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Pi day

Pi day is celebrated on March 14 in all the countries that use the MM/DD/YYYY date format (ie, the USA). Pi Approximation day is celebrated in the rest of the world on July 22. 

I’m proposing today for another one: π continued fraction day. Like the 22/7 festival it doesn’t depend on using base 10, and like American Pi day it is extensible when the stars align correctly. 

The continued fraction expansion of π is
Taking just a initial segment of this gives the best possible rational expansions: 3, 22/7, 355/113,…

In most years, π continued fraction day will be 3:7, giving the approximation 22/7, slightly better than 3.14. This year, however, it’s 3:7,15, the continued fraction giving \(\pi\approx 355/113\), with an error of about \(2.7\times 10^{-7}\). That’s two orders of magnitude smaller than the error in 3.1415, and it’s even smaller than the error when the US celebrates Correctly-Rounded Pi Day next year, 3/14/16