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  • JAMES GRIME: We're going to break a rule.

    我們'要打破一個規則。

  • We're break one of the rules of Numberphile.

    我們打破了Numberphile的規則之一。

  • We're talking about something that isn't a number.

    我們'說的是一些不是'數字的東西。

  • We're going to talk about infinity.

    我們'要談的是無窮大。

  • So infinity.

    所以無窮大。

  • Now like I said, infinity is not a number.

    就像我說的,無窮大不是一個數字。

  • It's a idea.

    這'是一種想法。

  • It's a concept.

    這是一個概念。

  • It's the idea of being endless, of going on forever.

    它'是無窮無盡的想法,永無止境地進行下去。

  • I think everyone's familiar with the idea of

    我想大家'的想法很熟悉。

  • infinity, even kids.

    無限,甚至孩子。

  • You start counting 1, 2, 3, 4, 5--

    你開始數1,2,3,4,5 --

  • you might be five years old, but already you're thinking,

    你可能是五歲,但你已經'想。

  • what's the biggest number I can think of.

    什麼'是我能想到的最大的數字。

  • And you go, oooh, it's 20.

    你去,哦,是'20。

  • You get a bit older, and you go, maybe it's a million.

    你再大一點,你去,也許是'萬。

  • It never ends, does it? 'Cause you can keep adding 1.

    它永遠不會結束,不是嗎'因為你可以繼續增加1。

  • So that's the idea of infinity.

    所以,這就是無限的想法。

  • The numbers go on forever.

    數字永遠在繼續。

  • But I'm going to tell you one of the more surprising facts

    但我'要告訴你一個比較驚人的事實

  • about infinity.

    關於無窮大。

  • There are different kinds of infinity.

    無間有不同的種類。

  • Some infinities are bigger than others.

    有的無限大,有的無限大。

  • Let's have a look.

    讓我們'看看吧。

  • The first type of infinity is called countable.

    第一種類型的無窮大叫做可數。

  • And I don't like the name countable.

    而且我不喜歡可數這個名字。

  • And Brady gave me a little bit of a hmm, just then.

    而布雷迪也給了我一個小小的暗示,就在這時。

  • Because if you're talking about infinity, you can't

    因為如果你說的是無限大,你不能

  • count infinity, can you?

    數無窮大,你會嗎?

  • Because it goes on forever.

    因為它永遠在繼續。

  • I think it's a terrible name.

    我認為這是一個可怕的名字。

  • I prefer to call it listable.

    我更喜歡叫它可上市。

  • Can we list these numbers?

    我們可以列出這些數字嗎?

  • All right.

    好吧,我知道了

  • Let's do these simple numbers, 1, 2, 3--

    讓我們來做這些簡單的數字,1,2,3------。

  • BRADY HARAN: You're not gonna do all of them, are you James?

    BRADY HARAN:你'不打算做所有的人,是詹姆斯?

  • JAMES GRIME: 4.

    JAMES GRIME: 4.

  • How long have we got?

    我們還有多少時間?

  • BRADY HARAN: (LAUGHING) 10 minutes.

    十分鐘。

  • JAMES GRIME: Right.

    對。

  • 5, 6--

    5, 6--

  • so you can list the whole numbers.

    所以你可以列出整數。

  • So this is called countable.

    所以這叫可數。

  • Listable, I prefer.

    可上市,我更喜歡。

  • What about the integers?

    那整數呢?

  • All the integers.

    所有的整數。

  • That's all the negative numbers as well.

    這'也都是負數。

  • So there's 0.

    所以有'的0。

  • Let's have that.

    讓'的有。

  • But there's 1 and minus 1, there's 2 and minus 2, there's

    但是,有1和負1,有2和負2,有

  • 3, and minus 3.

    3,和負3。

  • Now, that is an infinity as well.

    現在,這也是一個無窮大的問題。

  • And in some sense, it's twice as big, because there seems to

    而在某種意義上,它的兩倍大,因為似乎有。

  • be twice as many numbers.

