We have identified in Python some of the elements that must appear in any powerful programming language:

1. Numbers and arithmetic operations are primitive built-in data values and functions.
2. Nested function application provides a means of combining operations.
3. Binding names to values provides a limited means of abstraction.

Now we will learn about function definitions, a much more powerful abstraction technique by which a name can be bound to compound operation, which can then be referred to as a unit.

We begin by examining how to express the idea of squaring. We might say, "To square something, multiply it by itself." This is expressed in Python as

>>> def square(x):
        return mul(x, x)
        

which defines a new function that has been given the name square. This user-defined function is not built into the interpreter. It represents the compound operation of multiplying something by itself. The x in this definition is called a formal parameter, which provides a name for the thing to be multiplied. The definition creates this user-defined function and associates it with the name square.

How to define a function. Function definitions consist of a def statement that indicates a <name> and a comma-separated list of named <formal parameters>, then a return statement, called the function body, that specifies the <return expression> of the function, which is an expression to be evaluated whenever the function is applied:

def <name>(<formal parameters>):
    return <return expression>

The second line must be indented — most programmers use four spaces to indent. The return expression is not evaluated right away; it is stored as part of the newly defined function and evaluated only when the function is eventually applied.

Having defined square, we can apply it with a call expression:

>>> square(21)
441
>>> square(add(2, 5))
49
>>> square(square(3))
81

We can also use square as a building block in defining other functions. For example, we can easily define a function sum_squares that, given any two numbers as arguments, returns the sum of their squares:

>>> def sum_squares(x, y):
        return add(square(x), square(y))

>>> sum_squares(3, 4)
25

User-defined functions are used in exactly the same way as built-in functions. Indeed, one cannot tell from the definition of sum_squares whether square is built into the interpreter, imported from a module, or defined by the user.

Both def statements and assignment statements bind names to values, and any existing bindings are lost. For example, g below first refers to a function of no arguments, then a number, and then a different function of two arguments.

>>> def g():
        return 1
>>> g()
1
>>> g = 2
>>> g
2
>>> def g(h, i):
        return h + i
>>> g(1, 2)
3