A History of Pi by Petr Beckmann

Author:Petr Beckmann
Language: eng
Format: epub
ISBN: 9781466887169
Publisher: St. Martin's Press

10

THE DIGIT HUNTERS

So, Nat’ralists observe, a Flea

Hath smaller Fleas that on him prey.

And these have smaller Fleas to bite’em,

And so proceed ad infinitum.

JONATHAN SWIFT

(1667-1745)

IT has already been pointed out that the invention of decimal fractions and logarithms greatly facilitated numerical calculations in the late 1500’s and the early 1600’s, and this is reflected in the history of π, for about this time people started to calculate its value to an ever increasing number of decimal places, each new digit increasing the accuracy of the former approximation by no less than 10 times. The process continued beyond any possible practical use of so many decimal places; by the end of the 16th century, π was known to 30 decimal places, by the end of the 18th century it was known to 140 places, by the end of the 19th century it had been calculated to 707 places (though later only 526 of them proved to be correct), and the digital computer of the 20th century has raised this number to a whopping 500,000 — perhaps more by the time you read this (see Chapter 18).

Archimedes calculated π to the equivalent of two decimal places, and at first the hunt for greater accuracy may have been dictated by practical needs. Later, especially after the advent of the differential calculus and infinite series, the number of decimal places may have been used to demonstrate the quality of the method of calculation. Perhaps some investigators hoped to discover a periodicity in the ever lengthening sequence of digits. Had this been so, they would have been able to express π as the ratio of two integers, for if a certain sequence of d decimal digits kept recurring, then the fractional part would be the sum of the geometric series

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