When I first learned about π, I was told that a good approximation was 22/7. Even when I was 12 years old, I thought this was utterly pointless. 22/7 agrees with π to two decimal places (so three matching digits): Since there are three digits to remember in 22 and 7, what have you gained? You have just as much to remember, but have lost the notion that π is “just over 3”. Is there a better rational approximation where we actually get out more digits than we put in? Here is a brief and rather low-brow investigation (and the chance to win something if you can do better).
Computation & Analysis