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  • 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.

JAMES GRIME: Today, we're going to talk about one of the

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ゼロファクター - 数字マニア (Zero Factorial - Numberphile)

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