Tutorial: Using the Sage notebook, navigating the help system, first exercises

This worksheet is based on William Stein’s JPL09__intro_to_sage.sws worksheet and the Sage days 20.5_demo worksheet and aims to be an interactive introduction to Sage through exercises. You will learn how to use the notebook and call the help.

Making this help page into a worksheet

If you are browsing this document as a static web page, you can see all the examples; however you need to copy-paste them one by one to experiment with them. Use the Upload worksheet button of the notebook and copy-paste the URL of this page to obtain an editable copy in your notebook.

If you are browsing this document as part of Sage’s live documentation, you can play with the examples directly here; however your changes will be lost when you close this page. Use Copy worksheet from the File... menu at the top of this page to get an editable copy in your notebook.

Both in the live tutorial and in the notebook, you can clear all output by selecting Delete All Output from the Action... menu next to the File... menu at the top of the worksheet.

Entering, Editing and Evaluating Input

To evaluate code in the Sage Notebook, type the code into an input cell and press shift-enter or click the evaluate link. Try it now with a simple expression (e.g., \(2+3\)). The first time you evaluate a cell takes longer than subsequent times since a new Sage process is started:

sage: 2 + 3

sage: # edit here

sage: # edit here

To create new input cells, click the blue line that appears between cells when you move your mouse around. Try it now:

sage: 1 + 1

sage: # edit here

You can go back and edit any cell by clicking in it (or using the arrow keys on your keyboard to move up or down). Go back and change your \(2+3\) above to \(3+3\) and re-evaluate it. An empty cell can be deleted with backspace.

You can also edit this text right here by double clicking on it, which will bring up the TinyMCE Javascript text editor. You can even put embedded mathematics like this $sin(x) - y^3$ by using dollar signs just like in TeX or LaTeX.

Help systems

There are various ways of getting help in Sage.

  • navigate through the documentation (there is a link Help at the top right of the worksheet),
  • tab completion,
  • contextual help.

We detail below the latter two methods through examples.

Completion and contextual documentation

Start typing something and press the tab key. The interface tries to complete it with a command name. If there is more than one completion, then they are all presented to you. Remember that Sage is case sensitive, i.e. it differentiates upper case from lower case. Hence the tab completion of klein won’t show you the KleinFourGroup command that builds the group \(\ZZ/2 \times \ZZ/2\) as a permutation group. Try it on the next cells:

sage: klein<tab>

sage: Klein<tab>

To see documentation and examples for a command, type a question mark ? at the end of the command name and press the tab key as in:

sage: KleinFourGroup?<tab>
sage: # edit here

Exercise A

What is the largest prime factor of \(600851475143\)?

sage: factor?<tab>
sage: # edit here

In the above manipulations we have not stored any data for later use. This can be done in Sage with the = symbol as in:

sage: a = 3
sage: b = 2
sage: print a+b

This can be understood as Sage evaluating the expression to the right of the = sign and creating the appropriate object, and then associating that object with a label, given by the left-hand side (see the foreword of Tutorial: Objects and Classes in Python and Sage for details). Multiple assignments can be done at once:

sage: a,b = 2,3
sage: print a,b
2 3

This allows us to swap the values of two variables directly:

sage: a,b = 2,3
sage: a,b = b,a
sage: print a,b
3 2

We can also assign a common value to several variables simultaneously:

sage: c = d = 1
sage: c, d
(1, 1)
sage: d = 2
sage: c, d
(1, 2)

Note that when we use the word variable in the computer-science sense we mean “a label attached to some data stored by Sage”. Once an object is created, some methods apply to it. This means functions but instead of writing f(my_object) you write my_object.f():

sage: p = 17
sage: p.is_prime()

See Tutorial: Objects and Classes in Python and Sage for details. To know all methods of an object you can once more use tab-completion. Write the name of the object followed by a dot and then press tab:

sage: a.<tab>

sage: # edit here

Exercise B

Create the permutation 51324 and assign it to the variable p.

sage: Permutation?<tab>
sage: # edit here

What is the inverse of p?

sage: p.inv<tab>

sage: # edit here

Does p have the pattern 123? What about 1234? And 312? (even if you don’t know what a pattern is, you should be able to find a command that does this).

