Did You Know: A Chronological History and Some Interesting Facts about Pi (π)
Did You Know: A Chronological History and Some Interesting Facts about Pi (π)

In my previous article I talked about the history of Pi (π): How various formulae have been evolved till date to reach the maximum accuracy of π. In this article I will jot down the very same thing in nothing but a tabular format but more systematically. Below is the table.

Pre computer calculations of π

Sl. No. Mathematician Date Places Comments Notes
1 Rhind papyrus 2000 BC 1 3.16045 (= 4(8/9)2)
2 Archimedes 250 BC 3 3.1418 (average of the bounds) It is claimed that in a text which is
now lost Archimedes gave better
bounds whose average gives the
value 3.141596 for π, correct to
seven places.
3 Vitruvius 20 BC 1 3.125(= 25/8) Marcus Vitruvius Pollio was a Roman
architect. He made his calculation of
π by measuring the distance a wheel
of a given diameter moved through
one revolution.
4 Chang Hong 130 1 3.1622(= √10) Chang Hong (78-139) was an
astrologer from China. He was
one of the first to use this
approximation. He deduced it
from his value for the ratio of the
volume of a cube to that of the
inscribed sphere was 8/5.
5 Ptolemy 150 3 3.14166 Ptolemy used a regular 360 polygon
to approximate π. He actually obtained
the number 3 + 8/60 + 30/602 which
written as a decimal is 3.1416666…
6 Wang Fan 250 1 3.155555(= 142/45) Wang Fan (229-267) was a Chinese
astronomer. No method of arriving
at his value of π is recorded, although
it has been conjectured that he came
up with the value 142/45 “knowing
that 3.14 was too low”.
7 Liu Hui 263 5 3.14159 Liu Hui was a Chinese mathematician.
He calculated π using an inscribed
regular polygon with 192 sides.
8 Zu Chongzhi 480 7 3.141592920(= 355/113) Zu Chongzhi produced a value of π
which was not bettered for nearly one
thousand years. In fact although he
gives 355/113 as a “very close” ratio, he
actually calculated
3.1415926 < π < 3.1415927 which, taking
an average, would give π correct to 8 places.
9 Aryabhata 499 4 3.1416(= 62832/20000) Aryabhata made his approximation using
an inscribed regular polygon with 384 sides.
10 Brahmagupta 640 1 3.1622 (= √10) Brahmagupta criticises Aryabhata for his
earlier, and much better, value of π. He
gave 3 as the approximate value for
practical calculation and √10 as the
exact value.
11 Al-Khwarizmi 800 4 3.1416 Al’Khwarizmi states in his Algebra that
the practical man uses 22/7 for π, the
geometer uses √10 for π, while the
astronomer uses 3.1416.
12 Fibonacci 1220 3 3.141818 Fibonacci gave the approximation
864/275 = 3.14181818… He used inscribed
and circumscribed regular polygons of
96 sides, taking the mean, but made no
reference to Archimedes.
13 Madhava 1400 11 3.14159265359 Madhava was an Indian mathematician
who discovered Gregory’s series long
before Gregory and was the first to
use a series to calculate π.
14 Al-Kashi 1430 14 3.141592653589
79
Al’Kashi obtained his remarkably accurate
value using a regular polygon with
3×228 = 805306368 sides. His aim
was to calculate a value which was
accurate enough to allow the calculation
of the circumference of the universe to
within a hair’s breadth. He takes the
universe to be a sphere 600000 times
the diameter of the earth. Note that in
fact 39 places of π will compute the
circumference of the universe (known
in 2000) to an accuracy of the radius
of the hydrogen atom.
15 Otho 1573 6 3.1415929 Valentin Otho, a pupil of Rheticus, was
a German engineer. He gave the
approximation 355/113.
16 Viète 1593 9 3.1415926536 Viète used Archimedes method with
polygons of 6×216 = 393216 sides to
obtain 3.1415926535 <π < 3.1415926537.
He is also famed as the first to find an
infinite series for π.
17 Romanus 1593 15 3.141592653589
793
Romanus (or Adriaan van Roomen) used
a regular polygon with 230 = 1073741824
sides. He gaveπ to 17 places but only 15
are correct.
18 Van Ceulen 1596 20 3.141592653589
79323846
Van Ceulen used inscribed and
circumscribed polygons with 60 233 =
515396075520 sides. He also used what
in modern notation would be essentially
