> their best known approximation had pi ≈ 25/8, which is too small

I'm curious if they had a better one that we don't know of yet—their best known approximation of sqrt(2) is significantly more accurate.

Finite decimal representations being of course the most convenient notation for inequalities like this.

Note that "expanding them to some number of decimal places" gives a somewhat misleading idea about how this works. What you're actually doing is computing a good enough approximation of pi, and expressing that as a decimal. But this is not the same kind of simple process that naturally gives decimals as it is for a rational fraction. Instead, you have to find some series with rational elements which converges to pi, and then compute enough terms of that series that you have a good enough approximation of pi for your purpose. Ideally, since you're interested in an inequality, you'd pick a series which is monotoniclaly increasing or decreasing, so that you know that computing more terms can't put you below or above the target number after you've reached a conclusion. But there is no canonical answer, there are numerous series which converge to pi that you could use, and they would givw you different decimal expansions as you are computing them.