sage: p.pat<tab>

sage: # edit here

Some linear algebra

Exercise C

Use the matrix() command to create the following matrix.

\[\begin{split}M = \left(\begin{array}{rrrr} 10 & 4 & 1 & 1 \\ 4 & 6 & 5 & 1 \\ 1 & 5 & 6 & 4 \\ 1 & 1 & 4 & 10 \end{array}\right)\end{split}\]
sage: matrix?<tab>
sage: # edit here

Then, using methods of the matrix,

  1. Compute the determinant of the matrix.
  2. Compute the echelon form of the matrix.
  3. Compute the eigenvalues of the matrix.
  4. Compute the kernel of the matrix.
  5. Compute the LLL decomposition of the matrix (and lookup the documentation for what LLL is if needed!)
sage: # edit here

sage: # edit here

Now that you know how to access the different methods of matrices,

  1. Create the vector \(v = (1,-1,-1,1)\).
  2. Compute the two products: \(M\cdot v\) and \(v\cdot M\). What mathematically borderline operation is Sage doing implicitly?
sage: vector?<tab>
sage: # edit here


Vectors in Sage are row vectors. A method such as eigenspaces might not return what you expect, so it is best to specify eigenspaces_left or eigenspaces_right instead. Same thing for kernel (left_kernel or right_kernel), and so on.

Some Plotting

The plot() command allows you to draw plots of functions. Recall that you can access the documentation by pressing the tab key after writing plot? in a cell:

sage: plot?<tab>
sage: # edit here

Here is a simple example:

sage: var('x')   # make sure x is a symbolic variable
sage: plot(sin(x^2), (x,0,10))

Here is a more complicated plot. Try to change every single input to the plot command in some way, evaluating to see what happens:

sage: P = plot(sin(x^2), (x,-2,2), rgbcolor=(0.8,0,0.2), thickness=3, linestyle='--', fill='axis')
sage: show(P, gridlines=True)

Above we used the show() command to show a plot after it was created. You can also use P.show instead:

sage: P.show(gridlines=True)

Try putting the cursor right after P.show( and pressing tab to get a list of the options for how you can change the values of the given inputs.

sage: P.show(

Plotting multiple functions at once is as easy as adding them together:

sage: P1 = plot(sin(x), (x,0,2*pi))
sage: P2 = plot(cos(x), (x,0,2*pi), rgbcolor='red')
sage: P1 + P2

Symbolic Expressions

Here is an example of a symbolic function:

sage: f(x) = x^4 - 8*x^2 - 3*x + 2
sage: f(x)
x^4 - 8*x^2 - 3*x + 2

sage: f(-3)

This is an example of a function in the mathematical variable \(x\). When Sage starts, it defines the symbol \(x\) to be a mathematical variable. If you want to use other symbols for variables, you must define them first:

sage: x^2
sage: u + v
Traceback (most recent call last):
NameError: name 'u' is not defined

sage: var('u v')
(u, v)
sage: u + v
u + v

Still, it is possible to define symbolic functions without first defining their variables:

sage: f(w) = w^2
sage: f(3)

In this case those variables are defined implicitly:

sage: w

Exercise D

Define the symbolic function \(f(x) = x \sin(x^2)\). Plot \(f\) on the domain \([-3,3]\) and color it red. Use the find_root() method to numerically approximate the root of \(f\) on the interval \([1,2]\):

sage: # edit here

Compute the tangent line to \(f\) at \(x=1\):

sage: # edit here

Plot \(f\) and the tangent line to \(f\) at \(x=1\) in one image:

sage: # edit here

Exercise E (Advanced)

Solve the following equation for \(y\):
\[y = 1 + x y^2\]

There are two solutions, take the one for which \(\lim_{x\to0}y(x)=1\). (Don’t forget to create the variables \(x\) and \(y\)!).

sage: # edit here

Expand \(y\) as a truncated Taylor series around \(0\) and containing \(n=10\) terms.

sage: # edit here

Do you recognize the coefficients of the Taylor series expansion? You might want to use the On-Line Encyclopedia of Integer Sequences, or better yet, Sage’s class OEIS which queries the encyclopedia:

sage: oeis?<tab>
sage: # edit here

Congratulations for completing your first Sage tutorial!