Free online mathematical software

I propose Sage. In my opinion it is the best free open-source mathematics software system.

Under development but already working very nicely is Sympy, a symbolic math package for Python.

The usage is similar to other math software, there is even a live version available. A session might look like the following:

>>> from sympy import *
>>> x = symbols('x')
>>> f = cos(x)
>>> f.series()
1 - x**2/2 + x**4/24 + O(x**6)
>>> f.series(n=12)
1 - x**2/2 + x**4/24 - x**6/720 + x**8/40320 - x**10/3628800 + O(x**12)
>>> latex(f.series(n=12))

$1 - \frac{x^{2}}{2} + \frac{x^{4}}{24} - \frac{x^{6}}{720} + \frac{x^{8}}{40320} - \frac{x^{10}}{3628800} + \mathcal{O}\left(x^{12}\right)$

>>> diff(f, x)
−sin(x)

An overview of the possible functions is given in the online documentation. It is free software and has quite readable source code.

I recomend MAXIMA. Of home page of this CAS we have the apresentation

"Maxima is a system for the manipulation of symbolic and numerical expressions, including differentiation, integration, Taylor series, Laplace transforms, ordinary differential equations, systems of linear equations, polynomials, and sets, lists, vectors, matrices, and tensors. Maxima yields high precision numeric results by using exact fractions, arbitrary precision integers, and variable precision floating point numbers. Maxima can plot functions and data in two and three dimensions."