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Free mathematics software (freeware) 
The below are open source tools and mostly available on GNU/Linux, and can run on other platforms too:
- GeoGebra - open source software for dynamic geometry, algebra and calculus for middle/high school and college. Exports interactive web pages.
- Kig A GUI based geometry editor. Includes some analytical functions and conic sections. Exports to several formats including SVG.
- Newt - free graphing program for windows
- Maxima - open source symbolic maths program, like Mathematica and Maple
- Axiom - another open source symbolic maths program
- pdf document - Other OS math software
- r- open source version of S-plus, a statistics program
- TeXmacs - WYSIWYG LaTeX
- Calc - an interactive RPN calculator package for numerical and symbolic calculations in Emacs
- gnuplot is a free graphing program for Linux, Windows (and Mac?)
- GraphCalc - A free, open source 2D/3D graphing application. (Windows)
- Graph - Graph is an open source application used to draw mathematical graphs in a coordinate system (Windows)
- GNU Octave - Matlab clone for Linux, Windows (and Mac?)
- Python numarray, powerful numerical calculations in a powerful programming language
- scilab - Matlab clone
- Qalculate - Multipurpose and very powerful desktop calculator
- SAGE - is a free open-source computer algebra system that includes programs for research and teaching in algebra, geometry, number theory, and cryptography. SAGE is mainly written in Python and provides interfaces to other mathematics packages.
- SciPy - SciPy is an open source library of scientific tools for Python
- Formulator Tarsia - is a free mathematical editor designed for teachers of mathematics creating the activities in a form of jigsaws, rectangular cards or dominos for later use in a class.
- Armadillo - Fast C++ matrix library with optional interfaces to LAPACK and ATLAS
- Pari/GP - number theory computations
Non-free software 
- Mathematica 5.2 - Advanced commercial math program. (Linux, Macintosh, Windows)
- Maple - Mature commercial math program. (Linux, Macintosh, Windows)
- Maple Studio Free online worksheet for plotting, based on Maple. (Browser with Java-Plugin like Mozilla, Internet Explorer, Safari)
- MuPad - Another advanced math program with free versions available. (Linux, Macintosh, Windows)
- Magma - Powerful package for solving computational problems in algebra.