In this chapter we will talk about modules and functions. A function is a block of code that is used to perform a single action. A module is a Python file containing variables, functions and many more things.

Start up your Python REPL and let’s use the "math" module which provides access to mathematical functions:

>>> import math
>>> math.cos(0.0)
1.0
>>> math.radians(275)
4.799655442984406

Functions are sequences of instructions that are executed when the function is invoked. The following defines the "do_hello" function that prints two messages when invoked:

>>> def do_hello():
...     print("Hello")
...     print("World")
...
>>> do_hello()
Hello
World

Make sure that you insert a tab before both print expressions in the previous function. Tabs and spaces in Python are relevant and define that a block of code is somewhat dependent on a previous instruction. For instance, the print expressions are "inside" the "do_hello" function therefore must have a tab.

Functions can also receive parameters and return values (using the "return" keyword):

>>> def add_one(val):
...     print("Function got value", val)
...     return val + 1
...
>>> value = add_one(1)
Function got value 1
>>> value
2

Use the Python documentation about the math module (https://docs.python.org/3/library/math.html) to solve the following exercises:

1. Find the greatest common divisor of the following pairs of numbers: (15, 21), (152, 200), (1988, 9765).

2. Compute the base-2 logarithm of the following numbers: 0, 1, 2, 6, 9, 15.

3. Use the "input" function to ask the user for a number and show the result of the sine, cosine and tangent of the number. Make sure that you convert the user input from string to a number (use the int() or the float() function).

1. Implement the "add2" function that receives two numbers as arguments and returns the sum of the numbers. Then implement the "add3" function that receives and sums 3 parameters.

2. Implement a function that returns the greatest of two numbers given as parameters. Use the "if" statement to compare both numbers: https://docs.python.org/3/tutorial/controlflow.html#if-statements.

3. Implement a function named "is_divisible" that receives two parameters (named "a" and "b") and returns true if "a" can be divided by "b" or false otherwise. A number is divisible by another when the remainder of the division is zero. Use the modulo operator ("%").

4. Create a function named "average" that computes the average value of a list passed as parameter to the function. Use the "sum" and "len" functions.

In computer programming, a recursive function is simply a function that calls itself. For instance take the factorial function.

\begin{equation} f(x)=\begin{cases} 1, & \text{if $x=0$}.\ x \times f(x-1), & \text{otherwise}. \end{cases} \end{equation}

As an example, take the factorial of 5:

\begin{equation} \begin{split} 5! &= 5 \times 4! \ &= 5 \times 4 \times 3! \ &= 5 \times 4 \times 3 \times 2! \ &= 5 \times 4 \times 3 \times 2 \times 1 \ &= 120 \end{split} \end{equation}

Basically, the factorial of 5 is 5 times the factorial of 4, etc. Finally, the factorial of 1 (or of zero) is 1 which breaks the recursion. In Python we could write the following recursive function:

def factorial(x):
    if x == 0:
        return 1
    else:
        return x * factorial(x-1)

The trick with recursive functions is that there must be a "base" case where the recursion must end and a recursive case that iterates towards the base case. In the case of factorial we know that the factorial of zero is one, and the factorial of a number greater that zero will depend on the factorial of the previous number until it reaches zero.

1. Implement the factorial function and test it with several different values. Cross-check with a calculator.

2. Implement a recursive function to compute the sum of the (n) first integer numbers (where (n) is a function parameter). Start by thinking about the base case (the sum of the first 0 integers is?) and then think about the recursive case.

3. The Fibonnaci sequence is a sequence of numbers in which each number of the sequence matches the sum of the previous two terms. Given the following recursive definition implement (fib(n)).

   \begin{equation} fib(n)=\begin{cases} 0, & \text{if $n=0$}.\ 1, & \text{if $n=1$}.\ fib(n-1) + fib(n-2), & \text{otherwise}. \end{cases} \end{equation}

   Check your results for the first numbers of the sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...