1 – cos A = 2sin2 A/2. He had discovered
this result.
19 Van Ceulen 1596 35 3.141592653589
7932384626433
832795029
Van Ceulen used a polygon with 262 = 4611686018427387904 sides to obtain
π to 35 places of accuracy. He spent most
of his life on this calculation which my
computer now gives essentially instantly:
3.1415926535897932384626433832795029.
Snell in 1621 showed that more accurate
values of π could be obtained by a clever
variation in the construction so that he
needed only to use a polygon with 230 sides
to obtain the accuracy which Van Ceulen
obtained by using ones with 262 sides. It
was around this time that variants of
Archimedes’ method stopped being used
and future calculations used infinite series
expansions.
20 Newton 1665 16 3.141592653589
7932
Isaac Newton used the series arcsin(x) =
x +1/(2.3) x3 + 1.3/(2.4.5) x5 +
1.3.5/(2.4.6.7) x7 + … putting x = 1/2 to
obtain a series for arcsin(1/2) = π/6.
This requires taking about 40 terms.
21 Sharp 1699 71 Abraham Sharp (1651-1742), an English
mathematician, calculated π at the
suggestion of Halley. He used James
Gregory’s series arctan(x) =xx3/3 + x5/5 –
… Putting x = 1/√3 gives a series for
arctan(1/√3) = π/6. To obtain π correct
to 71 places he used nearly 300 terms of
the series.
22 Seki Kowa 1700 10
23 Kamata 1730 25
24 Machin 1706 100 John Machin used the formula π/4 = 4
arctan(1/5) – arctan(1/129) and James
Gregory’s series for arctan(x).
25 De Lagny 1719 127 Only 112 correct De Lagny used the same method as Sharp.
26 Takebe 1723 41 Takebe Hikojiro Kenko (1664-1739) was
a Japanese mathematician.
27 Matsunaga 1739 50 Matsunaga Ryohitsu was a Japanese
mathematician. He used the same method
as Newton. This requires taking about 160
terms in the series.
28 Von Vega 1794 140 Only 136 correct Baron Georg von Vega used the relation
π/4 = 5 arctan(1/7) + 2 arctan (3/79), and
James Gregory’s series for arctan(x).
Euler had calculated π correct to 20
places in an hour in 1755 using this method.
29 Rutherford 1824 208 Only 152 correct William Rutherford published his result
in the Transactions of the Royal Society
in 1841. He used the relation π/4 = 4
arctan(1/5) – arctan(1/70) + arctan(1/99)
which Euler published in 1764.
30 Strassnitzky, Dase 1844 200 Von Strassnitzky from Vienna gave the
formula π/4 = arctan(1/2) + arctan (1/5) +
arctan(1/8) to Zacharias Dase, a calculating
prodigy. He used took about two months
to calculate π to 200 places using the formula.
31 Clausen 1847 248 Thomas Clausen wanted to find the correct
value of π to 200 places since he knew that
Rutherford and Dase gave different values
from the 153rd place. Clausen used Machin’s
relation π/4 = 4 arctan(1/5) – arctan(1/129).
He discovered that Dase was correct and
extended the number of correct places to 248.
32 Lehmann 1853 261 W Lehmann from Potsdam used the
formula π/4 = arctan(1/2) + arctan(1/3)
which was published by Charles Hutton
in 1776. The formula π/4 = 2 arctan(1/3)
+ arctan(1/7) was also given by Hutton
in 1776 and Euler, independently, in 1779.
33 Rutherford 1853 440 William Rutherford, having made an
error in his published value of π in 1841,
made another calculation. This time he
used Machin’s formula.
34 Shanks 1874 707 Only 527 correct Shanks was a pupil of William Rutherford.
He also used Machin’s formula. The first
to suspect that Shanks’s work on π may
not be correct was De Morgan who
discovered that in an version with 607
places given by Shanks, there were too
few occurrences of the digit 7.
35 Ferguson 1946 620 D F Ferguson who worked at the Royal
Naval College in England used the formula
π/4 = 3 arctan(1/4) + arctan(1/20) +
arctan(1/1985) to compute π. He worked
on it from May 1944 until May 1945 by
which time he had calculated 530 places
and found that Shanks was wrong after
place 528. He continued with his efforts
and published 620 correct places of π in
July 1946. Ferguson then continued with
his calculations, but after this he used a
desk calculator. This marks the point at
which hand calculations of π ended
and computer assisted calculations began.

