Why is zero factorial equal to 1?

Zero Factorial is equal to 1. Or in equation, 0! = 1.

I can understand why many of us have a hard time accepting the fact that the value of zero factorial is equal to one. It comes across as an absurd statement that there’s no way it can be true. We have a common perception of zero for being notorious because there’s something about it that can make any number associated with it either vanish or misbehave.

For instance, a large number such as 1,000 multiplied by zero becomes zero. It disappears! On the other hand, a nice number such as 5 divided by zero becomes undefined. It misbehaves. So it is okay to be skeptical about why zero “suddenly” becomes one, a nice number, after treating it with some special operation.

There are other ways to show why the statement is true. For this one, we will use the definition of factorial itself. To be honest, with this method the justification is simple and requires little math.

Simple “Proof” Why Zero Factorial is Equal to One

Let [latex]n[/latex] be a whole number, where [latex]n![/latex] is defined as the product of all whole numbers less than [latex]n[/latex] and including [latex]n[/latex] itself.

What it means is that you first start writing the whole number [latex]n[/latex] then count down until you reach the whole number [latex]1[/latex].

The general formula of factorial can be written in fully expanded form as

n! = n·(n-1)·(n-2)·...·3·2·1

or in partially expanded form as

n! = n · (n-1)!

We know with absolute certainty that 1!=1, where [latex]n = 1[/latex]. If we substitute that value of [latex]n[/latex] into the second formula which is the partially expanded form of [latex]n![/latex], we obtain the following:

1! = 1 · 0!

For the equation to be true, we must force the value of zero factorial to equal 1, and no other. Otherwise, 1!≠1 which is a contradiction.

So yes, 0! = 1 is correct because mathematicians agreed to define it that way (nothing more and nothing less) in order to be consistent with the rest of mathematics.

You may also be interested in these related math lessons or tutorials:

Factorial Notation, Formula, and Basic Examples
Dividing Factorials
Simplifying Factorials with Variables