Why is 0! (zero factorial) equal to 1?

This post was originally written at the end of 2013.

Today I had a very typical Further Maths A Level lesson. Someone asked a very simple question, I started to answer it, and ten minutes later we were talking about how many imaginary sheep there were in the classroom.

Like I said, a typical lesson.

The question that I was asked was, "Why is zero factorial one?". This was asked by a female student I will refer to as H (to protect her identity- she probably doesn't want to be associated with this nerdy conversation). I'm not entirely sure why H asked me this, as she was supposed to be working on the Secant Method (otherwise known as the most painful mathematical process of all time).

But anyway, she asked me this, and my immediate answer was that very useful mathematical phrase: "by convention".

She responded with, "What do you mean?" to which another student, who I will call J, replied, "To make everyone happy", which pretty much sums it up. Zero factorial was defined as one to make everyone happy. What a lovely answer!

But I couldn't just leave it there, could I? Oh no, my geek sense was tingling. I tried to get my head back to where it should be (doing the register) but I just couldn't. Before I knew it a board pen had somehow leapt into my hand and I was on my feet.

Let's take a look at the factorial function.
3! = 3 x 2 x 1 = 6
2! = 2 x 1 = 2
1! = 1
0! =

Notice the deliberately blank space next to 0! =. Because that's what the answer is. A blank space.
"Three factorial is three times two times one".
"Two factorial is two times one".
"One factorial is one".
 "Zero factorial is ...[silence]".

So the question is, what number is "...[silence]"? By the way, when you say "...[silence]" you should accompany this with a hand movement kind of like "ta-da!" but less dramatic. I might post a video up here later so you can see what I mean.

Sorry, I was saying, what number is "...[silence]" *hand movement* ?

Well in my opinion, it's one. To me, it's obviously one. It's not zero. Zero has too much meaning. Zero is a very definite nothing. Zero is a dangerous number - it can ruin all kinds of calculations. I think the "blank" number is one.

Here's a reason why:

What's 3x - 2x?
Answer: x.
What's the coefficient of x?
Answer: 1.
But where's the 1?
Answer: you don't need it.

The blank space in front of the x means one.

Another example:

Say you had some algebraic fractions to simplify by cancelling common factors. Look at the first two examples. Using similar logic, surely the answer to c) is a blank space? But we know the answer is 1.

It is easy to see why 0! has to be 1 when we look at combinatorics. 5C0 ("5 choose 0") means how many ways are there of choosing zero items from a choice of 5. The answer to this is one. Why? Well if you have to pick zero items, how many ways are there to do this? Well the only way to do it is to not do it, which is one way, so the answer is one. The formula for the nCr (choose) function involves factorials, and the only way for nC0 to equal one is if 0! = 1. So 0! has to be 1, or the formula won't work.

In other words, it's 1 to keep everyone happy. I should have just listened to J.

And in case you're wondering, I never did get round to doing my register.


PS No I will not explain the imaginary sheep thing. You had to be there.