Arithmetic:
sage: 1 + 1 2 sage: 1 + 3 4 sage: ( 1 + 2 * (3 + 5)^2 ) * 2 258 sage: 20/14 10/7 sage: 2^1000 107...376 sage: numerical_approx(20/14) 1.42857142857143 sage: 20.0/14 sage: numerical_approx(pi, 10000) 3.1415926535897932384626...
Editing the worksheet!
Polynomials:
sage: factor(x^100 - 1) (x - 1)*(x + 1)*(x^2 + 1)*(x^4 - x^3 + x^2 - x + 1)*(x^4 + x^3 + x^2 + x + 1)*(x^8 - x^6 + x^4 - x^2 + 1)*(x^20 - x^15 + x^10 - x^5 + 1)*(x^20 + x^15 + x^10 + x^5 + 1)*(x^40 - x^30 + x^20 - x^10 + 1)
sage: %display latex sage: factor(x^100 - 1) (x - 1)*(x + 1)*(x^2 + 1)*(x^4 - x^3 + x^2 - x + 1)*(x^4 + x^3 + x^2 + x + 1)*(x^8 - x^6 + x^4 - x^2 + 1)*(x^20 - x^15 + x^10 - x^5 + 1)*(x^20 + x^15 + x^10 + x^5 + 1)*(x^40 - x^30 + x^20 - x^10 + 1)
Symbolic calculations:
sage: var('x,y')
(x, y)
sage: f = sin(x) - cos(x*y) + 1 / (x^3+1)
sage: f
1/(x^3 + 1) - cos(x*y) + sin(x)
sage: f.integrate(x)
1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) ...
sage: expr = sin(x) + sin(2 * x) + sin(3 * x) sage: solve(expr, x) [sin(3*x) == -sin(2*x) - sin(x)]
sage: find_root(expr, 0.1, pi) 2.0943951023931957
.. todo:: arbitrary precision numerical approximation of the solution
sage: f = expr.simplify_trig(); f 2*(2*cos(x)^2 + cos(x))*sin(x) sage: solve(f, x) [x == 0, x == 2/3*pi, x == 1/2*pi]
Statistics:
sage: print(r.summary(r.c(1,2,3,111,2,3,2,3,2,5,4)))
.. todo:: other examples from MuPAD-Combinat/lib/DOC/demo/mupad.tex