This tutorial should take at most 3-4 hours to fully work through. You can read it in HTML or PDF versions, or from the Sage notebook click Help, then click Tutorial to interactively work through the tutorial from within Sage.

Though much of Sage is implemented using Python, no Python background is needed to read this tutorial. You will want to learn Python (a very fun language!) at some point, and there are many excellent free resources for doing so: the Python Beginner’s Guide [PyB] lists many options. If you just want to quickly try out Sage, this tutorial is the place to start. For example:

sage: 2 + 2
sage: factor(-2007)
-1 * 3^2 * 223

sage: A = matrix(4,4, range(16)); A
[ 0  1  2  3]
[ 4  5  6  7]
[ 8  9 10 11]
[12 13 14 15]

sage: factor(A.charpoly())
x^2 * (x^2 - 30*x - 80)

sage: m = matrix(ZZ,2, range(4))
sage: m[0,0] = m[0,0] - 3
sage: m
[-3  1]
[ 2  3]

sage: E = EllipticCurve([1,2,3,4,5]);
sage: E
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5
over Rational Field
sage: E.anlist(10)
[0, 1, 1, 0, -1, -3, 0, -1, -3, -3, -3]
sage: E.rank()

sage: k = 1/(sqrt(3)*I + 3/4 + sqrt(73)*5/9); k
36/(20*sqrt(73) + 36*I*sqrt(3) + 27)
sage: N(k)
0.165495678130644 - 0.0521492082074256*I
sage: N(k,30)      # 30 "bits"
0.16549568 - 0.052149208*I
sage: latex(k)
\frac{36}{20 \, \sqrt{73} + 36 i \, \sqrt{3} + 27}


If you do not have Sage installed on a computer and just want to try some commands, use it online at

See the Sage Installation Guide in the documentation section of the main Sage webpage [SA] for instructions on installing Sage on your computer. Here we merely make a few comments.

  1. The Sage download file comes with “batteries included”. In other words, although Sage uses Python, IPython, PARI, GAP, Singular, Maxima, NTL, GMP, and so on, you do not need to install them separately as they are included with the Sage distribution. However, to use certain Sage features, e.g., Macaulay or KASH, you must have the relevant programs installed on your computer already.

  2. The pre-compiled binary version of Sage (found on the Sage web site) may be easier and quicker to install than the source code version. Just unpack the file and run sage.

  3. If you’d like to use the SageTeX package (which allows you to embed the results of Sage computations into a LaTeX file), you will need to make SageTeX known to your TeX distribution. To do this, see the section “Make SageTeX known to TeX” in the Sage installation guide (this link should take you to a local copy of the installation guide). It’s quite easy; you just need to set an environment variable or copy a single file to a directory that TeX will search.

    The documentation for using SageTeX is located in $SAGE_ROOT/venv/share/texmf/tex/latex/sagetex/, where “$SAGE_ROOT” refers to the directory where you installed Sage – for example, /opt/sage-9.6.

Ways to Use Sage#

You can use Sage in several ways.

Longterm Goals for Sage#

  • Useful: Sage’s intended audience is mathematics students (from high school to graduate school), teachers, and research mathematicians. The aim is to provide software that can be used to explore and experiment with mathematical constructions in algebra, geometry, number theory, calculus, numerical computation, etc. Sage helps make it easier to interactively experiment with mathematical objects.

  • Efficient: Be fast. Sage uses highly-optimized mature software like GMP, PARI, GAP, and NTL, and so is very fast at certain operations.

  • Free and open source: The source code must be freely available and readable, so users can understand what the system is really doing and more easily extend it. Just as mathematicians gain a deeper understanding of a theorem by carefully reading or at least skimming the proof, people who do computations should be able to understand how the calculations work by reading documented source code. If you use Sage to do computations in a paper you publish, you can rest assured that your readers will always have free access to Sage and all its source code, and you are even allowed to archive and re-distribute the version of Sage you used.

  • Easy to compile: Sage should be easy to compile from source for Linux, OS X and Windows users. This provides more flexibility for users to modify the system.

  • Cooperation: Provide robust interfaces to most other computer algebra systems, including PARI, GAP, Singular, Maxima, KASH, Magma, Maple, and Mathematica. Sage is meant to unify and extend existing math software.

  • Well documented: Tutorial, programming guide, reference manual, and how-to, with numerous examples and discussion of background mathematics.

  • Extensible: Be able to define new data types or derive from built-in types, and use code written in a range of languages.

  • User friendly: It should be easy to understand what functionality is provided for a given object and to view documentation and source code. Also attain a high level of user support.