You might use it just as a "calculator" (type "math" in a terminal) to differentiate
D[Sin[5*x]^2,x]
or integrate
Integrate[ 2*(x+3)/Sqrt[x^2+1],x]
solve differential equations symbolically
DSolve[y''[x]==5*y[x],y[x],x]
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
N[Pi*E*Sin[Sqrt[2]],100]
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.
