An ENIAC Determination of π and e to more than 2000 Decimal Places

Abstract

Early in June, 1949, Professor John von Neumann expressed an interest in the possibility that the ENIAC might sometime be employed to determine the value of π and e to many decimal places with a view toward obtaining a statistical measure of the randomness of distribution of the digits, suggesting the employment of one of the formulas:

$$ \begin{array}{*{20}{c}} {\pi /4 = 4\arctan 1/5 - \arctan 1/239} \\ {\pi /4 = 8\arctan 1/10 - 4\arctan 1/515 - \arctan 1/239} \\ {\pi /4 = 3\arctan 1/4 + \arctan 1/20 + \arctan 1/1985} \\ \end{array} $$

in conjunction with the Gregory series

$$ \arctan x = \sum\limits_{n = 0}^\infty {{{( - 1)}^n}{{(2n + 1)}^{ - 1}}{x^{2n + 1}}} $$

.