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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

• 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

A2 初級

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

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VoiceTube に公開 2021 年 01 月 14 日