    是數字的兩倍。

  • But it is infinity as well.

    但它也是無窮大的。

  • They're both infinity, and they're both the

    他們都是無限的,他們都是那個

  • same type of infinity.

    同類型的無窮大。

  • They both can be listed.

    他們都可以上市。

  • Perhaps more surprisingly, the fractions can

    也許更令人驚訝的是,分數可以是

  • be listed as well.

    也被列入。

  • But you have to be a bit clever about this.

    但是,你必須要有一點聰明的做法。

  • Let's try and list the fractions.

    讓我們'試著列出分數。

  • I'm going to write out a rectangle.

    我'要寫出一個長方形。

  • 1 divided by 1.

    1除以1。

  • That's a fraction.

    那是一小部分。

  • [INAUDIBLE].

    [INAUDIBLE].

  • Let's have 1 divided by 2, 1/3, 1/4, 1/7--

    讓我們有1除以2,1/3,1/4,1/7--。

  • OK, that goes on.

    好,那就繼續。

  • Let's do the next row and have two at the top.

    讓'我們做下一排,上面有兩個。

  • 2/1, 2/2, 2/3, 2/4.

    2/1, 2/2, 2/3, 2/4.

  • Let's do the next one.

    讓我們'做下一個。

  • 3/1, 3/2.

    3/1, 3/2.

  • 4/6, 4/7.

    4/6, 4/7.

  • That goes on and we can keep going.

    這樣下去,我們可以繼續走下去。

  • So here, I've made some sort of an infinite rectangle array

    所以在這裡,我做了一個無限的矩形數組。

  • of fractions.

    的分數。

  • Now if I want to make it a list like this, though, If I

    現在,如果我想把它做成這樣的清單,雖然,如果我。

  • went row by row, you're going to have a problem.

    了一排一排,你'會有問題。

  • If you go row by row, I'll go--

    如果你一排一排地走,我就... ...

  • there's 1, 1/2, 1/3, 1/5, 1/6, 1/7-- and

    有1,1/2,1/3,1/5,1/6,1/7-----------。

  • I'll keep going forever.

    我'會一直走下去的。

  • And I'm never going to reach the second row.

    而我'永遠也到不了第二排。

  • I can't list them.

    我不能列出它們。

  • Not that way.

    不是這樣的。

  • You can't list them that way.

    你不能用這種方式列出它們。

  • You'll never reach the second row.

    你'永遠也到不了第二排。

  • This is how you list them.

    這就是你列出它們的方式。

  • Slightly more clever than that.

    比這稍微聰明一點。

  • You take the diagonal lines.

    你把對角線。

  • Now, I can guarantee that every fraction will appear on

    現在,我可以保證每一個分數都會出現在。

  • one of those diagonal lines.

    其中一條對角線。

  • And you list them diagonal by diagonal.

    而你卻把它們一斜一斜地列出來。

  • So that's the first diagonal.

    所以這就是第一條對角線。

  • Then you list the second diagonal-- there it is.

    然後你列出第二條對角線... 就是這樣。

  • Then you list the third diagonal, then you take the

    然後你列出第三條對角線,然後你以

  • fourth diagonal, and the fifth.

    第四條對角線,和第五條。

  • So eventually, you are going to do this every fraction.

    所以最終,你要做到每一個分數。

  • Every faction appears on a diagonal, and you're

    每個派別都出現在對角線上,你'。

  • going to list them.

    要把它們列出來。

  • Now, if you take all the numbers, right?

    現在,如果你把所有的數字,對嗎?

  • That's the whole number line.

    這'是整個數字線。

  • Let's try that.

    讓我們試試吧。

  • Look, I'm going to draw it.

    你看,我打算畫它。

  • It's a continuous line of numbers.

    這是一條連續的數字線。

  • These are all your decimals.

