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: 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)
0.6931471805599453094172321215 # 32-bit
0.69314718055994530941723212145817656807 # 64-bit
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]
>>> 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)
0.6931471805599453094172321215 # 32-bit
0.69314718055994530941723212145817656807 # 64-bit
>>> 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: def is_even(n):
....: return n%2 == 0
sage: is_even(2)
True
sage: is_even(3)
False
>>> 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: 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
>>> 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: is_divisible_by(6, divisor=5)
False
sage: is_divisible_by(divisor=2, number=6)
True
>>> 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: def even(n):
....: v = []
....: for i in range(3,n):
....: if i % 2 == 0:
....: v.append(i)
....: return v
Syntax Error:
return v
>>> 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: 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]
>>> 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: a = 5; b = a + 3; c = b^2; c
64
>>> 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: 2 + \
....: 3
5
>>> 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: for i in range(3):
....: print(i)
0
1
2
>>> 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: for i in range(2,5):
....: print(i)
2
3
4
>>> 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: for i in range(1,6,2):
....: print(i)
1
3
5
>>> 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: 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
>>> 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: list(range(2,10))
[2, 3, 4, 5, 6, 7, 8, 9]
>>> 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: v = [1, "hello", 2/3, sin(x^3)]
sage: v
[1, 'hello', 2/3, sin(x^3)]
>>> 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: v[0]
1
sage: v[3]
sin(x^3)
>>> 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: 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]
>>> 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: d = {'hi':-2, 3/8:pi, e:pi}
sage: d['hi']
-2
sage: d[e]
pi
>>> 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: 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."
>>> 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: e = Evens(10)
sage: e
Even positive numbers up to n.
>>> 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: list(e)
[2, 4, 6, 8, 10]
>>> 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: e.n
10
sage: e[2]
6
>>> from sage.all import *
>>> e.n
10
>>> e[Integer(2)]
6