And this is clearly a Friday.

而這顯然是一個星期五。

And this is the end of week one.

而這是一個星期的末端。

So you may recall that we left off, last time, with a cliffhanger of sorts.

所以，你可能還記得，我們​​不放過， 最後一次，有各種各樣的懸念。

Whereby we expose this lie, that no matter what you've been taught growing

我們藉以揭露這個謊言，沒有 不管你被教導成長

up, 1 divided by 10 is not, in fact, 0.1 or 0.100 or 0.10000.

向上，1除以10不是在 事實上，0.1或0.100或0.10000。

In fact, if we compile this program, as we did yesterday, with,

事實上，如果我們編譯這個 計劃，因為我們沒有昨天，有，

make imprecision, and then do dot, slash, imprecision,

使不精確性，然後 做點，斜線，不精確，

we learned that this is what's 1 divided by 10 actually is.

我們了解到，這是什麼 1除以10實際上是。

So that's not really quite the case.

所以這不是真的相當的情況下。

But this does hint at some fundamental limitation of computers.

但是，這確實暗示了一些 電腦的基本限制。

And indeed, among the things we're going to do

事實上，中 我們將要做的事情

today is take a look at why this has happened, what implications this has,

今天就來看看為什麼這有 事情發生了，有什麼意義這有，

how humanity has failed to grasp this in some very

人類如何失敗 掌握這一些很

serious situations-- the result of the which

嚴重situations-- 在其結果

has been quite tragic and expensive-- and also

已經相當慘烈 和expensive--也

take a look at how we can actually defend

看看我們如何 實際上可以捍衛

against these kinds of limitations.

針對這些種類的限制。

So intuitively, perhaps, why is 1 divided by 10,

所以直覺，也許， 為什麼是1除以10，

according to my computer here, not just 1/10, 0.10?

根據我的電腦 這裡，不只是的1/10，0.10？

Yeah.

是啊。

What do you think?

你怎麼看？

>> AUDIENCE MEMBER: The radix is different?

>> 觀眾：基數是不同的？

>> DAVID J MALAN: The what is?

>> 大衛·馬蘭：該是什麼？

Oh, the radix is different?

呵呵，基數是不同的？

So not quite.

因此，不完全是。

It's actually more fundamental to the hardware.

它實際上更多 基本的硬件。

Other thoughts?

其他的想法？

Yeah.

是啊。

AUDIENCE MEMBER: They represent numbers differently?

觀眾：他們 表示數字不同？

DAVID J MALAN: So they-- Right.

大衛·馬蘭：所以they--權。

They represent numbers differently.

他們代表的數字不同。

Inaccurately, apparently.

不準確，很明顯。

So that is-- Well, differently from what?

這樣is--好， 不同於什麼？

Or from whom?

或者從誰？

From us?

從我們呢？

AUDIENCE MEMBER: Yeah.

觀眾：是啊。

They don't use the decimal system to [INAUDIBLE].

它們不使用小數 系統[聽不清]。

DAVID J MALAN: OK.

大衛·馬蘭：OK。

So in a sense, they don't use the decimal system.

因此，從某種意義上說，他們不 使用十進制系統。

Underneath the hood, everything is, indeed, binary,

在系統底層， 一切，的確，二進制，

and that's related, in fact.

這就是相關的，其實。

But it's even a simpler explanation than that.

但是，它甚至更簡單 解釋比。

Yeah.

是啊。

>> AUDIENCE MEMBER: They only have so many bits.

>> 觀眾：他們 只有這麼多的位。

They can only store up to a certain extent, like with the decimals.

他們最多只能存儲一定 某種程度上，像小數。

>> DAVID J MALAN: Yeah.

>> 大衛·馬蘭：是的。

This is what's really getting at it.

這是什麼真正得到它。

It turns out that computers typically, will always only

事實證明，計算機 通常情況下，將永遠只有

use a finite amount of information to represent something.

使用的信息的有限數量 代表什麼。

Right?

對嗎？

After all, we only have a fixed amount of hard disk space.

畢竟，我們只有一個固定的 量的硬盤空間。

We only have a fixed amount of RAM, or computer memory.

