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Mathematica is a high level programming language with strong built-in symbolic, numerical and graphic capabilities. Mathematica code is in general short, intuitive and can be developed in a short time. You can download mathematica here go to" to request a password for the Licence number L2482-2405. The current version is Mathematica 6.0.
You might use it just as a "calculator" (type "math" in a terminal) to differentiate
or integrate
Integrate[ 2*(x+3)/Sqrt[x^2+1],x]
solve differential equations symbolically
or numerically (example lorentz.m ), solve partial differential equations numerically (example heat.m ), to plot surfaces (example surface.m , Level curves (example level.m ).
You might also just want to use it as a calculator with indefinite precision, like for real numbers
or integers Factor.m . It allows to implement and test efficiency of number theoretical algorithms as for example Pollard.m , try to crack encryption schemes (example Curley.m ), or for illustrations of topics in Mathematics: Lineintegral.m, Mandelbrot.m , Julia.m , Henon.m , Stellate.m , Shirpinsky.m , Menger.m etc.
A mini tutorial (HTML) .

Simplicity, Clarity, Generality B.W. Kernighan, R. Pike, in "The Practice of Programming".