    這些都是你的小數。

  • You've got 0 there in the middle, and you'll

    你有0在中間,你會

  • go 1 and 2 and 3.

    去1和2和3。

  • But it has a 1/3.

    但它有1/3。

  • It will contain pi, and e, and all the

    它將包含pi,和e,以及所有的。

  • irrational numbers as well.

    無理數也。

  • Can you list them?

    你能列出他們嗎?

  • How do you list them?

    你怎麼把它們列出來?

  • 0 to start with, and then 1?

    0開始,然後1?

  • But hang on.

    但是等一下

  • We've missed a half.

    我們'已經錯過了一半。

  • So we put in the half.

    所以我們把在半。

  • Hang on, we've missed the quarter.

    等等,我們已經錯過了這一季。

  • We put in the quarter.

    我們把四分之一。

  • But we've missed 0.237--

    但我們已經錯過了0.237----。

  • so how do you list the real numbers?

    那你怎麼列出實數呢?

  • It turns out you can't.

    事實證明,你不能'。

  • In fact, rather remarkably, I can show you that we can't

    事實上,相當驚人的是,我可以告訴你,我們可以'不

  • list them, even though were talking about something so

    列舉他們,即使是在談論這樣的事情。

  • complicated as infinity.

    複雜如無窮大。

  • BRADY HARAN: Do it, man!

    動手吧,夥計!

  • JAMES GRIME: We need paper.

    我們需要紙。

  • BRADY HARAN: We need an infinite amount of

    我們需要無限量的... ...

  • paper here, I think.

    我想,這裡的紙。

  • JAMES GRIME: (LAUGHING) It's a big topic.

    JAMES GRIME: (LAUGHING) It's a big topic.

  • Imagine we could list all the decimals, right?

    想象一下,我們可以列出所有的小數,對嗎?

  • We can't, actually.

    其實,我們不能'。

  • But pretend we can.

    但是假裝我們可以。

  • What sort of--

    什麼樣的...

  • what would it look like?

    會是什麼樣子?

  • We'll start with all the 0-point decimals.

    我們'將從所有的0點小數開始。

  • Let's pick some decimals.

    讓我們'選擇一些小數。

  • 0.121--

    0.121--

  • dot dot dot dot dot.

    點點滴滴。

  • Let's pick the next one.

    讓我們'挑下一個。

  • Let's say the next one is 0.221--.

    比方說下一個是0.221--。

  • Next one, let's do 0.31111129--.

    下一個,讓我們做0.31111129--。

  • And let's take another one, here.

    我們再來一個,這裡。

  • 0.00176--.

    0.00176--.

  • Now I'm going to make a number.

    現在我'要做一個數字。

  • This is the number I'm going to make.

    這是我要做的號碼。

  • I'm going to take the diagonals here.

    我'要在這裡採取對角線。

  • I'm going to take this number and this number and this

    我要把這個號碼和這個號碼和這個

  • number and this number and this number.

    這個數字和這個數字,這個數字。

  • And I am going to write that down.

    而我要把這些寫下來。

  • So what's that number I've made?

    那麼,我做的那個號是什麼呢'?

  • It's 0.12101--

    這是0.12101--。

  • something, something, something.

    東西,東西,東西。

  • Now this is my rule.

    這是我的原則。

  • I'm going to make a whole new number from that one.

    我'要從那個號上做一個全新的號。

  • This is the number I'm going to make.

    這是我要做的號碼。

  • If it has a 1, I'm going to change it to a 2.

    如果它有1,我'要把它改成2。

  • And if it has a 2 or anything else, I will change it to a 1.

    如果它有2或其他什麼,我會把它改成1。

  • So let's try that.

    所以讓我們試試吧。

  • So I'm going to turn this into--

    所以我打算把這個變成... ...

  • 0-point.

    0分

  • So if it has a 1, I'm going to turn it into a 2.

    所以如果它有1,我'要把它變成2。

  • If it's anything else, I'm going to turn it into a 1.

