Ludolph van Ceulen (1540-1610) spent most of his life calculating the first 36 digits of pi (which were named the Ludolphine Number). According to legend, these numbers were engraved on his now lost tombstone.c
- William Shanks (1812-1882) worked for years by hand to find the first 707 digits of pi. Unfortunately, he made a mistake after the 527th place and, consequently, the following digits were all wrong.c
- In 2002, a Japanese scientist found 1.24 trillion digits of pi using a powerful computer called the Hitachi SR 8000, breaking all previous records.e
- Pi is the secret code in Alfred Hitchcock’s Torn Curtain and in The Net starring Sandra Bullock.d
- Since there are 360 degrees in a circle and pi is intimately connected with the circle, some mathematicians were delighted to discover that the number 360 is at the 359th digit position of pi.d
- Computing pi is a stress test for a computer—a kind of “digital cardiogram.”d
- Umberto Eco’s famed book Foucault’s Pendulum associates the mysterious pendulum in the novel with the intrigue of pi.d
- Pi has been studied by the human race for almost 4,000 years. By 2000 B.C., Babylonians established the constant circle ratio as 3-1/8 or 3.125. The ancient Egyptians arrived at a slightly different value of 3-1/7 or 3.143.a
- One of the earliest known records of pi was written by an Egyptian scribe named Ahmes (c. 1650 B.C.) on what is now known as the Rhind Papyrus. He was off by less than 1% of the modern approximation of pi (3.141592).l
- The Rhind Papyrus was the first attempt to calculate pi by “squaring the circle,” which is to measure the diameter of a circle by building a square inside the circle.l
- The “squaring the circle” method of understanding pi has fascinated mathematicians because traditionally the circle represents the infinite, immeasurable, and even spiritual world while the square represents the manifest, measurable, and comprehensive world.e
- In 1888, a Indiana country doctor named Edwin Goodwin claimed he had been “supernaturally taught” the exact measure of the circle and even had a bill proposed in the Indiana legislature that would copyright his mathematical findings. The bill never became law thanks to a mathematical professor in the legislature who pointed out that the method resulted in an incorrect value of pi.d
- The first million decimal places of pi consist of 99,959 zeros, 99,758 1s, 100,026 2s, 100,229 3s, 100,230 4s, 100,359 5s, 99,548 6s, 99,800 7s, 99,985 8s, and 100,106 9s.a
- ”Pi Day” is celebrated on March 14 (which was chosen because it resembles 3.14). The official celebration begins at 1:59 p.m., to make an appropriate 3.14159 when combined with the date. Albert Einstein was born on Pi Day (3/14/1879) in Ulm Wurttemberg, Germany.d
- The Bible alludes to pi in 1 Kings 7:23 where it describes the altar inside Solomon’s temple: “And he made a molten sea of ten cubits from brim to brim . . . and a line of thirty cubits did compass it round about.” These measurements procure the following equation: 333/106 = 3.141509.k
- Pi was first rigorously calculated by one of the greatest mathematicians of the ancient world, Archimedes of Syracuse (287-212 B.C.). Archimedes was so engrossed in his work that he did not notice that Roman soldiers had taken the Greek city of Syracuse. When a Roman soldier approached him, he yelled in Greek “Do not touch my circles!” The Roman soldier simply cut off his head and went on his business.f
- A refined value of pi was obtained by the Chinese much earlier than in the West. The Chinese had two advantages over most of the world: they used decimal notations and they used a symbol for zero. European mathematicians would not use a symbolic zero until the late Middle Ages through contact with Indian and Arabic thinkers.f
- Al-Khwarizmi, who lived in Baghdad around A.D. 800, worked on a value of pi calculated to four digits: 3.1416. The term “algorithm” derives from his name, and his text Kitab al-Jabr wal-Muqabala (The Book of Completion Concerning Calculating by Transposition and Reduction) gives us the word “algebra” (from al-Jabr, which means “completion” or “restoration”).c
- Ancient mathematicians tried to compute pi by inscribing polygons with more and more sides that would more closely approach the area of a circle. Archimedes used a 96-sided polygon. Chinese mathematicians Liu Hui inscribed a 192-sided polygon and then a 3,072-sided polygon to calculate pi to 3.14159. Tsu Ch’ung and his son inscribed polygons with as many as 24,576 sides to calculate pi (the result had only an 8-millionth of 1% difference from the now accepted value of pi).f
- William Jones (1675-1749) introduced the symbol “π” in the 1706, and it was later popularized by Leonhard Euler (1707-1783) in 1737.c
- The π symbol came into standard use in the 1700s, the Arabs invented the decimal system in A.D. 1000, and the equal sign (=) appeared in 1557.e
- Before the π symbol was used, mathematicians would describe pi in round-about ways such as “quantitas, in quam cum multipliectur diameter, proveniet circumferential,” which means “the quantity which, when the diameter is multiplied by it, yields the circumference.”c
- Leonardo da Vinci (1452-1519) and artist Albrecht Durer (1471-1528) both briefly worked on “squaring the circle,” or approximating pi.d
- There are no occurrences of the sequence 123456 in the first million digits of pi—but of the eight 12345s that do occur, three are followed by another 5. The sequence 012345 occurs twice and, in both cases, it is followed by another 5.i
- Some scholars claim that humans are programmed to find patterns in the world because it’s the only way we can give meaning to the world and ourselves. Hence, the obsessive search to find patterns in π.e
- The father of calculus (meaning “pebble used in counting” from calx or “limestone”), Isaac Newton calculated pi to at least 16 decimal places.l
- Pi is also referred to as the “circular constant,” “Archimedes’ constant,” or “Ludolph’s number.”d
- In the seventeenth century, pi was freed from the circle and applied also to curves, such as arches and hypocycloids, when it was found that their areas could also be expressed in terms of pi. In the twentieth century, pi has been used in many areas, such as number theory, probability, and chaos theory.a
- The first six digits of pi (314159) appear in order at least six times among the first 10 million decimal places of pi.i
- Thirty-nine decimal places of pi suffice for computing the circumference of a circle girding the known universe with an error no greater than the radius of a hydrogen atom.i
- John Donne’s (1572-1631) poem “Upon the Translations of the Psalms by Sir Philip Sidney, and the Countess of Pembroke, His Sister” condemns attempts to find an exact value of pi, or to “square a circle,” which Donne views as an attempt to rationalize God:
Eternal God—for whom who ever dare
Seek new expressions, do the circle square,
And thrust into straight corners of poor wit
Thee, who art cornerless and infinite—k - Many mathematicians claim that it is more correct to say that a circle has an infinite number of corners than to view a circle as being cornerless.e
- Plato (427-348 B.C.) supposedly obtained for his day a fairly accurate value for pi: √2 + √3 = 3.146.a
- A Web site titled “The Pi-Search Page” finds a person’s birthday and other well known numbers in the digits of pi.j
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