sage: a = RR(sqrt(2)); a 1.41421356237310 sage: b = sqrt(RealField(100)(2)); b 1.4142135623730950488016887242 sage: (ab).parent() Real Field with 53 bits of precision sage: b.parent() Real Field with 100 bits of precision sage: c = RealField(100)(a); c 1.4142135623730951454746218587 sage: bc 9.6672933134529652200516048587e17 
There are different types of numbers available. There are three major groups for floating
point arithmetic:
Most important are the Sage specific types. They use specific libraries and have more
functionality. Important are RR and CC, because their behavior is independent of the
underlying system and architecture using the MPFR library. RDF and CDF use the GSL library and are very fast and
compatible with the Sage framework.
The example on the left shows how to construct different types of floating point numbers and
interactions between them.

Algebra 


sage: M = IntegerModRing(7) sage: M(2) + M(8) 3 sage: M.list() [0, 1, 2, 3, 4, 5, 6] sage: A.<a,b,c> = AbelianGroup([2,2,3]); A Multiplicative Abelian Group isomorphic to C2 x C2 x C3 sage: A.order() 12 sage: A.list() [1, c, c^2, b, b*c, b*c^2, a, a*c, a*c^2, a*b, a*b*c, a*b*c^2] sage: c^5*b*a^4*c b 
Sage is built on an object oriented programming language. It uses this feature to describe
categories of mathematical objects. A good example are algebraic objects like groups, rings
and fields.
On the left side you can see some examples on how to construct and use them. The first one picks
two integers out of the ring of integers modulo 7. The list()
method lists all elements of that ring. Similarly, the second example constructs an abelian
group and assigns its generators to the letters a, b and c.

Combinatorics 

sage: X = species.SingletonSpecies() sage: Y = species.BinaryTreeSpecies() sage: L = CombinatorialSpecies() sage: L.define(X+X*Y*Y+Y*L) sage: L.generating_series().coefficients(10) [0, 1, 1, 3, 8, 23, 70, 222, 726, 2431] sage: L.structures([1,2,3]).count() 18 
The demonstration on the left hand side shows how Sage is able to work with combinatorial
objects from the theory of Combinatorial Species.

Embedding Sage into LaTeX 

\section{SageTex Examples} This is a small calculation: The sum of $1+2+\sqrt{3} = \sage{1+2+sqrt(3)}$. Here you can see a $sin()$Function: \sageplot{plot(sin(x), x, 0, 2*pi)} 
It is possible to call Sage commands from inside a LaTeX document. The
SageTex package provides special LaTeX commands that translate Sage code into a Python
file. Then this file is evaluated by Sage and the results of each calculation are
written back into the LaTeX file. This even works for graphics, sourcecode and saved
objects.
The example on the left shows how you can embed a formula like $$1+2+\sqrt{3}$$ and a plot
in a LaTeX document.
This package is part of the Sage distribution in the /examples/latex_embed subdirectory together with the documentation. You can
also obtain it online: CTAN: SageTex
Package
