What is the closest rational approximation to pi?

Every irrational number can be approximated by rationals as closely as we want, since rational numbers are everywhere. But what makes an approximation better than another one? Two things: - How close is it to the target - How big is the denominator. For example, 314159/100000 is closer to pi than 22/7, but 22/7 has a smaller denominator. The best approximations are called 'convergents', and they can be easily found using a continuous fraction expression of the target number. How 'good' an approximation is can be formalized using a theorem of Legendre. This is very useful in quantum computing,