2.2 Assignment, Equality, Functions, Indentation, and Counting

sage: a = 5
sage: a
5
sage: 2 == 2
True
sage: 2 == 3
False
sage: 2 < 3
True
sage: a == 5
True

To define a 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

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

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

In Python, blocks of code are not indicated by curly braces or begin and end blocks like in Java or 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

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]

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

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

sage: 2 + \
...      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

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

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

The most basic data structure in Sage is the list, which is - as the name suggests - just a list of arbitrary objects. For example, the range command that we used creates a list:

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

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

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

Use len(v) to get the length of

, use v.append(obj) to append a new object to the end of

, and use del v[i] to delete the

th entry of

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]

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

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

; 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."

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

Note that

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]

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