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The desire to calculate the value of π as exactly as possible dates from antiquity. With modern computing power, it is now possible to calculate π to incredibly high degrees of accuracy. Why do mathematicians continue to calculate π to so many decimal places? It's not for accuracy in calculations; 10 decimal places or so of π are sufficient for almost every practical application.
Prior to the impossibility of the
quadrature of the circle
being proved, some calculations were made to show proposed calculations to be false.
In modern times, another reason is to demonstrate the power of computer hardware and software. Another is to attempt to shed insight on the digits of π. Are they random? Is there some pattern in them? The answer to this question is not known.
Here are some significant highlights in calculating the value of
- 1650 B.C.
- Ahmes, the ancient Egyptian
scribe, implies that π = 256/81 =
- 250 B.C.
- Archimedes determines that
3 10⁄71 < π 3 1/7.
The average of these two values is 3.1418, correct to 3 decimal places.
- ca. 200 A.D.
- Claudius Ptolemy uses
π = 377/120 = 3.14166...,
correct to four decimal places.
- In China, Tsu Ch'ung-chih establishes the value of
355⁄113, which is π to six decimal places.
This fraction is the smallest fraction that approximates π so well.
In the West, this approximation was not discovered as an approximation
to π for another millennium.
- Fibonacci uses
π = 864/275 = 3.141818...
- Al-Kashi calculates π to 10 sexagecimal (base 60) places,
equivalent to 16 decimal places.
- Adriaen Romanus finds π to 15 decimal places.
- Ludolph Van Ceulen calculates π to 20 decimal places.
In 1610, he continued his calculation to 15 more decimal places,
making 35 in total.
- Willebrord Snell calculates π to 34 decimal places.
- Grienberger calculates π to 39 decimal places based on Snell's work.
- Abraham Sharp calculates π to 72 decimal places (71 correct).
- John Machin calculates π to 100 decimal places.
- De Lagny calculates π to 127 decimal places (112 correct).
- Vega calculates π to 140 decimal places (136 correct).
- Dase calculates π to 205 decimal places (200 correct).
- Clausen calculates π to 250 decimal places (248 correct).
- Rutherford calculates π to 440 places of decimals.
- William Shanks publishes his calculation of π to 707 decimal places.
- D. F. Ferguson calculates 808 decimal places of π.
In doing so, he discovers that Shanks' calculation was wrong from the
527th place onwards. Ferguson used a desk calculator.
- ENIAC, an early computer, computes 2,037 decimal places of π in a little under three days.
- Daniel Shanks and John Wrench use an IBM 7090 computer to compute 100,200 decimal places. The calculation took 8.72 hours.
- Jean Guillord and M. Bouyer use a CDC 7600 to compute
1 million decimal places in 23.3 hours.
- Y. Tamura and Y. Kanada use a HITAC M-280H to compute
16 million digits in under thirty hours.
- David and Gregory Chudnovsky find 480 million digits, and,
later in the year, 1 billion digits. They would calculate over 8 billion
digits in 1996.
- Kanada and Takahashi calculate 51.5 billion digits
on a Hitachi SR2201 in just over 29 hours.
- Shigeru Kondo calculates π to ten trillion decimal digits.
You can view the first 10,000 digits of π here.
Much of this material was obtained from the book The Joy of π
by David Blatner (see
my bibliography page).