字幕列表 影片播放
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Welcome to our first official video on our new channel called Numberphile,
歡迎來到我們正式推出的新頻道《數字狂》,今天是第一集
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all about different numbers.
新推出的頻道是有關各種不同的數字
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We've decided to make out first video on a very special date,
我們決定第一集要拍一個非常特別的日期
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that's the 11th of November 2011.
也就是 2011 年 11 月 11日
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That's 11/11/11.
在英國是 11/11/11 ( 註:日 / 月/ 年 )
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Or if you're American, that's 11/11/11, right?
如果你是美國人的話,則是 11/11/11 ( 註:月 / 日 / 年 )
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So we've come to Nottingham Forest Football Stadium, which Brady insisted on.
我們來到英國諾汀漢森林隊的主場 ─ 城市足球場,Brady 堅持要來這個地方
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11 football players in a team.
一個足球隊伍有 11 個 足球員
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Personally, I more think of 11 Doctor Who's, but never mind.
我個人是想到英國影集《異世奇人》11 位博士,不過這也無所謂
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And we've just come in through Entrance 11.
我們剛從第 11 號入口進來會場
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If we walk this way up here to row 11, and I'm going to sit
我們從這裡往上走到第 11 排,我就坐在
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right here in seat number 11, so 11, 11, 11.
11 號位置,所以也是第 11 號入口的第 11 排 11 號座位
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So guess what we're going to talk about today?
猜猜看我們今天的主題要談什麼呢?
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BRADY HARAN: What?
Brady Haran:是什麼主題?
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JAMES GRIME: 11.
James Grime:數字 11
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Yes, so recently we've had a lot of dates with ones and zeroes in them.
我們近來許多日期含有數字 1 和 0
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They're called binary days.
稱為二進制日
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They're like binary numbers.
它們就好像二進位
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And this is the last binary day until next century.
直到下個世紀前,這是最後的二進制日
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When Brady first asked me to talk about the number 11,
當初 Brady 問我要先談談數字 11 的時候
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especially for this date, I was worried.
尤其是 11 月 11 日,我當時蠻擔心的
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I thought, well, it's a prime number, but I don't know what else I could talk about.
我那時只想到 11 是質數,但我還真的不知道還有哪些內容可以談
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And then I realized there's a way that we use the number 11 every day, all the time.
但我後來發現我們每天無時無刻都會使用到 11 這個數字
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We're going to use a book.
我們現在要用一本書來講解
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Now, you can do this yourself.
你也可以自己做做看
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If you've got a book nearby, grab it.
如果你身邊正好有一本書,隨手拿著它
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And here's my book, this is Johnny Ball's Second Thinks.
我手邊這本書,是強尼‧博爾撰寫的《思考的第二面向》
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Never leave home without it.
出門時必帶的書籍
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On the back of the book, there's a bar code.
書的背後有條碼
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And next to the barcode, there is an ISBN.
在條碼的旁邊有索書號
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That's an International Standard Book Number.
這是國家標準書號
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And that number is going to be 13 digits long, or it's going
每個書號最多有 13 位數字,最少也有
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to be 10 digits long.
10 位數字
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Actually, I'm not interested in the 13-digit number.
事實上,我對於 13 位數字的書號沒興趣
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I want the 10-digit number.
我要的是 10 位數字
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Does your book have a 10-digit number on the back?
你手上的書是 10 位數字的書號嗎?
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Now, here's something you can do with that number.
你現在可以對那串數字做一件事
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Let's write down my number here on the back--
我先來寫下我這本書背後的索書號
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0140318194.
0140318194
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That's the ISBN number for old Johnny Ball there.
這是強尼‧博爾的書的索書號
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Now, the first number here actually tells you which
索書號的第一個數字告訴你
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country the book was published.
這本書是在哪個國家出版的
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The next few numbers tell you the name of the publisher.
接下來的幾個數字告訴你出版商
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The few numbers after that is the book itself.
再接下來的幾個數字代表書籍本身
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And the very last number is called a check digit.