    如果它'別的,我'要把它變成1。

  • So that will be a 1.

    所以這將是一個1。

  • I'm going to change 1 here into a 2.

    我'要把這裡的1改成2。

  • I'm going to change that one into a 1.

    我'要把這個改成1。

  • I'm going to change that one into a 2-- that was my rule.

    我要把這個改成2--這是我的規則。

  • And I'll make something new.

    而我'會做出一些新的東西。

  • That does not appear on the list.

    這並沒有出現在名單上。

  • That number is completely different from anything else

    這個數字和其他東西完全不同

  • on the list, because it's not the first number, because it's

    因為它不是第一個數字,因為它是

  • different in the first place.

    首先是不同的。

  • It's not the second number, because it's different in

    這不是第二個數字,因為它的不同,在

  • second place.

    第二名。

  • It's not the third number, because it's different in the

    這不是第三個數字,因為它是不同的在。

  • third place.

    第三名。

  • It's not the fourth number because it's different in the

    這不是第四個數字,因為它是不同的在。

  • fourth place.

    第四名。

  • It's not the fifth number, because it's different in the

    這不是第五個數字,因為它是不同的在。

  • fifth place.

    第五名。

  • You've made a number that's not on that list.

    你'做了一個不在那個名單上的數字。

  • And so you can't list all the decimals, in which case it is

    所以你不能列出所有的小數,在這種情況下,它是

  • uncountable.

    不可計數的。

  • It is unlistable.

    它是無法上市的。

  • And that means it's a whole new type of infinity.

    而這意味著它'是一種全新的無窮大的類型。

  • A bigger type of infinity.

    更大類型的無窮大。

  • BRADY HARAN: Surely we could, James, because all we've got

    我們當然可以,詹姆斯,因為我們所有的'已經得到了

  • to do is keep doing your game and making them and adding

    要做的是繼續做你的遊戲,並使他們和添加。

  • them to the list.

    列表中。

  • And if we keep doing that, won't we get there eventually?

    如果我們繼續這樣做,我們最終不會到達那裡嗎?

  • JAMES GRIME: But you could then create another number

    但你可以再創造一個數字

  • that won't be on that list.

    那不會在那個名單上。

  • And so the guy who came up with is a German mathematician

    所以想出這個辦法的人是一個德國數學家。

  • called Cantor.

    稱為康托爾。

  • Cantor lived 'round about the turn of the 20th century.

    康托爾生活在'20世紀初左右。

  • He was ridiculed for this.

    他是以受到了嘲笑。

  • For this idea that there were different types of infinity,

    對於這種想法,有不同類型的無限性。

  • he was called a charlatan.

    他被稱為金光黨。

  • And he was called-- it was nonsense, it was called.

    他被稱為... ... 這是胡說八道,它被稱為。

  • And poor old Cantor was treated really badly by his

    而可憐的老康托爾被他的家人虐待

  • contemporaries, and he spent a lot of his later life in and

    同時代的人,而他晚年的很多時間都是在和。

  • out of mental institutions, where he died, in the end.

    出了精神病院,在那裡他死了,最後。

  • Near the end of his life, it was recognized.

    在他生命接近尾聲時,得到了認可。

  • It was true.

    這是真的。

  • It was recognized.

    它是公認的。

  • And he had all the recognition that he deserved.

    而他也得到了所有他應得的認可。

  • BRADY HARAN: And now he's on Numberphile.

    現在他在Numberphile上。

  • JAMES GRIME: And now he's on Numberphile, the greatest

    現在他在《Numberphile》上,最偉大的。

  • accolade of all.

    譽的所有。

  • Georg Cantor.

    Georg Cantor.

JAMES GRIME: We're going to break a rule.

我們'要打破一個規則。

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B1 中級 中文 數字 無窮 對角線 列出 無限 分數

無限是什麼?「無限」已經超乎你所能想像的了! Infinity is bigger than you think - Numberphile

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