Getting Help - Tutorial

Skip to content
Back to top
View this page
Edit this page

# Getting Help

Sage has extensive built-in documentation, accessible by typing the name of a function or a constant (for example), followed by a question mark:

Sage

    sage: tan?
    Type:        <class 'sage.calculus.calculus.Function_tan'>
    Definition:  tan( [noargspec] )
    Docstring:

        The tangent function

        EXAMPLES:
            sage: tan(pi)
            0
            sage: tan(3.1415)
            -0.0000926535900581913
            sage: tan(3.1415/4)
            0.999953674278156
            sage: tan(pi/4)
            1
            sage: tan(1/2)
            tan(1/2)
            sage: RR(tan(1/2))
            0.546302489843790
    sage: log2?
    Type:        <class 'sage.functions.constants.Log2'>
    Definition:  log2( [noargspec] )
    Docstring:

        The natural logarithm of the real number 2.

        EXAMPLES:
            sage: log2
            log2
            sage: float(log2)
            0.69314718055994529
            sage: RR(log2)
            0.693147180559945
            sage: R = RealField(200); R
            Real Field with 200 bits of precision
            sage: R(log2)
            0.69314718055994530941723212145817656807550013436025525412068
            sage: l = (1-log2)/(1+log2); l
            (1 - log(2))/(log(2) + 1)
            sage: R(l)
            0.18123221829928249948761381864650311423330609774776013488056
            sage: maxima(log2)
            log(2)
            sage: maxima(log2).float()
            .6931471805599453
            sage: gp(log2)
            0.693147180559945309417232121...
    sage: sudoku?
    File:        sage/local/lib/python2.5/site-packages/sage/games/sudoku.py
    Type:        <... 'function'>
    Definition:  sudoku(A)
    Docstring:

        Solve the 9x9 Sudoku puzzle defined by the matrix A.

        EXAMPLE:
            sage: A = matrix(ZZ,9,[5,0,0, 0,8,0, 0,4,9, 0,0,0, 5,0,0,
        0,3,0, 0,6,7, 3,0,0, 0,0,1, 1,5,0, 0,0,0, 0,0,0, 0,0,0, 2,0,8, 0,0,0,
        0,0,0, 0,0,0, 0,1,8, 7,0,0, 0,0,4, 1,5,0,   0,3,0, 0,0,2,
        0,0,0, 4,9,0, 0,5,0, 0,0,3])
            sage: A
            [5 0 0 0 8 0 0 4 9]
            [0 0 0 5 0 0 0 3 0]
            [0 6 7 3 0 0 0 0 1]
            [1 5 0 0 0 0 0 0 0]
            [0 0 0 2 0 8 0 0 0]
            [0 0 0 0 0 0 0 1 8]
            [7 0 0 0 0 4 1 5 0]
            [0 3 0 0 0 2 0 0 0]
            [4 9 0 0 5 0 0 0 3]
            sage: sudoku(A)
            [5 1 3 6 8 7 2 4 9]
            [8 4 9 5 2 1 6 3 7]
            [2 6 7 3 4 9 5 8 1]
            [1 5 8 4 6 3 9 7 2]
            [9 7 4 2 1 8 3 6 5]
            [3 2 6 7 9 5 4 1 8]
            [7 8 2 9 3 4 1 5 6]
            [6 3 5 1 7 2 8 9 4]
            [4 9 1 8 5 6 7 2 3]

Python

    >>> from sage.all import *
    >>> tan?
    Type:        <class 'sage.calculus.calculus.Function_tan'>
    Definition:  tan( [noargspec] )
    Docstring:

        The tangent function

        EXAMPLES:
    >>> tan(pi)
            0
    >>> tan(RealNumber('3.1415'))
            -0.0000926535900581913
    >>> tan(RealNumber('3.1415')/Integer(4))
            0.999953674278156
    >>> tan(pi/Integer(4))
            1
    >>> tan(Integer(1)/Integer(2))
            tan(1/2)
    >>> RR(tan(Integer(1)/Integer(2)))
            0.546302489843790
    >>> log2?
    Type:        <class 'sage.functions.constants.Log2'>
    Definition:  log2( [noargspec] )
    Docstring:

        The natural logarithm of the real number 2.

