As others have said, the convergents of the simple continued fraction representation give the best rational approximation [1] relative to the size of the fracti

As others have said, the convergents of the simple continued fraction representation give the best rational approximation [1] relative to the size of the fraction, in the strongest relevant metric: that of absolute error bound. Proofs can be found in any book on Diophantine Approximation; the classic (and readable) reference is Khinchin's Continued Fractions book. [1] https://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations https://en.wikipedia.org/wiki/Continued_fraction#Best_ration...