Why the factorial of 0 is 1

From all the information gathered, let’s start to define a factorial as it’s done with mathematics.

Factorial defined

We’ve seen that the factorial of N+1 is the product of the factorial of N and the number N+1.

(N+1)! = (N+1)•N!

Similarly one could also write the following:

N! = N•(N-1)!

That could be further written as

N! = N•(N-1)•(N-2)!

And this could continue to

N! = N•(N-1)•(N-2)•…•(3)•(2)•(1!)

The factorial of a number can thus be defined as the product of every natural number upto and including itself.

Now, here’s the thing. Once you arrived at an expression that stops at 1!, you can either just take that as equal to 1, or go one step further and substitute that with the product of 1 and 0!.

N! = N•(N-1)•(N-2)•…•(3)•(2)•(1)•(0!)

We can thus deduce what exactly the factorial of zero from this expression.

1! = 1•0!

Since the factorial of one is one and one times any number is that number we get

0! = 1

Thus, after a long and arduous process, we’ve proved that the factorial of zero is indeed one using mathematics.

Can it be made simpler?

Of course, it can. Knowledge of basic maths wasn’t really required here. As stated in the very first line, the factorial of a number is the total number of ways in which a set of that many objects can be arranged. One needn’t be a math enthusiast to know that there is only one way to arrange zero objects.

Thus, the factorial of zero is one.

Do let me know via the comments what you thought about this article. You can make suggestions for topics you think I should discuss. Thanks for reading.