我們只有一個固定量 的RAM，或計算機的內存。

And if you only have a fixed amount of something,

如果你只有一個 固定金額的東西，

surely you can't actually count up to infinity or any number you want.

當然你不能真正計數 到無窮大或任何你想要的數字。

You, kind of, have to pick and choose what range

你有種，有 挑選什麼範圍

of values you're going to support.

值你要支持。

>> And so for instance, a week or two ago, when we talked about ASCII,

>> 因此，例如，一兩個星期 以前，當我們談到ASCII，

and we talked about 8 bits, or a byte, so to speak,

我們談了8位， 或字節，可以這麼說，

the biggest number we could represent with 8 bits was what?

最大數我們可以 有8位表示是什麼呢？

255.

255。

And we can represent 256 total values but not if we spend one of them

我們可以表示256個總價值 但如果我們把其中的一個

on 0-- Well, but if we spend one of them on 0,

在0--好了，但如果我們 花其中的一個0，

then the biggest number is, of course, 255.

那麼最大的數 是，當然，255。

>> So it turns out, that this is the case too, in this context.

>> 所以，事實證明，這是 的情況下也一樣，在這種情況下。

We started talking, last time, about floating-point numbers, which

我們開始談，最後一次， 有關浮點數字，

are different from integers, in that they have a decimal point

來自整數不同，在 他們有一個小數點

and, hopefully, some numbers after that, but they are too.

並希望一些數字 在這之後，但是它們太。

A computer is only typically going to use 32 bits, maybe 64 bits

一台計算機通常只打算 使用32位，也許64位

to represent a floating point value.

來表示一個浮點值。

So even though we've grown up learning mathematics and knowing

因此，即使我們已經長大了 學習數學和了解

that you can absolutely have an infinite number of numbers

你絕對可以有 數的無限數量

after the decimal point, not if you have finite memory.

小數點後不 如果你有有限的內存。

You, kind of, have to round, or pick and choose which

你有種，有圓形， 或者挑選哪些

numbers you're going to represent.

你會數來表示。

And so you can think of, in a sense, this being the computer's closest

所以，你能想到的，從某種意義上說， 這是計算機的最接近

approximation to the value 1/10 that it can get, with just 32 or so bits.

近似​​的值的1/10 它可以得到，只有32左右位。

And it's not just 1/10.

而且它不只是1/10。

>> For instance, if I change this to 1/3, which we think is even simpler.

>> 舉例來說，如果我改變這 1/3，我們認為這是更簡單。

So 1 divided by 3.

因此，1除以3。

Let me save the file.

讓我保存文件。

Let me go ahead and recompile it.

讓我繼續前進，重新編譯它。

And let me re-run it.

讓我重新運行它。

Here too, apparently, 1/3 is not 0.3 and then an infinite number of 3's

在這裡，顯然，三分之一沒有0.3 然後3的無限數量

thereafter.

其後。

You have this imprecision at the end.

你有這樣的不精確性在最後。

So we humans are correct, and what you learned is, in fact, correct,

所以，我們人類是正確的，什麼 你了解，其實，正確的，

but we're bumping up against some limitations.

但是我們撞了 針對一些限制。

>> And what I thought we'd do today, is begin

>> 而我的想法 我們今天會做，是開始

by looking at, frankly, the tragic consequences of this

通過看，坦率地說，在 這種悲慘的後果

sometimes, when mankind does not quite implement

有時，人類的時候 並不完全實現

for this reality and these limitations.

針對這一現實，這些限制。

And we'll see a series of vignettes from the History Channel that takes

而且我們會看到一系列的小插圖 從歷史頻道的需要

a look at how things have gone wrong.

一起來看看如何事情已經出了問題。

It's about 8 minutes long, and we'll come back after this

這是約8分鐘之久， 我們將在這之後回來

and take a look at exactly what else can go wrong.

看一看在完全相同 還有什麼可能出錯。

If we could dim the lights--

如果我們能昏暗的lights--

>> [VIDEO PLAYBACK]

>> [視頻回放]

>> -Computers, we've all come to accept the often frustrating problems that

>> -Computers，我們都逐漸接受 經常令人沮喪的問題，

go with them.