最後一位數字被稱為檢查數元
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Let me show you something you can do.
讓我先來示範怎麼做
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I'm going to take the first digit, multiply it by 10.
我現在將第一個數字乘以 10
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OK, so 0 times 10.
0 乘以 10
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That's quite easy, that's a 0.
答案是 0,這很簡單
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The second digit there I'm going to times by 9.
第二位數字乘以 9
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So 1 times 9.
1 乘以 9
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Quite easy again, that's 9.
答案是 9,這也很簡單
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The third digit I multiply by 8.
第三個數字乘以 8
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That's going to be four times 8.
4 乘以 8
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That's 32.
答案是 32
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The next digit I times by 7, and that's another
下一個數字乘以 7,這對我來說
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easy one for me.
也是非常簡單的乘法
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That's a 0.
答案是 0
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And you can keep going.
你可以照著這樣的規則繼續計算
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And the very last digit you multiply by 1.
最後一個數字乘以 1
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So that's going to be 4 times 1, that's the number 4.
4 乘以 1,答案是 4
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Now, what you do is you add these together so you're going
現在你把加乘後的所有數字加總
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to make a number.
得到一個數字
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It's going to be something like 100, 200, 300,
數字有可能是 100,200,或是 300 等等
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that sort of size.
大約是這個範圍
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If I add these numbers together-- and I've done this
我加總這些數字,我也已經算好了
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already, I've done this in my head.
我剛才用心算方式計算
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If you add these numbers together, you will get here
如果你加總這些數字,你會得到
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the number 121.
121 這個答案
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And this number is a multiple of 11.
121 便是 11 乘以 11
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And it always will be.
不管是哪個數字排列,答案都是相同
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If you try this yourself, you will get a multiple of 11.
如果你用自己的書去試算,你也會得到 11 的加乘
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In this case, this was 11 squared, 11 times 11.
以這本書為例,這是 11 的平方,也就是 11 乘以 11
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Whatever you get, it's going to be a multiple of 11.
不論你的答案是哪個數字,它都會是 11 的多次方
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And they do this on purpose so that they can detect if
這是人為設計的數字排列方式,以便於偵測
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there's been an error.
數字排列有否錯誤
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If you inputted the book code in wrongly, if you swapped two
如果你輸入條碼錯誤,如果你不小心將 2 個數字
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digits over by mistake, it will be
互相調換,這系統
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able to detect a problem.
可以及時偵測到這問題
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It's called an error detection code.
這方法稱為錯誤偵測條碼
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The very last digit is chosen so that this will work.
通常最後一個數字決定這偵測方法的成功
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Now, the very last digit will be a number between 0 and 10.
最後一個數字可以是 0 到 10
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So for the first 9 digits, that's not a problem.
書號上呈現前面 9 位數字都不會是個問題
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0, 1, 2, 3, 4, 5.
0,1,2,3,4,5
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The very last digit, the number 10, well, we don't have
但對於最後一個數字 10
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a single digit for a 10.
我們沒有一個數字就能代表 10
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So instead, you may see the letter X, which is the Roman
所以你會看到英文字母 X,代表的是
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numeral for 10.
羅馬數字的 10
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That's used for 10 instead.
用以代表數字 10
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So this part, well, that could be anything that
所以這部分可以是任何數字
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they want it to be.
只要他們想要
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But this number is completely engineered to make this work,
但最後的數字必須是由程式去挑選
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to make a multiple of 11.
才會是 11 的多次方
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If you have one of the 13-digit book codes, they use
如果你的書籍是使用 13 位數字的索書碼
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a slightly different operation.
運算的程式略有不同
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It's actually based on the number 10 instead
它是用 10 為基礎,而非
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of the number 11.
11 為基礎去運算
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OK, so this is the view from 11, 11, 11.
這便是 11,11,11 的其中一個說法
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The number 11 is used for these error correction codes.
數字 11 可以作為錯誤偵測條碼之用途
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But error correction code, that idea is used
不過錯誤偵測條碼的用途
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all over the place.
