Python Tutorial Part 1  >  More Functions

Fruitful functions

Return values

Some of the built-in functions we have used, such as the math functions, have
produced results. Calling the function generates a new value, which we usually
assign to a variable or use as part of an expression.
e = math.exp(1.0)
height = radius * math.sin(angle)
But so far, none of the functions we have written has returned a value.
In this chapter, we are going to write functions that return values, which we will
call fruitful functions, for want of a better name. The first example is area,
which returns the area of a circle with the given radius:
import math
def area(radius):
temp = math.pi * radius**2
return temp
We have seen the return statement before, but in a fruitful function the return
statement includes a return value. This statement means: 'Return immediately
from this function and use the following expression as a return value.' The expression
provided can be arbitrarily complicated, so we could have written this
function more concisely:
def area(radius):
return math.pi * radius**2
50 Fruitful functions
On the other hand, temporary variables like temp often make debugging easier.
Sometimes it is useful to have multiple return statements, one in each branch of
a conditional:
def absoluteValue(x):
if x < 0:
return -x
return x
Since these return statements are in an alternative conditional, only one will be
executed. As soon as one is executed, the function terminates without executing
any subsequent statements.
Code that appears after a return statement, or any other place the flow of execution
can never reach, is called dead code.
In a fruitful function, it is a good idea to ensure that every possible path through
the program hits a return statement. For example:
def absoluteValue(x):
if x < 0:
return -x
elif x > 0:
return x
This program is not correct because if x happens to be 0, neither condition is
true, and the function ends without hitting a return statement. In this case, the
return value is a special value called None:
>>> print absoluteValue(0)
As an exercise, write a compare function that returns 1 if x
> y, 0 if x == y, and -1 if x < y.

Program development

At this point, you should be able to look at complete functions and tell what
they do. Also, if you have been doing the exercises, you have written some small
functions. As you write larger functions, you might start to have more difficulty,
especially with runtime and semantic errors.
To deal with increasingly complex programs, we are going to suggest a technique
called incremental development. The goal of incremental development is to
avoid long debugging sessions by adding and testing only a small amount of code
at a time.
As an example, suppose you want to find the distance between two points, given by
the coordinates (x1, y1) and (x2, y2). By the Pythagorean theorem, the distance
distance = p(x2 - x1)2 + (y2 - y1)2
The first step is to consider what a distance function should look like in Python.
In other words, what are the inputs (parameters) and what is the output (return
In this case, the two points are the inputs, which we can represent using four
parameters. The return value is the distance, which is a floating-point value.
Already we can write an outline of the function:
def distance(x1, y1, x2, y2):
return 0.0
Obviously, this version of the function doesn't compute distances; it always returns
zero. But it is syntactically correct, and it will run, which means that we can test
it before we make it more complicated.
To test the new function, we call it with sample values:
>>> distance(1, 2, 4, 6)
We chose these values so that the horizontal distance equals 3 and the vertical
distance equals 4; that way, the result is 5 (the hypotenuse of a 3-4-5 triangle).
When testing a function, it is useful to know the right answer.
At this point we have confirmed that the function is syntactically correct, and we
can start adding lines of code. After each incremental change, we test the function
again. If an error occurs at any point, we know where it must be'in the last line
we added.
A logical first step in the computation is to find the differences x2-x1 and y2-y1.
We will store those values in temporary variables named dx and dy and print them.
def distance(x1, y1, x2, y2):
dx = x2 - x1
dy = y2 - y1
print "dx is", dx
print "dy is", dy
return 0.0

If the function is working, the outputs should be 3 and 4. If so, we know that
the function is getting the right arguments and performing the first computation
correctly. If not, there are only a few lines to check.
Next we compute the sum of squares of dx and dy:
def distance(x1, y1, x2, y2):
dx = x2 - x1
dy = y2 - y1
dsquared = dx**2 + dy**2
print "dsquared is: ", dsquared
return 0.0
Notice that we removed the print statements we wrote in the previous step. Code
like that is called scaffolding because it is helpful for building the program but
is not part of the final product.
Again, we would run the program at this stage and check the output (which should
be 25).
Finally, if we have imported the math module, we can use the sqrt function to
compute and return the result:
def distance(x1, y1, x2, y2):
dx = x2 - x1
dy = y2 - y1
dsquared = dx**2 + dy**2
result = math.sqrt(dsquared)
return result
If that works correctly, you are done. Otherwise, you might want to print the
value of result before the return statement.
When you start out, you should add only a line or two of code at a time. As
you gain more experience, you might find yourself writing and debugging bigger
chunks. Either way, the incremental development process can save you a lot of
debugging time.
The key aspects of the process are:
1. Start with a working program and make small incremental changes. At any
point, if there is an error, you will know exactly where it is.
2. Use temporary variables to hold intermediate values so you can output and
check them.
3. Once the program is working, you might want to remove some of the scaffolding
or consolidate multiple statements into compound expressions, but
only if it does not make the program difficult to read.
As an exercise, use incremental development to write a function called
hypotenuse that returns the length of the hypotenuse of a right triangle
given the lengths of the two legs as arguments. Record each stage of
the incremental development process as you go.


