Does sage also handle numerical stuff? The impression I got was that it was a bit more general than Maxima.

However, I am terribly frightened by programs like this, as I suspect they are much smarter than me, so I tend to avoid them.

Yes, Sage includes Python and the Python numerical stack (numpy, scipy, etc). It also has a lot of new code related to interval arithmetic (arbitrary precision numerical computation with proven error bounds).

I do numerical linear algebra stuff, but every time I hear about interval arithmetic my ears perk up. It just seems like our error bounds require caveats (a-hah, I actually gave you the backward error, nyaah!) while theirs are just numbers.