Computer calculations of π

Sl. No. Mathematician Date Decimal Places Type of computer
1 Ferguson Jan 1947 710 Desk calculator
2 Ferguson, Wrench Sept 1947 808 Desk calculator
3 Smith, Wrench 1949 1120 Desk calculator
4 Reitwiesner et al. 1949 2037 ENIAC
5 Nicholson, Jeenel 1954 3092 NORAC
6 Felton 1957 7480 PEGASUS
7 Genuys Jan 1958 10000 IBM 704
8 Felton May 1958 10021
9 Guilloud 1959 16167 IBM 704
10 Shanks, Wrench 1961 100265 IBM 7090
11 Guilloud, Filliatre 1966 250000 IBM 7030
12 Guilloud, Dichampt 1967 500000 CDC 6600
13 Guilloud, Bouyer 1973 1001250 CDC 7600
14 Miyoshi, Kanada 1981 2000036 FACOM M-200
15 Guilloud 1982 2000050
16 Tamura 1982 2097144 MELCOM 900II
17 Tamura, Kanada 1982 4194288 HITACHI M-280H
18 Tamura, Kanada 1982 8388576 HITACHI M-280H
19 Kanada, Yoshino, Tamura 1982 16777206 HITACHI M-280H
20 Ushiro, Kanada Oct 1983 10013395 HITACHI S-810/20
21 Gosper Oct 1985 17526200 SYMBOLICS 3670
22 Bailey Jan 1986 29360111 CRAY-2
23 Kanada, Tamura Sept 1986 33554414 HITACHI S-810/20
24 Kanada, Tamura Oct 1986 67108839 HITACHI S-810/20
25 Kanada, Tamura, Kubo Jan 1987 134217700 NEC SX-
26 Kanada, Tamura Jan 1988 201326551 HITACHI S-820/80
27 Chudnovskys May 1989
28 Chudnovskys June 1989 525229270
29 Kanada, Tamura July 1989 536870898
30 Chudnovskys Aug 1989 1011196691
31 Kanada, Tamura Nov 1989 1073741799
32 Chudnovskys Aug 1991 2260000000
33 Chudnovskys May 1994 4044000000
34 Kanada, Tamura June 1995
35 Kanada Aug 1995 4294967286
36 Kanada Oct 1995 6442450938
37 Kanada, Takahashi Aug 1997 51539600000 HITACHI SR2201
38 Kanada, Takahashi Sept 1999 206158430000 HITACHI SR8000
39 Kanada et. al. Nov 2002 1,241,100,000,000 HITACHI SR8000
40 Takahashi et. al. Apr 2009 2,576,980,377,524 T2K Open Supercomputer
41 Fabrice Bellard Dec 2009 2,699,999,990,000
  • Core i7 CPU at 2.93 GHz
  • 6 GiB (1) of RAM
  • 7.5 TB of disk storage using five 1.5 TB hard disks (Seagate Barracuda 7200.11 model)
  • 64 bit Red Hat Fedora 10 distribution
  • Computation of the binary digits: 103 days
  • Verification of the binary digits: 13 days
  • Conversion to base 10: 12 days
  • Verification of the conversion: 3 days
  • 131 days in total – The verification of the binary digits used a network of 9 Desktop PCs during 34 hours, Chudnovsky algorithm.
42 Shigeru Kondo Aug 2010 5,000,000,000,000
  • using y-cruncher by Alexander Yee
  • the Chudnovsky formula was used for main computation
  • verification used the Bellard & Plouffe formulas on different computers, both computed 32 hexadecimal digits ending with the 4,152,410,118,610th.
  • with 2 x Intel Xeon X5680 @ 3.33 GHz – (12 physical cores, 24 hyperthreaded)
  • 96 GB DDR3 @ 1066 MHz – (12 × 8 GB – 6 channels) – Samsung (M393B1K70BH1)
  • 1 TB SATA II (Boot drive) – Hitachi (HDS721010CLA332), 3 × 2 TB SATA II (Store Pi Output) – Seagate (ST32000542AS) 16 x 2 TB SATA II (Computation) – Seagate (ST32000641AS)
  • Windows Server 2008 R2 Enterprise x64
  • Computation of binary digits: 80 days
  • Conversion to base 10: 8.2 days
  • Verification of the conversion: 45.6 hours
  • Verification of the binary digits: 64 hours (primary), 66 hours (secondary)
  • Total Time: 90 days – The verification of the binary digits were done simultaneously on two separate computers during the main computation.
43 Shigeru Kondo Oct 2011 10,000,000,000,050
  • using y-cruncher by Alexander Yee
  • Computation: 371 days
  • Verification: 1.86 days and 4.94 days
  • Total time: 371 days
44 Shigeru Kondo Dec 2013 12,100,000,000,050
  • using y-cruncher by Alexander Yee
  • with 2 x Intel Xeon E5-2690 @ 2.9 GHz – (16 physical cores, 32 hyperthreaded)
  • 128 GB DDR3 @ 1600 MHz – 8 x 16 GB – 8 channels
  • Windows Server 2012 x64
  • Computation: 94 days
  • Verification: 46 hours
  • Total time: 94 days