和他們一起去。

Bugs, viruses, and software glitches are small prices

錯誤，病毒，和軟件 故障都是小的價格

to pay for the convenience.

支付的便利。

But in high-tech and high-speed military and space program applications,

但是，在高科技和高速 軍事和航天計劃的應用，

the smallest problem can be magnified into disaster.

可以最小的問題 被放大成災難。

On June 4, 1996, scientists prepared to launch an unmanned Ariane 5 rocket.

1996年6月4日，科學家準備 啟動一個無人阿麗亞娜5型火箭。

It was carrying scientific satellites designed

它攜帶的科學 衛星設計

to establish precisely how the Earth's magnetic field interacts

建立精確如何 地球的磁場相互作用

with solar winds.

與太陽風。

The rocket was built for the European Space Agency

火箭是專為 歐洲航天局

and lifted off from its facility on the coast of French Guiana.

並從其工廠升空 在法屬圭亞那的海岸。

>> -And about 30 seconds into the flight, they first

>> - 和大約30秒後進入 飛行中，他們首先

noticed something was going wrong.

發現了一些打算錯了。

That the nozzles were swiveling in a way they really shouldn't.

使噴嘴被旋轉 在某種程度上，他們真的不應該。

Around 40 seconds into the flight, clearly the vehicle was in trouble,

約40秒後進入飛行， 明確了車輛遇到了麻煩，

and that's when they made the decision to destroy it.

而當他們提出的 決定摧毀它。

A Range Safety Officer with tremendous guts pressed the button,

隨著一系列安全主任 巨大的勇氣按下按鈕，

blew up the rocket before it could become a hazard to public safety.

炸毀了火箭之前，它可能 成為危害公眾安全。

>> -This was the maiden voyage of the Ariane 5,

>> - 這是處女 阿麗亞娜5的航程，

and its destruction took place because of a flaw

它破壞了 地方，因為一個缺陷

embedded in the rocket's software.

嵌在火箭的軟件。

-The problem on the Ariane was that there

在-The問題 阿麗亞娜是，有

was a number that required 64 bits to express,

是一個數字，需要 64位表示，

and he wanted to convert it to a 16 bit number.

他想轉換 它以一個16位的數。

They assumed that the number was never going to be very big,

他們假定數 從來沒有將是非常大的，

that most of those digits in the 64 bit number were 0's.

大多數的這些數字中 64位數字是0。

They were wrong.

但他們錯了。

>> -The inability of one software program to accept

>> 一-The無力 軟件程序接受

the kind of number generated by another was at the root of the failure.

所產生的那種數 另一種是在失敗的根源。

Software development had become a very costly part of new technology.

軟件開發已經成為一個 的新技術非常昂貴的部分。

The Ariane 4 rocket had been very successful, so much of the software

阿麗亞娜-4火箭一直非常 成功的，那麼多的軟件

created for it was also used on the Ariane 5.

創造這也是 用在阿麗亞娜5。

>> -The basic problem was that the Ariane 5 was faster, accelerated faster.

>> -The基本的問題是，阿麗亞娜 5增快，加速快。

And the software hadn't accounted for that.

而且軟件早已不是 佔了這一點。

>> -The destruction of the rocket was a huge financial disaster,

>> 火箭-The破壞 是一個巨大的金融災難，

all due to a minute software error.

全因一分鐘，軟件錯誤。

But this wasn't the first time data conversion problems

但是，這是不是第一次 實時數據轉換問題

had plagued modern rocket technology.

一直困擾現代火箭技術。

>> -In 1991, with the start of the First Gulf War,

>> -in 1991，在開始 第一次海灣戰爭中，

the Patriot missile experienced a similar kind

愛國者導彈 經歷了類似的一種

of number conversion problem.

數轉換的問題。

As a result, 28 people, 28 American soldiers

其結果是，28人， 28美國士兵

were killed and about 100 others wounded,

被打死，約 100人受傷，

when the Patriot, which was supposed to protect against incoming scuds,

當愛國者，這是應該 以防止傳入的飛毛腿導彈，

failed to fire a missile.

