Machin's Computation of Pi

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Around 1706, John Machin found the arc tangent formula .

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He expanded it using Gregory's series to compute to over 100 digits. The convergence of this series is much more rapid than that for the simple Gregory–Leibniz series: .

Let and be the series for and . Manchin's formula is then .

Split and into four series according to the signs of their coefficients:

,

,

,

.

Machin used values truncated to 103 decimals, which means that only terms greater than are calculated.

To make sure that all terms of and with indices are less than , it is necessary that . For , this means .

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Contributed by: Izidor Hafner (May 2016)
Open content licensed under CC BY-NC-SA

Details

Let , be an arc of radius 1, and be perpendicular to , , , and . Construct points and on the arc , and and on the straight line , so that and .

Then and , .

So , .

See [1, pp. 8–9].

References

[1] V. F. Rickey. "Machin's Formula for Computing Pi." (Apr 28, 2016) fredrickey.info/talks/10-03-13-HPM-DC-Machin/10-03-13-HPM_DC_Machin.pdf.

[2] MacTutor. "John Machin." (Apr 28, 2016) www-history.mcs.st-andrews.ac.uk/Biographies/Machin.html.

[3] Wikipedia. "John Machin." (Apr 28, 2016) en.wikipedia.org/wiki/John_Machin.

[4] W. W. Rouse Ball, H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York: Dover Publications, 1987 p. 356.

Permanent Citation

Izidor Hafner

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