 

Various Formulas for Computing π

  1. Wallis:
    π/2=(2.2.4.4.6.6.8.8. …)/(1.3.3.5.5.7.7.9. …)
  2. Machin:
    π/4=4 arctan(1/5)-arctan(1/239)
  3. Ferguson:
    π/4 = 3 arctan(1/4)+arctan(1/20)+arctan(1/1985)
  4. Euler:
    π/4 = 5 arctan(1/7)+2 arctan(3/79)
  5. Euler:
    π2/6=1/22+1/32+1/42+1/52+ …
  6. Euler:
    e+1=0
  7. Borwein and Borwein:
    1/π = 12Ʃ[(-1)n(6n)!/(n!)3(3n)!][(A+nB)/Cn+1/2];
    where,
    A=212175710912√(61)+1657145277365;
    B=13773980892672√(61)+107578229802750;
    C=[5280(236674+30303√(61)]3
  8. Borwein, Bailey, and Plouffe:
    π = Ʃ[4/(8n+1)-2/(8n+4)-1/(8n+5)-1/(8n+6)](16)-n
    This formula enables one to calculate the nth digit of pi, in hexadecimal notation, without being forced to calculate the preceding n-1 digits.

 

Some interesting things about π

  1. In early work it was not known that the ratio of the area of a circle to the square of its radius and the ratio of the circumference to the diameter are the same. Some early texts use different approximations for these two “different” constants. For example, in the Indian text the Sulba Sutras the ratio for the area is given as 3.088 while the ratio for the circumference is given as 3.2.
  2. Euclid gives in the Elements XII Proposition 2: Circles are to one another as the squares on their diameters. He makes no attempt to calculate the ratio.
  3. Calculating π to many decimal places was used as a test for new computers in the early days.
  4. There is an algorithm by Bailey, Borwein and Plouffe, published in 1996, which allows the nth hexadecimal digit of π to be computed without the preceding n– 1 digits.
  5. Plouffe discovered a new algorithm to compute the nth digit of π in any base in 1997.
  6. 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: 3.14159265358979323846264…
    How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics. All of thy geometry, Herr Planck, is fairly hard…

 

Open questions about the number π

  1. Does each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 each occur infinitely often in π?
  2. Brouwer’s question: In the decimal expansion of π, is there a place where a thousand consecutive digits are all zero?
  3. Is π simply normal to base 10? That is does every digit appear equally often in its decimal expansion in an asymptotic sense?
  4. 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?
  5. 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.
  6. 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.

 

Hope this was a good read for you. You think to share anything else, Do not forget to write here.. Have a good time till my next article. Bye Bye…

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Scientific History Blog Writer • Art enthusiast and Illustrator • Amateur Photographer • Biker and Hiker • Beer Enthusiast • Electrical Engineer • Chicago

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