Chronology of Calculating π

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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 in most cases less, of π are sufficient for virtually every practical application imaginable. 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 π:

1650 B.C.

Ahmes, the ancient Egyptian scribe, implies that π = ²⁵⁶/₈₁ = 3.1605.

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.

450

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.

1220

Fibonacci uses π = ⁸⁶⁴/₂₇₅ = 3.141818...

1424

Al-Kashi calculates π to 10 sexagecimal (base 60) places, equivalent to 16 decimal places.

1593

Adriaen Romanus finds π to 15 decimal places.

1596

Ludolph Van Ceulen calculates π to 20 decimal places. In 1610, he continued his calculation to 15 more decimal places, making 35 in total.

1621

Willebrord Snell calculates π to 34 decimal places.

1630

Grienberger calculates π to 39 decimal places based on Snell's work.

1699

Abraham Sharp calculates π to 72 decimal places (71 correct).

1706

John Machin calculates π to 100 decimal places.

1719

De Lagny calculates π to 127 decimal places (112 correct).

1794

Vega calculates π to 140 decimal places (136 correct).

1844

Dase calculates π to 205 decimal places (200 correct).

1847

Clausen calculates π to 250 decimal places (248 correct).

1853

Rutherford calculates π to 440 places of decimals.

1873

William Shanks publishes his calculation of π to 707 decimal places.

1947

D. F. Ferguson calculates 808 decimal places of π. In doing so, he discovers that Shanks' calculation was wrong from the 527^th place onwards. Ferguson used a desk calculator.

1949

ENIAC, an early computer, computes 2,037 decimal places of π in a little under three days.

1961

Daniel Shanks and John Wrench use an IBM 7090 computer to compute 100,200 decimal places. The calculation took 8.72 hours.

1973

Jean Guillord and M. Bouyer use a CDC 7600 to compute 1 million decimal places in 23.3 hours.

1983

Y. Tamura and Y. Kanada use a HITAC M-280H to compute 16 million digits in under thirty hours.

1989

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.

1997

Kanada and Takahashi calculate 51.5 billion digits on a Hitachi SR2201 in just over 29 hours.

2011

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).