Note that writing sqrt(2) as 1.41 or 1.41421 or any other decimal expansion you might want to write is incorrect: you will always get some roundoff error. If you want to calculate that sqrt(2)*sqrt(2)=2 then you can’t do so by multiplying the decimal expansions.

Sure if a question asks for the escape velocity from Jupiter this has an approximate numerical value, but you don't just start by throwing numbers at a wall, you get the simplest equation which represents the value you're interested in an then evaluate it once you have a single equation for that parameter.

Yes sqrt(2)*pi has a numerical approximation but you don't want that right at the start of taking about something like spin orbitals or momenta of spinning disks. Doing the latter compounds errors.

It's no different to keeping around "i"/"j" until you need to express a phase or angle as it's cleaner and avoids compounding accuracy errors.