非常之廣
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It's used with computers.
可以用在電腦
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It's used with mobile phones if you're
也可以用在手機,當你試著
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trying to send a message.
傳送一則簡訊
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And, well, in transmission, you can
當然傳送訊息的過程中,你會
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lose some of the message.
遺失部分內容
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So you want to be able to reconstruct the message
所以你會希望能重新建立訊息
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without having to send it all over again.
卻能不用重新寄送簡訊
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And you can do this yourself.
你可以靠自己就能完成
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Again, this is with a CD.
我這裡有片 CD
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And if you burn a CD, if you hold it up to the light, you
如果你要燒錄一片 CD,把 CD 拿到光亮處
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can even see where the tracks are.
你可以看到其中的軌跡
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And if you use a black pen here, a black marker pen, I
假設使用一隻黑色麥克筆,我現在
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could put a dot on this CD, and this would represent
在 CD 上用筆弄一個黑點,把這黑點視作
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something like dirt or a scratch.
灰塵或是刮痕
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And this will still play.
這片 CD 還是可以播放
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It can compensate for this missing information.
它可以自動修復遺漏的訊息
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Well, in a book code, we might know there is a missing five.
我們知道書碼的數字可能有誤
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There's something we can do.
我們還有辦法補救
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In a CD, it uses a whole equation with all sorts of
CD 片的方程式透過
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numbers, and you can use that equation to work out which
各種數字,你可以利用方程式來找出
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number is missing and what it is.
遺漏的數字,以及是什麼數字
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So we've done this with my CD in my computer right now.
我們現在就要用我的 CD 在我的電腦做這件事
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I'm going to play you a song, a song all about numbers.
我要播送一首歌給你們聽,是一首有關數字的歌曲
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I thought it would be an appropriate choice.
我想這是不錯的選擇
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You can hear it does still play.
你可以聽到 CD 正常播放音樂
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If I pull it out, slightly more dramatic than a black
我現在直接拿出 CD,這比黑色麥克筆畫點
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marker pen.
更嚴重
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We actually drilled a hole through the CD.
事實上,我們剛才在 CD 上鑽了一個洞
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So the data is actually literally missing.
所以這部分的資料是真的不見了
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Not just a black mark, we actually took out the data.
不是一個黑點,而是我們真的把資料取出來
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Yet the CD player can reconstruct it and still play
但 CD 播放機仍然可以自動修復,而且繼續播放音樂
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the song because it will use this error detection
因為它可以透過錯誤偵測科技來修正
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technology, this mathematics that reconstructs the missing
數學運算方式可以重新建構遺失的
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information.
訊息
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In fact, this hole is about 2 millimeters.
這洞約有 2 毫米
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You can go up to about 2.5.
你可以鑽一個 2.5 毫米的洞
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If I go even further, I've got one here with 3 millimeters, a
如果我再鑽更大洞,大概是 3 毫米大小
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3-millimeter hole.
一個 3 毫米的洞
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This hole is too big.
這洞真的非常大
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Let's try it out.
我們現在再試一次
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I'm going to try and play this as well.
我來試試看可否播放
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You'll notice the difference.
你會發現其中的差別
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[MUSIC SKIPPING]
〔音樂跳針〕
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It's trying.
它正在試著修復
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It's trying its best.
它正在用盡全力試著修復
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It doesn't play.
它不能播放了
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Too much missing information.
太多遺失的資訊
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So 11 was just our starting point into a world of
11 只是我們踏入數學世界的
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mathematics and error correction code, something
起始點,錯誤偵測條碼的用途
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that is used everywhere that makes your CD work, that makes
隨處可見,可以讓 CD 正常運轉
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your mobile phone work, and, of course, makes that number
讓手機正常運作,其中當然包括讓書籍背後的
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on the back of a book work.
書碼數字排列正常
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And Brady insists that I remind you that, yes, it is
Brady 堅持要我提醒你們 11
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the number of players in a football team.
是一個足球隊伍裡足球員的總數