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Machin's formula [4] (for which the derivation is straightforward) is: = The benefit of the new formula, a variation on the Gregory–Leibniz series ( π / 4 = arctan 1), was that it had a significantly increased rate of convergence, which made it a much more practical method of calculation.
The formula is a special case of the Euler–Boole summation formula for alternating series, providing yet another example of a convergence acceleration technique that can be applied to the Leibniz series. In 1992, Jonathan Borwein and Mark Limber used the first thousand Euler numbers to calculate π to 5,263 decimal places with the Leibniz ...
For the first term in the Taylor series, all digits must be processed. In the last term of the Taylor series, however, there's only one digit remaining to be processed. In all of the intervening terms, the number of digits to be processed can be approximated by linear interpolation. Thus the total is given by:
is the power series for arctan(x) specialized to x = 1. It converges too slowly to be of practical interest. It converges too slowly to be of practical interest. However, the power series converges much faster for smaller values of x {\displaystyle x} , which leads to formulae where π {\displaystyle \pi } arises as the sum of small angles with ...
where C is the circumference of a circle, d is the diameter, and r is the radius.More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width.
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The first to use an electronic computer (the ENIAC) to calculate π [25] 70 hours 2,037: 1953: Kurt Mahler: Showed that π is not a Liouville number: 1954 S. C. Nicholson & J. Jeenel Using the NORC [26] 13 minutes 3,093: 1957 George E. Felton: Ferranti Pegasus computer (London), calculated 10,021 digits, but not all were correct [27] [28] 33 ...
Google engineer Emma Haruka Iwao has calculated pi to 31 trillion digits, breaking the world record.