This document is one of More SageMath Tutorials. You may edit it on github. $$\def\NN{\mathbb{N}}$$ $$\def\ZZ{\mathbb{Z}}$$ $$\def\QQ{\mathbb{Q}}$$ $$\def\RR{\mathbb{R}}$$ $$\def\CC{\mathbb{C}}$$

# Demonstration: Basics¶

Arithmetic:

sage: 1 + 1

sage: 1 + 3

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')
sage: f = sin(x) - cos(x*y) + 1 / (x^3+1)
sage: f

sage: f.integrate(x)

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