There is a rigorous notion about how well an irrational can be approximated by a rational, related to its continued fraction representation ( https://en.wikiped

There is a rigorous notion about how well an irrational can be approximated by a rational, related to its continued fraction representation ( https://en.wikipedia.org/wiki/Continued_fraction https://en.wikipedia.org/wiki/Continued_fraction ). I'm having trouble finding a better description than the note in that WP page, but the idea is if the cf representation has a "surprisingly large" value, then the rational number you get by truncating the sequence there will be a very good approximation, like 355/133 is for pi (see the comment in https://math.stackexchange.com/questions/435668/finding-the-value-of-a-continued-fraction https://math.stackexchange.com/questions/435668/finding-the-...).