Algebraic numbers can be very handy.

I was making myself toy, tapered kaleidoscopes using one piece cardboard plans. One needed to ensure that the dihedral angles between the mirrors be precise.

This is not easy to do with the usual middle-school geometry-box rulers and protractors. Graduations not fine enough, precise enough, extending long straight lines using a small ruler not straight enough ...

However the dimensions being all algebraic numbers one could use entirely straight edge and compass constructions. Had much better luck this way, with a pointy enough pencil.

A proper drafting board and a T-square or a drafter would have made things easier for parallel and perpendicular translations. But one can do those with compass too.