Not a mathematician. Used Sage for engineering purposes. But I think one example would be: you're investigating something and get a large linear system, a poly

Not a mathematician. Used Sage for engineering purposes. But I think one example would be: you're investigating something and get a large linear system, a polynomial, etc. Sage can solve equations with symbols. It saves a lot of manual labor. Various expressions can be manipulated, simplified, etc. It can readily verify equality of huge expressions (which can be error prone by hand), and so on. More specific fields have more specific tools that could be used to assist, so it would depend on the field I guess. See here: https://ask.sagemath.org/question/24866/how-to-simplify-with-sage/ https://ask.sagemath.org/question/24866/how-to-simplify-with... Edit: I also think it should be fairly common to need to perform an actual numerical calculation in mathematical work? I think it's quite better than a simple calculator for that too, since it gives exact e.g. rational results, and approximations would be unfit for a proof (unless extensive precautions are made). I remember I got an expression for an eigenvalue of a matrix (as a function of some parameters) that would be very laborious to get by hand.