This is a list of free software for mathematics.
TeX is a typesetting system written by Donald Knuth. TeX is designed for the typesetting of mathematical formulae. TeX Live is an easy-to-use TeX system runs on windows, linux and mac.
pari/gp is a wildly used computer algebra system. It is designed for fast computations in number theory. It runs on windows, linux, mac and android.
Boost Library is a set of C++ library that add various mathematical functions. It runs on most systems supporting C++11.
The following is a list of functions the author frequently uses.
UBASIC is a BASIC-like programming language. It is designed for fast computations in number theory. It run on MS-DOS.
GAP is a computer algebra system designed for computations in group theory. It runs on windows, linux and mac.