        EXAMPLES:
    >>> log2
            log2
    >>> float(log2)
            0.69314718055994529
    >>> RR(log2)
            0.693147180559945
    >>> R = RealField(Integer(200)); R
            Real Field with 200 bits of precision
    >>> R(log2)
            0.69314718055994530941723212145817656807550013436025525412068
    >>> l = (Integer(1)-log2)/(Integer(1)+log2); l
            (1 - log(2))/(log(2) + 1)
    >>> R(l)
            0.18123221829928249948761381864650311423330609774776013488056
    >>> maxima(log2)
            log(2)
    >>> maxima(log2).float()
            .6931471805599453
    >>> gp(log2)
            0.693147180559945309417232121...
    >>> sudoku?
    File:        sage/local/lib/python2.5/site-packages/sage/games/sudoku.py
    Type:        <... 'function'>
    Definition:  sudoku(A)
    Docstring:

        Solve the 9x9 Sudoku puzzle defined by the matrix A.

        EXAMPLE:
    >>> A = matrix(ZZ,Integer(9),[Integer(5),Integer(0),Integer(0), Integer(0),Integer(8),Integer(0), Integer(0),Integer(4),Integer(9), Integer(0),Integer(0),Integer(0), Integer(5),Integer(0),Integer(0),
        0,3,0, 0,6,7, 3,0,0, 0,0,1, 1,5,0, 0,0,0, 0,0,0, 0,0,0, 2,0,8, 0,0,0,
        0,0,0, 0,0,0, 0,1,8, 7,0,0, 0,0,4, 1,5,0,   0,3,0, 0,0,2,
        0,0,0, 4,9,0, 0,5,0, 0,0,3])
    >>> A
            [5 0 0 0 8 0 0 4 9]
            [0 0 0 5 0 0 0 3 0]
            [0 6 7 3 0 0 0 0 1]
            [1 5 0 0 0 0 0 0 0]
            [0 0 0 2 0 8 0 0 0]
            [0 0 0 0 0 0 0 1 8]
            [7 0 0 0 0 4 1 5 0]
            [0 3 0 0 0 2 0 0 0]
            [4 9 0 0 5 0 0 0 3]
    >>> sudoku(A)
            [5 1 3 6 8 7 2 4 9]
            [8 4 9 5 2 1 6 3 7]
            [2 6 7 3 4 9 5 8 1]
            [1 5 8 4 6 3 9 7 2]
            [9 7 4 2 1 8 3 6 5]
            [3 2 6 7 9 5 4 1 8]
            [7 8 2 9 3 4 1 5 6]
            [6 3 5 1 7 2 8 9 4]
            [4 9 1 8 5 6 7 2 3]

Sage also provides ‘Tab completion’: type the first few letters of a function and then hit the `Tab` key. For example, if you type `ta` followed by `Tab`, Sage will print `tachyon, tan, tanh, taylor`. This provides a good way to find the names of functions and other structures in Sage.

# Functions, Indentation, and Counting

To define a new function in Sage, use the `def` command and a colon after the list of variable names. For example:

Sage

    sage: def is_even(n):
    ....:     return n%2 == 0
    sage: is_even(2)
    True
    sage: is_even(3)
    False

Python

    >>> from sage.all import *
    >>> def is_even(n):
    ...     return n%Integer(2) == Integer(0)
    >>> is_even(Integer(2))
    True
    >>> is_even(Integer(3))
    False

Note: Depending on which version of the tutorial you are viewing, you may see three dots `....:` on the second line of this example. Do not type them; they are just to emphasize that the code is indented. Whenever this is the case, press [Return/Enter] once at the end of the block to insert a blank line and conclude the function definition.

You do not specify the types of any of the input arguments. You can specify multiple inputs, each of which may have an optional default value. For example, the function below defaults to `divisor=2` if `divisor` is not specified.

