355/113 is a convergent of the continued fraction for pi. The convergents of a continued fraction (for an irrational number) give the best rational approximatio

355/113 is a convergent of the continued fraction for pi. The convergents of a continued fraction (for an irrational number) give the best rational approximations to the number. I'm teaching this stuff this summer; here's a link to my notes on this: http://www.millersville.edu/~bikenaga/number-theory/approximation-by-rationals/approximation-by-rationals.html http://www.millersville.edu/~bikenaga/number-theory/approxim... For instance, 278/125 is the best rational approximation to the cube root of 11 having denominator less than 155. The proof is pretty easy --- you don't have to do it by trial and error. As for why 355/113 doesn't come up in other expressions for pi --- I'm not sure why it should. The fact that two expressions converge to pi is no reason that I can see to expect that they should involve the same integer coefficients, say. There are infinitely many integers. There are infinitely many expressions for pi.