Pi Approximation Day is observed on July 22 because 22/7 is the approximate value of Pi in the fractional form and in India they write the date first and then the month.
In 1914, the Indian Math wizard Ramanujan published a formula for PI that converges remarkably faster than many other approaches. See here.
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The above stamp depicting Srinivasa Ramanujan was issued by India in 2010.
In U.S. Pi day is observed on 14th of March (3.14 or 3/14 in the U.S. month/day date format), since 3, 1, and 4 are the three most significant digits of π in the decimal form.
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