JAMES GRIME: Today, we're going to talk about one of the
questions that we get sent in a lot at Numberphile, and the
question is-- well, Brady, what's the question?
BRADY HARAN: The question is, why does 0 factorial equal 1?
JAMES GRIME: Right.
Why does 0 factorial equal 1?
So let's start off with a quick recap of what a
factorial is.
For our whole number, let's pick a number n--
n factorial, which is written like that. n with an
exclamation mark.
This is equal to.
You multiply all the whole numbers less
than or equal to n.
It's n multiplied by n minus 1 multiplied by n minus 2
multiplied by--
and you keep going down, and you'll go down to 3
times 2 times 1.
Quick example.
Let's do 5 factorial.
5 times 4 times 3 times 2 times 1.
And you do that.
It's 120.
OK.
The question we've been asked is what is 0 factorial.
So the way you can answer this-- one of the ways you can
answer this is to complete the pattern.
Let's complete the pattern.
This pattern in particular, 4 factorial, is equal to 5
factorial divided by 5.
If you can see that, if I take 5 factorial here and divide by
5, that means I can knock off that 5, and you
end up with 4 factorial.
So 5 factorial divided by 5, or 120 divided
by 5, that's 24.
That's 4 factorial.
3 factorial is going to be 4 factorial divided by 4.
That's 24 divided by 4.
That's 6.
That's the answer to 3 factorial.
2 factorial, 3 factorial divided by 3, 6, which we've
just worked out, divided by 3, equals 2.
1 factorial.
Do it again.
It's 2 factorial divided by 2.
2 factorial is 2 divided by 2.
We've got 2 divided by 2.
That's equal to 1.
Now this is where it's getting exciting.
Do you feel the anticipation?
So 0 factorial.
We're going to complete the pattern.
0 factorial is 1 factorial divided by 1.
1 factorial is 1.
It's 1 divided by 1, and that is equal to 1.
So 0 factorial is equal to 1.
You complete the pattern.
BRADY HARAN: Who says the pattern has to be complete?
Where's that rule come from?
JAMES GRIME: I guess it doesn't necessarily have to be
a pattern that completes.
It is a pattern that competes, though.
Let me try another way to explain it.
BRADY HARAN: Let me continue the pattern first.
Does that mean negative 1 factorial would be next in
that sequence?
JAMES GRIME: Let's see what happens.
I'm not sure what's going to happen.
Let's try.
Minus 1 factorial.
So what shall I get?
0 factorial divided by 0.
1 divided by 0.
BRADY HARAN: Oh, divided by 0.
JAMES GRIME: You've broken maths, Brady.
Stop that.
Another way to explain what 0 factorial might be.
n factorial is the number of ways you
can arrange n objects.
Let me just try to show you what I mean.
Let's get some objects.
I'll get the wallet out.
I'll get some coins out.
See?
Who says mathematicians don't make a lot of money?
There's literally 50p here.
Let's pick a silver one and a 5p one.
Three objects here, and how many ways are there to arrange
three objects?
There's six ways to do it.
It's 3 factorial.
Let's just check them.
That's one, that's two, or we could have this one here--
that's three, that's four.
Or we could have--
I think it was that one we didn't have at the front.
So that would be five and six.
If we take one away, we have now two objects.
How many ways are there to arrange two objects?
That's one, that's two.
Take one away.
How many ways are there to arrange one object?
There it is.
There's one way to do it.
One way to arrange one object.
Now we're going to take the last coin away.
This is where it gets a little philosophical.
We have zero objects.
How many ways are there to arrange zero objects?
There's one way to do it.
There it is.
Do you want to see me do it again?
There it is.
Slightly philosophical, but we say there is one way to
arrange zero objects.
So again, the pattern holds.
0 factorial equals 1.
Just to continue the idea just a little bit further, if we're
talking about factorials, let's try and graph them.
So let's say let's have one, two, three, four, five.
1 factorial is 1, so if you call that 1.
2 factorial is 2, so somewhere about here.
3 factorial is 6.
I don't know.
Somewhere like this.
4 factorial is 24, so that's going to be actually quite
high up here.
And then 5 factorial is going quite high.
If we join these together, I've also said that 0
factorial is 1, so I reckon this is the graph.
So in theory, we should be able to get values for in
between, like, say, the number 1 and 1/2.
1 and 1/2 factorial.
What is 1 and 1/2 factorial?
So mathematicians have done that.
They generalize the idea.
And there is the idea of 1 and 1/2 factorial.
We call it gamma.
That's the Greek letter gamma.
We call it gamma of.
And the way we write it--
actually, now this is getting a bit more sophisticated.
We say gamma of n is equal to the integral between 0 and
infinity of--
let's pick something--
t to the power n minus 1, multiplied by e to the power
minus n dn.
Some people won't be familiar with that.
Some of you will be familiar with that.
Some of you won't be.
It's a much more complicated mathematical idea, but this
would agree with the factorials.
But it gives you in between values as well.
It plots this line.
There is something I do need to say.
It's slightly unexpected, but if we take a value for a whole
number, gamma of n, and n is whole, this actually gives you
n minus 1 factorial, so be careful of that.
That might catch you out.
Bit of a pain, that.
So what's the point of having a function that will give you
factorials in between whole numbers when you can't arrange
1 and 1/2 objects?
So it's a generalization, and it turns out to be quite
useful in many things.
Particularly, I'm thinking of probability.
You can use them in formulas that you find in probability
where you're thinking about continuous time instead of
just arranging objects in discreet probability.
You're now starting to think about continuous events.
Time is the best example.
Then you start to generalize the ideas, and therefore you
need a generalized factorial.
BRADY HARAN: 9, 6, and 3.
20.
44.