Sage Maths is free open source software for doing virtually every type of maths you can imagine. Not just numerical maths, but symbolic maths too – you can give Sage an equation and it will tell you what the equation of its integral or differential is, for example. And it will do numerical maths, plot graphs, analyze statistical information and solve equations or sets of equations. In fact, it will do virtually anything mathematical you can think of.
Sage was developed as an open source alternative to commercial systems like Mathematica and Matlab (it has most but not all of the functionality of both) because mathematicians and scientists need to be able to understand and review the algorithms their software uses – something not possible with a closed system.
Originally developed for graduate mathematicians, Sage is now at the stage where it is useful and interesting to professional and hobbyist mechanical and electronic engineers, amateur astronomers, business number crunchers, and people who just want to know more maths than they do. It runs on Linux, Windows and OS X, and lately people have managed to run it on both Apple iThings and Android smartphones.