A picture asking the question if zero factorial is equal to one

It is common knowledge that zero factorial is equal to one. Everyone learns this in math class in middle school. However, when we delve into it further, most of us might realise that our understanding of this concept is rather shaky.

The general challenge we have with mathematics is that we take things for granted. We often don’t wish to come across as stupid for asking ‘dumb’-sounding questions. If the teacher says 0! = 1, it is probably wise to just accept it and move on. Why question authority that has never been wrong before?

Well, why not? Why not ask WHY?! It doesn’t really have much to do with the teacher, but more to do with us. If we do not understand something fundamentally, we should not be fear asking why, even if the question sounds stupid. In any case, this is no classroom, and I am no teacher.

But I fully intend to delve deeper into this concept to explain why exactly is 0! = 1. I will be explaining why this is the case by solving easy, incremental puzzles that anyone can understand. If you read my article on why any number raised to the power zero = 1, then you should know that we are going to use a very similar approach here.

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What is a Factorial?

A factorial is a function in mathematics that is applied to non-negative integers. When this function is applied to an integer n, it results in a multiplicative series of all positive integers less than or equal to n. It is probably easier to look at a couple of examples:

Equations to ullustrate what a factorial is: 3! = 3*2*1 4! = 4*3*2*1 5! = 5*4*3*2*1 6! = 6*5*4*3*2*1 . . . n! = n*(n-1)*(n-2)*...*1
Math illustrated by the author

The key point to note here is that the number is multiplied with integers in decreasing order until 1 is reached. So far, regardless of which positive integer we start from, the multiplicative series has ended with 1. However, zero is a non-negative integer as well.

What happens when we need to compute the factorial function for zero? It isn’t quite obvious. But at the same time, I assure you that it’s not as complex as it appears. We’ll start by moving the factorial function up and down the integers – this will make it straightforward for us to understand what is going on here.

Factorials in Ascending Order

Let us consider the factorial function for 1. 1! = 1 (it is 1 because there is no positive integer that is lesser than 1). Let us say that I wish to transform this into 2! How can I do it?

Since we are moving up an integer, we just need to multiply by 2. That is: 2! = 2*1. Similarly, if we wish to transform this to 3!, all we need to do is multiply the existing series with a 3. Below, you can see a few examples worked out for clarity:

Zero factorial explanation Equations for ascending factorials: 1! = 1 (because 1 is the lowest positive integer) 2! = 2*1 = 2*1! 3! = 3*2*1 = 3*2! 4! = 4*3*2*1 = 4*3! 5! = 5*4*3*2*1 = 5*4!
Math illustrated by the author

Furthermore, what we see from the examples is that the factorial of any integer is just the integer multiplied by the factorial of the integer minus one. For example, 4! = 4*3! And 5! = 5*4! This observation will come in handy for us in the next step.

Factorials in Descending Order

Let us now do the exact opposite of what we did before. What if we start with 5! and work our way down? Let’s do just that. What do I need to do to 5! to get to 4!? Since we are moving down by one integer, we need to do the opposite of what we did last time; we just divide by 5. Similarly, if we wish to transform 4! to 3!, all we need to do is divide by 4. Here are the corresponding examples worked out:

Zero factorial explanation Equations for descending factorials: 5! = 5*4*3*2*1 = 5*4! 4! = 5!/5 = (5*4!)/5 = 4*3! 3! = 4!/4 = (4*3!)/4 = 3*2! 2! = 3!/3 = (3*2!)/3 = 2*1!
Math illustrated by the author

The key point to note here is that in order to lower the factorial function from n! to (n-1)!, we have to divide n! by n, i.e., n!/n.

Getting to Zero Factorial

Now that we have solved the puzzle up to the point of moving down integers using the factorial function, let us just keep going. What happens if we wish to transform 2! to 1!, and so on? Let’s find out.

Zero factorial illustration: 2! = 3!/3 = (3*2!)/3 = 2*1! 1! = 2!/2 = (2*1!)/2 = 1 (1 is the lowest positive integer) 0! = 1!/1 = 1/1 Therefore, 0! = 1
Math illustrated by the author

So, as we keep going down to zero, we realise why 0! equals 1. What 0! actually means is that 1! is being divided by 1 (1!/1). This makes logical sense under the definition of the factorial function. And we no longer have to take anything for granted.

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Further reading that might interest you: Does Division By Zero Really Lead To Infinity? and How To Use Mathematics To Choose A Life Partner?

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