[lifthrasiir]

> no square root, sin, and so on with rationals

Better statement would be "no standard definition possible for square root, sin, and so on with rationals", as rationals with unbounded denominators don't have an inherent precision limit---something like ulps in floating point systems---and the max denominator wouldn't be a good choice for precision even when denominator is limited. Every such function now has to be given an explicit denominator value or similar strategy, which complicates both the interface and implementation.