Geogebra is a free and multi-platform dynamic mathematics software

johndb

Geogebra is a free and multi-platform dynamic mathematics software for learning and teaching. Looks pretty good to me if you can't afford an advanced software program like Mathematica or Matlab.

Sankaku

I like Geogebra: cross platform, great fun for showing stuff to kids. Even does some very basic calc. Decent for quickly slapping together diagrams for other documents.
http://en.wikipedia.org/wiki/Geogebra

However, if you want an alternative to Mathematica, look at Maxima instead.
http://en.wikipedia.org/wiki/Maxima_(software)

Matlab has a few FOSS clones:
http://en.wikipedia.org/wiki/Scilab
http://en.wikipedia.org/wiki/GNU_Octave

Three very different problems need three very different types of program...