Sage

    sage: def is_divisible_by(number, divisor=2):
    ....:     return number%divisor == 0
    sage: is_divisible_by(6,2)
    True
    sage: is_divisible_by(6)
    True
    sage: is_divisible_by(6, 5)
    False

[Python]

    >>> from sage.all import *
    >>> def is_divisible_by(number, divisor=Integer(2)):
    ...     return number%divisor == Integer(0)
    >>> is_divisible_by(Integer(6),Integer(2))
    True
    >>> is_divisible_by(Integer(6))
    True
    >>> is_divisible_by(Integer(6), Integer(5))
    False

You can also explicitly specify one or either of the inputs when calling the function; if you specify the inputs explicitly, you can give them in any order:

Sage

    sage: is_divisible_by(6, divisor=5)
    False
    sage: is_divisible_by(divisor=2, number=6)
    True

Python

    >>> from sage.all import *
    >>> is_divisible_by(Integer(6), divisor=Integer(5))
    False
    >>> is_divisible_by(divisor=Integer(2), number=Integer(6))
    True

In Python, blocks of code are not indicated by curly braces or begin and end blocks as in many other languages. Instead, blocks of code are indicated by indentation, which must match up exactly. For example, the following is a syntax error because the `return` statement is not indented the same amount as the other lines above it.

Sage

    sage: def even(n):
    ....:     v = []
    ....:     for i in range(3,n):
    ....:         if i % 2 == 0:
    ....:             v.append(i)
    ....:    return v
    Syntax Error:
           return v

[Python]

    >>> from sage.all import *
    >>> def even(n):
    ...     v = []
    ...     for i in range(Integer(3),n):
    ...         if i % Integer(2) == Integer(0):
    ...             v.append(i)
    ...    return v
    Syntax Error:
           return v

If you fix the indentation, the function works:

Sage

    sage: def even(n):
    ....:     v = []
    ....:     for i in range(3,n):
    ....:         if i % 2 == 0:
    ....:             v.append(i)
    ....:     return v
    sage: even(10)
    [4, 6, 8]

Python

    >>> from sage.all import *
    >>> def even(n):
    ...     v = []
    ...     for i in range(Integer(3),n):
    ...         if i % Integer(2) == Integer(0):
    ...             v.append(i)
    ...     return v
    >>> even(Integer(10))
    [4, 6, 8]

Semicolons are not needed at the ends of lines; a line is in most cases ended by a newline. However, you can put multiple statements on one line, separated by semicolons:

Sage

    sage: a = 5; b = a + 3; c = b^2; c
    64

[Python]

    >>> from sage.all import *
    >>> a = Integer(5); b = a + Integer(3); c = b**Integer(2); c
    64

If you would like a single line of code to span multiple lines, use a terminating backslash:

Sage

    sage: 2 + \
    ....:    3
    5

Python

    >>> from sage.all import *
    >>> Integer(2) +    Integer(3)
    5

In Sage, you count by iterating over a range of integers. For example, the first line below is exactly like `for(i=0; i<3; i++)` in C++ or Java:

Sage

    sage: for i in range(3):
    ....:     print(i)
    0
    1
    2

[Python]

    >>> from sage.all import *
    >>> for i in range(Integer(3)):
    ...     print(i)
    0
    1
    2

The first line below is like `for(i=2;i<5;i++)`.

Sage

    sage: for i in range(2,5):
    ....:     print(i)
    2
    3
    4

Python

    >>> from sage.all import *
    >>> for i in range(Integer(2),Integer(5)):
    ...     print(i)
    2
    3
    4

The third argument controls the step, so the following is like `for(i=1;i<6;i+=2)`.

Sage

    sage: for i in range(1,6,2):
    ....:     print(i)
    1
    3
    5

[Python]

    >>> from sage.all import *
    >>> for i in range(Integer(1),Integer(6),Integer(2)):
    ...     print(i)
    1
    3
    5

Often you will want to create a nice table to display numbers you have computed using Sage. One easy way to do this is to use string formatting. Below, we create three columns each of width exactly 6 and make a table of squares and cubes.

