Maybe it's better to formalize this in terms of having a rational interval as the approximation of an irrational - for example the interval [19/7, 87/32] approx

Maybe it's better to formalize this in terms of having a rational interval as the approximation of an irrational - for example the interval [19/7, 87/32] approximates e. That's an interval of width 1/224. Then if [a/b, c/d] contains the irrational number x, then: - either [a/b, (a+c)/(b+d)] or [(a+c)/(b+d), b/d] contains x, and - whichever one contains x is smaller than the original. Written that way it's trivial, though. The real meat, I suppose, is that using the mediant keeps the numerators and denominators small. In this case, [106/39, 87/32] has width 1/1248 and contains e, whereas using the mean would give you the interval [1217/448, 87/32] of width 1/448.