Calculate a Factorial With Python - Iterative and Recursive

Introduction

By definition, a factorial is the product of a positive integer and all the positive integers that are less than or equal to the given number. In other words, getting a factorial of a number means to multiply all whole numbers from that number, down to 1.

0! equals 1 as well, by convention.

A factorial is denoted by the integer and followed by an exclamation mark.

5! denotes a factorial of five.

And to calculate that factorial, we multiply the number with every whole number smaller than it, until we reach 1:

5! = 5 * 4 * 3 * 2 * 1
5! = 120

Keeping these rules in mind, in this tutorial, we will learn how to calculate the factorial of an integer with Python, using loops and recursion. Let's start with calculating the factorial using loops.

Calculating Factorial Using Loops

We can calculate factorials using both the while loop and the for loop. The general process is pretty similar for both. All we need is a parameter as input and a counter.

Let's start with the for loop:

def get_factorial_for_loop(n):
    result = 1
    if n > 1:
        for i in range(1, n+1):
            result = result * i
        return result
    else:
        return 'n has to be positive'

You may have noticed that we are counting starting from 1 to the n, whilst the definition of factorial was from the given number down to 1. But mathematically:

1∗2∗3∗4...∗n=n∗(n−1)∗(n−2)∗(n−3)∗(n−4)...∗(n−(n−1))1∗2∗3∗4...∗n=n∗(n−1)∗(n−2)∗(n−3)∗(n−4)...∗(n−(n−1))

To simplify, (n - (n-1)) will always be equal to 1.

That means that it doesn't matter in which direction we're counting. It can start from 1 and increase towards the n, or it can start from n and decrease towards 1. Now that's clarified, let's start breaking down the function we've just wrote.

Our function takes in a parameter n which denotes the number we're calculating a factorial for. First, we define a variable named result and assign 1 as a value to it.

Why assign 1 and not 0 you ask?

Because if we were to assign 0 to it then all the following multiplications with 0, naturally would result in a huge 0.

Then we start our for loop in the range from 1 to n+1. Remember, the Python range will stop before the second argument. To include the last number as well, we simply add an additional 1.

Inside the for loop, we multiply the current value of result with the current value of our index i.

Finally, we return the final value of the result. Let's test our function print out the result:

inp = input("Enter a number: ")
inp = int(inp)

print(f"The result is: {get_factorial_for_loop(inp)}")

If you'd like to read more about how to get user input, read our Getting User Input in Python.

It will prompt the user to give input. We'll try it with 4:

Enter a number: 4
The result is: 24

You can use a calculator to verify the result:

4! is 4 * 3 * 2 * 1, which results 24.

Now let's see how we can calculate factorial using the while loop. Here's our modified function:

def get_factorial_while_loop(n):
    result = 1
    while n > 1:
        result = result * n
        n -= 1
    return result

This is pretty similar to the for loop. Except for this time we're moving from n towards the 1, closer to the mathematical definition. Let's test our function:

inp = input("Enter a number: ")
inp = int(inp)

print(f"The result is: {get_factorial_while_loop(inp)}")

We'll enter 4 as an input once more:

Enter a number: 4
The result is: 24

Although the calculation was 4 * 3 * 2 * 1 the final result is the same as before.

Calculating factorials using loops was easy. Now let's take a look at how to calculate the factorial using a recursive function.

Calculating Factorial Using Recursion

A recursive function is a function that calls itself. It may sound a bit intimidating at first but bear with us and you'll see that recursive functions are easy to understand.

In general, every recursive function has two main components: a base case and a recursive step.

Base cases are the smallest instances of the problem. Also a break, a case that will return a value and will get out of the recursion. In terms of factorial functions, the base case is when we return the final element of the factorial, which is 1.

Without a base case or with an incorrect base case, your recursive function can run infinitely, causing an overflow.

Recursive steps - as the name implies- are the recursive part of the function, where the whole problem is transformed into something smaller. If the recursive step fails to shrink the problem, then again recursion can run infinitely.

Consider the recurring part of the factorials:

• 5! is 5 * 4 * 3 * 2 * 1.

But we also know that:

• 4 * 3 * 2 * 1 is 4!.

In other words 5! is 5 * 4!, and 4! is 4 * 3! and so on.

So we can say that n! = n * (n-1)!. This will be the recursive step of our factorial!

A factorial recursion ends when it hits 1. This will be our base case. We will return 1 if n is 1 or less, covering the zero input.

Let's take a look at our recursive factorial function:

def get_factorial_recursively(n):
    if n <= 1:
        return 1
    else:
        return n * get_factorial_recursively(n-1)

As you see the if block embodies our base case, while the else block covers the recursive step.

Let's test our function:

inp = input("Enter a number: ")
inp = int(inp)

print(f"The result is: {get_factorial_recursively(inp)}")

We will enter 3 as input this time:

Enter a number:3
The result is: 6

We get the same result. But this time, what goes under the hood is rather interesting:

You see, when we enter the input, the function will check with the if block, and since 3 is greater than 1, it will skip to the else block. In this block, we see the line return n * get_factorial_recursively(n-1).

We know the current value of n for the moment, it's 3, but get_factorial_recursively(n-1) is still to be calculated.

Then the program calls the same function once more, but this time our function takes 2 as the parameter. It checks the if block and skips to the else block and again encounters with the last line. Now, the current value of the n is 2 but the program still must calculate the get_factorial_recursively(n-1).

So it calls the function once again, but this time the if block, or rather, the base class succeeds to return 1 and breaks out from the recursion.

Following the same pattern to upwards, it returns each function result, multiplying the current result with the previous n and returning it for the previous function call. In other words, our program first gets to the bottom of the factorial (which is 1), then builds its way up, while multiplying on each step.

Also removing the function from the call stack one by one, up until the final result of the n * (n-1) is returned.

This is generally how recursive functions work. Some more complicated problems may require deeper recursions with more than one base case or more than one recursive step. But for now, this simple recursion is good enough to solve our factorial problem!