Sage

    sage: for i in range(5):
    ....:     print('%6s %6s %6s' % (i, i^2, i^3))
         0      0      0
         1      1      1
         2      4      8
         3      9     27
         4     16     64

Python

    >>> from sage.all import *
    >>> for i in range(Integer(5)):
    ...     print('%6s %6s %6s' % (i, i**Integer(2), i**Integer(3)))
         0      0      0
         1      1      1
         2      4      8
         3      9     27
         4     16     64

The most basic data structure in Sage is the list, which is – as the name suggests – just a list of arbitrary objects. For example, using `range`, the following command creates a list:

Sage

    sage: list(range(2,10))
    [2, 3, 4, 5, 6, 7, 8, 9]

[Python]

    >>> from sage.all import *
    >>> list(range(Integer(2),Integer(10)))
    [2, 3, 4, 5, 6, 7, 8, 9]

Here is a more complicated list:

Sage

    sage: v = [1, "hello", 2/3, sin(x^3)]
    sage: v
    [1, 'hello', 2/3, sin(x^3)]

Python

    >>> from sage.all import *
    >>> v = [Integer(1), "hello", Integer(2)/Integer(3), sin(x**Integer(3))]
    >>> v
    [1, 'hello', 2/3, sin(x^3)]

List indexing is 0-based, as in many programming languages.

Sage

    sage: v[0]
    1
    sage: v[3]
    sin(x^3)

[Python]

    >>> from sage.all import *
    >>> v[Integer(0)]
    1
    >>> v[Integer(3)]
    sin(x^3)

Use `len(v)` to get the length of `v`, use `v.append(obj)` to append a new object to the end of `v`, and use `del v[i]` to delete the \(i^{th}\) entry of `v`:

Sage

    sage: len(v)
    4
    sage: v.append(1.5)
    sage: v
    [1, 'hello', 2/3, sin(x^3), 1.50000000000000]
    sage: del v[1]
    sage: v
    [1, 2/3, sin(x^3), 1.50000000000000]

Python

    >>> from sage.all import *
    >>> len(v)
    4
    >>> v.append(RealNumber('1.5'))
    >>> v
    [1, 'hello', 2/3, sin(x^3), 1.50000000000000]
    >>> del v[Integer(1)]
    >>> v
    [1, 2/3, sin(x^3), 1.50000000000000]

Another important data structure is the dictionary (or associative array). This works like a list, except that it can be indexed with almost any object (the indices must be immutable):

Sage

    sage: d = {'hi':-2,  3/8:pi,   e:pi}
    sage: d['hi']
    -2
    sage: d[e]
    pi

[Python]

    >>> from sage.all import *
    >>> d = {'hi':-Integer(2),  Integer(3)/Integer(8):pi,   e:pi}
    >>> d['hi']
    -2
    >>> d[e]
    pi

You can also define new data types using classes. Encapsulating mathematical objects with classes is a powerful technique that can help to simplify and organize your Sage programs. Below, we define a class that represents the list of even positive integers up to n; it derives from the builtin type `list`.

Sage

    sage: class Evens(list):
    ....:     def __init__(self, n):
    ....:         self.n = n
    ....:         list.__init__(self, range(2, n+1, 2))
    ....:     def __repr__(self):
    ....:         return "Even positive numbers up to n."

Python

    >>> from sage.all import *
    >>> class Evens(list):
    ...     def __init__(self, n):
    ...         self.n = n
    ...         list.__init__(self, range(Integer(2), n+Integer(1), Integer(2)))
    ...     def __repr__(self):
    ...         return "Even positive numbers up to n."

The `__init__` method is called to initialize the object when it is created; the `__repr__` method prints the object out. We call the list constructor method in the second line of the `__init__` method. We create an object of class `Evens` as follows:

Sage

    sage: e = Evens(10)
    sage: e
    Even positive numbers up to n.

[Python]

    >>> from sage.all import *
    >>> e = Evens(Integer(10))
    >>> e
    Even positive numbers up to n.

Note that `e` prints using the `__repr__` method that we defined. To see the underlying list of numbers, use the `list` function:

Sage

    sage: list(e)
    [2, 4, 6, 8, 10]

Python

    >>> from sage.all import *
    >>> list(e)
    [2, 4, 6, 8, 10]

We can also access the `n` attribute or treat `e` like a list.

Sage

    sage: e.n
    10
    sage: e[2]
    6

[Python]

    >>> from sage.all import *
    >>> e.n
    10
    >>> e[Integer(2)]
    6