As you should expect by now, you can call one function from within another. This
ability is called composition.
As an example, we'll write a function that takes two points, the center of the circle
and a point on the perimeter, and computes the area of the circle.
Assume that the center point is stored in the variables xc and yc, and the perimeter
point is in xp and yp. The first step is to find the radius of the circle, which is
the distance between the two points. Fortunately, there is a function, distance,
that does that:
radius = distance(xc, yc, xp, yp)
The second step is to find the area of a circle with that radius and return it:
result = area(radius)
return result
Wrapping that up in a function, we get:
def area2(xc, yc, xp, yp):
radius = distance(xc, yc, xp, yp)
result = area(radius)
return result
We called this function area2 to distinguish it from the area function defined
earlier. There can only be one function with a given name within a given module.
The temporary variables radius and result are useful for development and debugging,
but once the program is working, we can make it more concise by composing
the function calls:
def area2(xc, yc, xp, yp):
return area(distance(xc, yc, xp, yp))
As an exercise, write a function slope(x1, y1, x2, y2) that returns
the slope of the line through the points (x1, y1) and (x2, y2). Then use
this function in a function called intercept(x1, y1, x2, y2) that
returns the y-intercept of the line through the points (x1, y1) and
(x2, y2).

Boolean functions

Functions can return boolean values, which is often convenient for hiding complicated
tests inside functions. For example:
def isDivisible(x, y):
if x % y == 0:
return True
return False
The name of this function is isDivisible. It is common to give boolean functions
names that sound like yes/no questions. isDivisible returns either True or
False to indicate whether the x is or is not divisible by y.
We can make the function more concise by taking advantage of the fact that the
condition of the if statement is itself a boolean expression. We can return it
directly, avoiding the if statement altogether:
def isDivisible(x, y):
return x % y == 0
This session shows the new function in action:
>>> isDivisible(6, 4)
>>> isDivisible(6, 3)
Boolean functions are often used in conditional statements:
if isDivisible(x, y):
print "x is divisible by y"
print "x is not divisible by y"
It might be tempting to write something like:
if isDivisible(x, y) == True:
But the extra comparison is unnecessary.
As an exercise, write a function isBetween(x, y, z) that returns
True if y = x = z or False otherwise.

More recursion

So far, you have only learned a small subset of Python, but you might be interested
to know that this subset is a complete programming language, which means
that anything that can be computed can be expressed in this language. Any program
ever written could be rewritten using only the language features you have
learned so far (actually, you would need a few commands to control devices like
the keyboard, mouse, disks, etc., but that's all).
Proving that claim is a nontrivial exercise first accomplished by Alan Turing, one
of the first computer scientists (some would argue that he was a mathematician,
but a lot of early computer scientists started as mathematicians). Accordingly, it
is known as the Turing Thesis. If you take a course on the Theory of Computation,
you will have a chance to see the proof.
To give you an idea of what you can do with the tools you have learned so far, we'll
evaluate a few recursively defined mathematical functions. A recursive definition is
similar to a circular definition, in the sense that the definition contains a reference
to the thing being defined. A truly circular definition is not very useful:
frabjuous: An adjective used to describe something that is frabjuous.
If you saw that definition in the dictionary, you might be annoyed. On the other
hand, if you looked up the definition of the mathematical function factorial, you
might get something like this:
0! = 1
n! = n(n - 1)!
This definition says that the factorial of 0 is 1, and the factorial of any other value,
n, is n multiplied by the factorial of n - 1.
So 3! is 3 times 2!, which is 2 times 1!, which is 1 times 0!. Putting it all together,
3! equals 3 times 2 times 1 times 1, which is 6.
If you can write a recursive definition of something, you can usually write a Python
program to evaluate it. The first step is to decide what the parameters are for
this function. With little effort, you should conclude that factorial has a single
def factorial(n):

