Since nobody's spelled it out directly: Let an approximant be a rational number p/q which is near the desired irrational x. Then, let p/q be a best approximant iff no other rational with denominator smaller than q is closer to x.

It's a delightfully beautiful result that the best approximants for x are given by x's continued fraction: https://en.wikipedia.org/wiki/Continued_fraction#Best_ration...