The math software I use is Mathematica (Home Ed), Sage, Octave, R, and TeXShop. All open source except for Mathematica.
Mathematica has a few advantages. The input notation can look more like traditional mathematics: adjacency instead of * for multiplication, superscript instead of ^ for exponentiation, fraction notation. I think that cuts down on errors. Variables are symbolic in Mathematica by default. The Mathematica language is something of a defacto standard and WolframAlpha understands it to a certain extent, so you have access for free if you have internet. WolframAlpha mobile gives you CAS on your phone.
That said, an open source version of Mathematica is desirable and Sage should be supported. I keep a table of equivalences at http://hyperpolyglot.org/math to try to lessen the mental burden of using two equivalent products.
R is sweet. Octave has its ups and downs--I'm thinking Numpy/SciPy/Matlibplot may be a better route for my own personal computing (still learning though)--especially since Python has bindings practically for everything.
I agree that Sage still needs work (the 100+ bugs submitted daily to the RSS support feed says so!) but it works really well.
Octave is free and I used MATLAB in school so it made sense to install it. I've never had problems with Octave though I've heard others complain about it. I don't use it that much. For graphics I always use R.
Maybe I should have mentioned Coq in my previous list. I installed it a few days ago. Haven't learned to use it yet.
Octave does not have a pre-packaged optimization routine or a usable toolbox of which I am aware, and I find GnuPlot abysmal (but this is likely due to my lack of experience with it).
These are probably easily corrected. But solutions already exist within the FLOSS community with other packages such as Sage (and, of course, Numpy and the rest)--reinventing the wheel is something few have time for! :-) This is a big reason I like Sage: it is designed like the Borg in that it uses bindings to connect into other software (both proprietary and open source).
Yes! Star Trek reference and SageMath in the same sentence. It is a good night.
Mathematica has a few advantages. The input notation can look more like traditional mathematics: adjacency instead of * for multiplication, superscript instead of ^ for exponentiation, fraction notation. I think that cuts down on errors. Variables are symbolic in Mathematica by default. The Mathematica language is something of a defacto standard and WolframAlpha understands it to a certain extent, so you have access for free if you have internet. WolframAlpha mobile gives you CAS on your phone.
That said, an open source version of Mathematica is desirable and Sage should be supported. I keep a table of equivalences at http://hyperpolyglot.org/math to try to lessen the mental burden of using two equivalent products.