If the argument happens to be 0, all we have to do is return 1:
def factorial(n):
if n == 0:
return 1
Otherwise, and this is the interesting part, we have to make a recursive call to
find the factorial of n - 1 and then multiply it by n:
def factorial(n):
if n == 0:
return 1
recurse = factorial(n-1)
result = n * recurse
return result
The flow of execution for this program is similar to the flow of countdown in
Section 4.9. If we call factorial with the value 3:
Since 3 is not 0, we take the second branch and calculate the factorial of n-1...
Since 2 is not 0, we take the second branch and calculate the factorial
of n-1...
Since 1 is not 0, we take the second branch and calculate
the factorial of n-1...
Since 0 is 0, we take the first branch and return 1
without making any more recursive calls.
The return value (1) is multiplied by n, which is 1, and the
result is returned.
The return value (1) is multiplied by n, which is 2, and the result is
The return value (2) is multiplied by n, which is 3, and the result, 6, becomes the
return value of the function call that started the whole process.
Here is what the stack diagram looks like for this sequence of function calls:
n 3 recurse 2
recurse 1
recurse 1 return 1
return 2
return 6
n 2
n 1
n 0
The return values are shown being passed back up the stack. In each frame, the
return value is the value of result, which is the product of n and recurse.
Notice that in the last frame, the local variables recurse and result do not exist,
because the branch that creates them did not execute.

Leap of faith

Following the flow of execution is one way to read programs, but it can quickly
become labyrinthine. An alternative is what we call the 'leap of faith.' When
you come to a function call, instead of following the flow of execution, you assume
that the function works correctly and returns the appropriate value.
In fact, you are already practicing this leap of faith when you use built-in functions.
When you call math.cos or math.exp, you don't examine the implementations of
those functions. You just assume that they work because the people who wrote
the built-in functions were good programmers.
The same is true when you call one of your own functions. For example,
we wrote a function called isDivisible that determines whether
one number is divisible by another. Once we have convinced ourselves that this
function is correct'by testing and examining the code'we can use the function
without looking at the code again.
The same is true of recursive programs. When you get to the recursive call, instead
of following the flow of execution, you should assume that the recursive call works
(yields the correct result) and then ask yourself, 'Assuming that I can find the
factorial of n - 1, can I compute the factorial of n?' In this case, it is clear that
you can, by multiplying by n.
Of course, it's a bit strange to assume that the function works correctly when you
haven't finished writing it, but that's why it's called a leap of faith!

One more example

In the previous example, we used temporary variables to spell out the steps and
to make the code easier to debug, but we could have saved a few lines:
def factorial(n):
if n == 0:
return 1
return n * factorial(n-1)
From now on, we will tend to use the more concise form, but we recommend that
you use the more explicit version while you are developing code. When you have
it working, you can tighten it up if you are feeling inspired.
After factorial, the most common example of a recursively defined mathematical
function is fibonacci, which has the following definition:
fibonacci(0) = 1
fibonacci(1) = 1
fibonacci(n) = fibonacci(n - 1) + fibonacci(n - 2);
Translated into Python, it looks like this:
def fibonacci (n):
if n == 0 or n == 1:
return 1
return fibonacci(n-1) + fibonacci(n-2)
If you try to follow the flow of execution here, even for fairly small values of n,
your head explodes. But according to the leap of faith, if you assume that the
two recursive calls work correctly, then it is clear that you get the right result by
adding them together.
What happens if we call factorial and give it 1.5 as an argument?
>>> factorial (1.5)
RuntimeError: Maximum recursion depth exceeded
It looks like an infinite recursion. But how can that be? There is a base case'
when n == 0. The problem is that the values of n miss the base case.
In the first recursive call, the value of n is 0.5. In the next, it is -0.5. From there,
it gets smaller and smaller, but it will never be 0.
We have two choices. We can try to generalize the factorial function to work
with floating-point numbers, or we can make factorial check the type of its
argument. The first option is called the gamma function and it's a little beyond
the scope of this book. So we'll go for the second.
We can use the built-in function isinstance to verify the type of the argument.
While we're at it, we also make sure the argument is positive:
def factorial (n):
if not isinstance(n, int):
print "Factorial is only defined for integers."
return -1
elif n < 0:
print "Factorial is only defined for positive integers."
return -1
elif n == 0:
return 1
return n * factorial(n-1)
Now we have three base cases. The first catches nonintegers. The second catches
negative integers. In both cases, the program prints an error message and returns
a special value, -1, to indicate that something went wrong:
>>> factorial ("fred")
Factorial is only defined for integers.
>>> factorial (-2)
Factorial is only defined for positive integers.
If we get past both checks, then we know that n is a positive integer, and we can
prove that the recursion terminates.
This program demonstrates a pattern sometimes called a guardian. The first two
conditionals act as guardians, protecting the code that follows from values that
might cause an error. The guardians make it possible to prove the correctness of
the code.