questions that we get sent in a lot at Numberphile, and the

就是今天的題目

question is-- well, Brady, what's the question?

Brady，那他們想問的是？

BRADY HARAN: The question is, why does 0 factorial equal 1?

BRADY HARAN: 他們問：「爲什麼0的階乘是1？」

JAMES GRIME: Right.

JAMES GRIME: 對

Why does 0 factorial equal 1?

爲什麼0的階乘等於1？

So let's start off with a quick recap of what a

我們首先要認識什麼是階乘

factorial is.

設自然數n

For our whole number, let's pick a number n--

n的階乘是這樣的

n factorial, which is written like that. n with an

在n後面加一個感歎號

exclamation mark.

這個數等於

This is equal to.

你要把n乘以大於或You multiply all the whole numbers less

You multiply all the whole numbers less

等於n的自然數

than or equal to n.

n乘以(n-1)乘以(n-2)

It's n multiplied by n minus 1 multiplied by n minus 2

乘以——

multiplied by--

然後不斷重複 直至

and you keep going down, and you'll go down to 3

3乘2乘1

times 2 times 1.

來個例子

Quick example.

就5的階乘吧

Let's do 5 factorial.

5乘4乘3乘2乘1

5 times 4 times 3 times 2 times 1.

結果是

And you do that.

120

It's 120.

OK

OK.

問題就是 0的階乘是什麼

The question we've been asked is what is 0 factorial.

其中一個作答的方法是

So the way you can answer this-- one of the ways you can

看看階乘的規律

answer this is to complete the pattern.

這個規律就是

Let's complete the pattern.

例如4的階乘等於 4 factorial, is equal to 5

This pattern in particular, 4 factorial, is equal to 5

5的階乘除以5

factorial divided by 5.

將這個數除以5的話

If you can see that, if I take 5 factorial here and divide by

就可以拿走這裏的5了

5, that means I can knock off that 5, and you

結果就是4的階乘

end up with 4 factorial.

5的階乘除5 或者120除5

So 5 factorial divided by 5, or 120 divided

商是24

by 5, that's 24.

也就是4的階乘

That's 4 factorial.

3的階乘就是4的階乘除4

3 factorial is going to be 4 factorial divided by 4.

24除以4

That's 24 divided by 4.

答案是6

That's 6.

也就是3的階乘

That's the answer to 3 factorial.

2的階乘是3的階乘除以3

2 factorial, 3 factorial divided by 3, 6, which we've

即是6除以3 答案是2

just worked out, divided by 3, equals 2.

1的階乘

1 factorial.

與前者相同

Do it again.

2的階乘除2

It's 2 factorial divided by 2.

2除以2

2 factorial is 2 divided by 2.

對 2除以2

We've got 2 divided by 2.

答案是1

That's equal to 1.

然後就來戲玉了

Now this is where it's getting exciting.

你們期待嗎？

Do you feel the anticipation?

來了 0的階乘

So 0 factorial.

我們要跟着規律

We're going to complete the pattern.

0的階乘是1的階乘除1

0 factorial is 1 factorial divided by 1.

1的階乘是1

1 factorial is 1.

1除以1等於1

It's 1 divided by 1, and that is equal to 1.

所以0的階乘等於1

So 0 factorial is equal to 1.

只要依照規律完成便行

You complete the pattern.

BRADY HARAN: 誰說規律一定要依？

BRADY HARAN: Who says the pattern has to be complete?

這個規則誰定的？

Where's that rule come from?

JAMES GRIME: 其實階乘不一定要是

JAMES GRIME: I guess it doesn't necessarily have to be

完整的規律

a pattern that completes.

但這真的是完整的規律

It is a pattern that competes, though.

容許我用另一個方式解釋

Let me try another way to explain it.

BRADY HARAN: 不如讓我繼續完成規律吧

BRADY HARAN: Let me continue the pattern first.

那負1不就是下一個數嗎？

Does that mean negative 1 factorial would be next in

JAMES GRIME: 就看看什麼會發生

that sequence?

我不太肯定

JAMES GRIME: Let's see what happens.

一起來試

I'm not sure what's going to happen.

負1的階乘

Let's try.

答案就是

Minus 1 factorial.

0的階乘除以0

So what shall I get?

1除以0

0 factorial divided by 0.

BRADY HARAN: 噢 除以0

1 divided by 0.

JAMES GRIME: Brady，不要再弄壞數學了

BRADY HARAN: Oh, divided by 0.

另一個解釋0的階乘的方法是

JAMES GRIME: You've broken maths, Brady.

n的階乘是n個物件排列的組合

Stop that.

讓我闡述一下

Another way to explain what 0 factorial might be.

我要拿一些物件

n factorial is the number of ways you

就拿出我的錢包

can arrange n objects.

找些硬幣

Let me just try to show you what I mean.

誰告訴您數學家賺錢很少？

Let's get some objects.

這裏可是有50英鎊

I'll get the wallet out.

