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

    JAMES GRIME: 我們Numberphile經常收到這個問題

  • 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.

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

JAMES GRIME: 我們Numberphile經常收到這個問題

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