沒有發射導彈。

>> -When Iraq invaded Kuwait and America launched Desert Storm in early 1991,

>> - 當伊拉克入侵科威特和美國 在1991年年初推出沙漠風暴，

Patriot missile batteries were deployed to protect Saudi Arabia and Israel

愛國者導彈部署 保護沙特阿拉伯和以色列

from Iraqi Scud missile attacks.

從伊拉克飛毛腿導彈攻擊。

The Patriot is a US, medium-range surface-to-air system

愛國者是美國，中程 表面對空氣系統

manufactured by the Raytheon company.

由雷神公司製造。

>> -The size of the Patriot interceptor, itself, is about, roughly 20 feet long.

>> 愛國者攔截-The大小， 本身就是約，約20英尺長。

And it weighs about 2000 pounds.

而且它的重量約2000磅。

And it carries a warhead of about-- I think it's roughly 150 pounds.

同時還可以進行about--的彈頭 我認為這是約150磅。

And the warhead, itself, is a high explosive,

和彈頭本身， 是一種高爆炸藥，

which has fragments around it.

它有它周圍的碎片。

The casing of the warhead is designed to act like buckshot.

彈頭的外殼是 設計像鹿彈。

>> -The missiles are carried, 4 per container,

>> -The導彈 進行，4％的集裝箱，

and are transported by a semi-trailer.

並通過半拖車被運送。

>> -The Patriot anti missile system goes back at least 20 years now.

>> -The愛國者反導彈系統 追溯到現在至少需要20年。

It was originally designed as an air defense missile

它最初的設計 作為防空導彈

to shoot down enemy airplanes.

擊落敵人的飛機。

In the First Gulf War, when that war came along,

在第一次海灣戰爭， 當戰爭來了，

the Army wanted to use it to shoot down scuds, not airplanes.

陸軍希望用它來 擊落飛毛腿導彈，而不是飛機。

The Iraqi Force was not so much of a problem,

伊拉克隊還是沒有 這麼多的問題，

but the Army was worried about scuds.

但陸軍擔心飛毛腿導彈。

And so they tried to upgrade the Patriot.

因此，他們試圖 升級愛國者。

>> -Intercepting an enemy missile traveling at mach five

>> -Intercepting敵人 導彈行駛在馬赫5

was going to be challenging enough.

將要進行足夠的挑戰。

But when the Patriot was rushed into service,

但是愛國者的時候 被衝進服務，

the Army was not aware of an Iraqi modification that made

陸軍不知道的 ，使得伊拉克修改

their scuds nearly impossible to hit.

他們的飛毛腿導彈幾乎是不可能的打擊。

>> -What happened, is the scuds that were coming in, were unstable,

>> - 什麼事，是飛毛腿的 要來的，是不穩定的，

they were wobbling.

他們搖晃。

The reason for this, was the Iraqis, in order

這樣做的原因， 是伊拉克人，為了

to get 600 kilometers out of a 300 kilometer range missile,

讓600公里出 300公里的射程導彈，

took weight out of the front warhead and made the warhead lighter.

把重出前彈頭 並提出彈頭更輕。

So now, the Patriot's trying to come at the scud, and most of the time,

所以，現在，愛國者的苦思 在飛毛腿，並且大多數時候，

the overwhelming majority of the time, it would just fly by the scud.

絕大多數的 時間，它只是由飛毛腿飛。

>> -Once the Patriot system operators realized the Patriot missed its target,

>> - 一旦愛國者系統運營商 實現了愛國者偏離目標，

they detonated the Patriot's warhead, to avoid possible casualties if it

他們引爆了愛國者彈頭， 為了避免可能的人員傷亡，如果它

was allowed to fall to the ground.

被允許倒在地上。

>> -That was what most people saw, those big fireballs in the sky,

>> - 那是大多數人所看到的， 那些大火球在天空中，

and misunderstood as intercepts of Scud warheads.