我就拿出這個銀色的 還有這個5鎊

I'll get some coins out.

這裏有3個物件 要排列這3個物件

See?

總共有幾多個方法？

Who says mathematicians don't make a lot of money?

總共有6個方法

There's literally 50p here.

就是3的階乘

Let's pick a silver one and a 5p one.

看看我對不對

Three objects here, and how many ways are there to arrange

第一個組合 第二個 或者這樣做

three objects?

第三個組合 第四個

There's six ways to do it.

又可以這樣排列

It's 3 factorial.

這個硬幣放在頭位

Let's just check them.

那就是五 六個組合了

That's one, that's two, or we could have this one here--

拿走一個硬幣 剩下兩個

that's three, that's four.

2個物件有多少個組合？

Or we could have--

一個組合 兩個組合

I think it was that one we didn't have at the front.

再次拿走一個

So that would be five and six.

有多少個組合？

If we take one away, we have now two objects.

對 就只有一個

How many ways are there to arrange two objects?

排列1個物件只有1個方法

That's one, that's two.

不如也拿走最後的硬幣

Take one away.

接下來的有點哲學

How many ways are there to arrange one object?

這裏有0個物件

There it is.

排列0個物件有多少個組合？

There's one way to do it.

只有一個組合

One way to arrange one object.

就這樣

Now we're going to take the last coin away.

要我再來一次？

This is where it gets a little philosophical.

就這樣

We have zero objects.

有點哲學 但要排列0個物件

How many ways are there to arrange zero objects?

的確只有一個方法

There's one way to do it.

那就證明規律對了

There it is.

0的階乘等於1

Do you want to see me do it again?

其實有第三個方法

There it is.

就是畫圖表

Slightly philosophical, but we say there is one way to

1 2 3 4 5

arrange zero objects.

1的階乘是1 這就是1

So again, the pattern holds.

2的階乘是2 大約是這裏

0 factorial equals 1.

3的階乘是6

Just to continue the idea just a little bit further, if we're

我不知道 大概是這裏吧

talking about factorials, let's try and graph them.

4的階乘是24 所以會在這個圖表挺高的位置

So let's say let's have one, two, three, four, five.

然後5的階乘更高

1 factorial is 1, so if you call that 1.

把這些點連起來 之前提到0的階乘是1

2 factorial is 2, so somewhere about here.

所以這就是圖表了

3 factorial is 6.

理論上，整數之間的數字也有階乘的

I don't know.

例如1又2份1

Somewhere like this.

1又2份1的階乘

4 factorial is 24, so that's going to be actually quite

1又2份1的階乘是什麼？

high up here.

數學家已經解答過

And then 5 factorial is going quite high.

他們把歸納這個概念

If we join these together, I've also said that 0

這就是1又2份1階乘的概念

factorial is 1, so I reckon this is the graph.

稱爲gamma

So in theory, we should be able to get values for in

是希臘字母gamma

between, like, say, the number 1 and 1/2.

我們稱之爲「~」的gamma

1 and 1/2 factorial.

這樣寫出來

What is 1 and 1/2 factorial?

接下來的會比較複雜

So mathematicians have done that.

n的gamma等於0至無限的積分

They generalize the idea.

隨便算個數

And there is the idea of 1 and 1/2 factorial.

t的負1次方 乘以e的負t次方 dn

We call it gamma.

不是人人也熟悉這個概念

That's the Greek letter gamma.

有些人熟悉

We call it gamma of.

其他不會

And the way we write it--

這個數學概念比較複雜

actually, now this is getting a bit more sophisticated.

但與階乘很有關係

We say gamma of n is equal to the integral between 0 and

用這個便能取得整數之間的階乘

infinity of--

就是這條線上的點

let's pick something--

但我要強調一點

t to the power n minus 1, multiplied by e to the power

這點有些意想不到 但如過我取一個整數

minus n dn.

n的gamma 而n是整數

Some people won't be familiar with that.

這條函數的答案會是(n-1)的階乘 要注意這點

Some of you will be familiar with that.

可能會誤導你

Some of you won't be.

有點困難

It's a much more complicated mathematical idea, but this

既然你不能排列1又2份1個物件

would agree with the factorials.

那這條計算非整數階乘的函數又有什麼用途？

But it gives you in between values as well.

這只是概括 這條函數其實於不同範疇很有用

It plots this line.

例如概率

There is something I do need to say.

你能把這條函數用於有關概率的公式

It's slightly unexpected, but if we take a value for a whole

例如連續時間 有別於排列一件件物件

number, gamma of n, and n is whole, this actually gives you

這個函數就可以用於連續的事件

n minus 1 factorial, so be careful of that.

時間是最好的例子

That might catch you out.

當你概括不同概念

Bit of a pain, that.

那就需要概括階乘的概念

So what's the point of having a function that will give you

BRADY HARAN: 9 6 和3

factorials in between whole numbers when you can't arrange

20

1 and 1/2 objects?

44

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.