Here is the e-mail I wrote last year, on 22/7, 2009:
Greetings all! and a very happy international pi approximation day to each and every one of you!
22/7 is an amazing fractional approximation of the mathematical constant (pi), which itself cannot be expressed as a fraction of whole numbers.
- 22/7 = 3.142857142857142857…, while
- pi = 3.141592653589793238…
–These numbers are so close together that their difference is just 0.04% the value of pi!
What makes this even more special is that 7 is such a low denominator. Since we use the decimal number system, it would seem natural to us to truncate pi’s decimal expansion to get fractions with powers of ten in the denominator. However, 22/7 is closer to pi than 31/10=3.1, and 22/7 is even closer to pi than 314/100=3.14! The main reason 22/7 gets so close to pi for such a tiny denominator is because it comes from a truncation of pi’s continued fraction: 3 + 1/7 (the continued fraction for pi can be seen here).
When I was growing up, I learned that the 22nd of July was European Pi Approximation Day, seeing as how in most of Europe they abbreviate the date as day/month. For today, that is 22/7. In particular, I first learned about this in my Spanish class, where today’s date would be el veintidos de julio; my friends in French class told me they abbreviated the date in a similar way. However, since then I have realized that most of the world, not just Europe, abbreviates the date in this format: only we from the USA and a few other countries are the oddballs (see here). So perhaps a better name is International Pi Approximation Day.
I hope that you all took some time today (or if this message reaches you too late–as is almost certain–, that you take some time tomorrow), to ponder the beauty of the number pi and to reflect on the simultaneous elegance and utility of mathematics. Best wishes to you and yours on this day of approximate mathematical celebration!