和誤解為 飛毛腿彈頭的攔截。

Although in the night skies, Patriots appeared

雖然在夜間 天空，愛國者亮相

to be successfully destroying scuds, at Dhahran,

要成功地 摧毀飛毛腿導彈，在達蘭，

there could be no mistake about its performance.

有可能是沒有錯 關於它的性能。

There, the Patriot's radar system lost track of an incoming Scud,

在那裡，愛國者的雷達系統 失去了進入的飛毛腿的軌道，

and never launched, due to a software flaw.

永不推出， 由於軟件缺陷。

It was the Israelis who first discovered that the longer the system was on,

這是誰首先發現了以色列人 該系統不再是上，

the greater the time discrepancy became, due to a clock embedded

越大時間差異 成為，由於一個時鐘的嵌入式

in the system's computer.

在系統的計算機。

>> -About 2 weeks before the tragedy in Dhahran,

>> - 關於前2週 悲劇發生在宰赫蘭，

the Israelis reported to the Defense Department,

以色列人報 國防部，

that the system was losing time, that after about 8 hours of running,

該系統是浪費時間， 該後約8小時的運行，

they noticed that the system was becoming noticeably less accurate.

他們注意到，該系統 變得明顯不太準確。

The Defense Department responded by telling all of the Patriot batteries

國防部回應 告訴所有的愛國者電池

to not leave the systems on for a long time.

不離開系統 上很長一段時間。

They never said what a long time was.

他們從來不說什麼很長一段時間了。

8 hours?

8小時？

10 hours?

10小時？

1000 hours?

千小時？

Nobody knew.

沒有人知道。

-The Patriot battery stationed at the barracks

-The愛國者電池 駐紮在軍營

at Dhahran, and its flawed internal clock, had been on over 100 hours

在宰赫蘭的，其有缺陷的內部 時鐘，已經在超過100小時

on the night of February 25.

2月25日的晚上。

>> -It tracked time to an accuracy of about 1/10 of a second.

>> - 它跟踪時間精度 約一秒的1/10。

Now 1/10 of a second is an interesting number

現在1/10秒是 一個有趣的數字

because it can't be expressed in binary, exactly.

因為它不能 按二進制，沒錯。

Which means, it can't be expressed, exactly,

這意味著，它不能 表示，準確地說，

in any modern digital computer.

在任何現代數字計算機。

It's hard to believe, but use this as an example.

很難相信，但 使用這個作為一個例子。

Let's take the number, 1/3.

讓我們來數的1/3。

1/3 cannot be expressed in decimal, exactly.

三分之一不能表示 十進制，沒錯。

1/3 is 0.333 going on for infinity.

1/3是0.333持續了無窮大。

There's no way to do that, with absolute accuracy, in decimal.

有沒有辦法做到這一點，有 絕對精度，十進制。

That's exactly the same kind of problem that happened in the Patriot.

這正是相同種類的 問題發生在愛國者。

The longer the system ran, the worse the time error became.

該系統運行更長的 更糟糕的時間誤差成了。

>> -After 100 hours of operation, the error in time was only about 1/3 of a second.

>> -After運行100小時，誤差 在時間只有一秒鐘的約1/3。

But in terms of targeting a missile traveling at mach 5,

但是，在針對一個方面 導彈5馬赫旅行，

it resulted in a tracking error of over 600 meters.

它導致了跟踪 對600多萬米的錯誤。

It would be a fatal error for the soldiers at Dhahran.

這將是一個致命的錯誤 對於戰士在宰赫蘭。

>> -What happened, is a Scud launch was detected by early-warning satellites,

>> - 什麼事，是飛毛腿發射是 通過預警衛星探測，

and they new that the Scud was coming in their general direction.

他們新的飛毛腿是 即將在他們的大方向。

They didn't know where it was coming.

他們不知道有人來了。

>> -It was now up to the radar component of the Patriot system,

>> 現在 - 它要由雷達 愛國者系統的組成部分，

defending Dhahran, to locate and keep track of the incoming enemy missile.

保衛達蘭，定位和保持 跟踪來襲的敵方導彈。

>> -The radar was very smart.

>> -The雷達是非常聰明的。

It would actually track the position of the Scud

它實際上跟踪 飛毛腿的位置