I just closed the circuit with my tongue and I felt all 9 of the volts.

1呎。 2.3公斤。 9伏特。

So what do all these things have in common?

(啊!!!!)我感受到九伏特電流在我舌頭所形成的封閉迴路。所以這些東西

They're units. Yes, but they're also absolutely, completely arbitrary.

有什麼相同的地方嗎? 它們都是"單位"!

[Theme Music]

沒錯。 但是它們是可以徹底互相變換的

You know who decides how much a kilogram weighs?

[開場音樂] [燈燈等等~燈燈等等等等登~燈燈等等等登]

A hunk of platinum and iridium known as the International Prototype Kilogram or IPK.

你知道誰決定一公斤是多少嗎? 是塊叫做"國際公斤原器"(IPK)

The IPK isn't just how much a kilogram weighs. In a very real sense the IPK is the kilogram.

的鉑銥合金所決定的

Every other kilogram is exactly the same as the IPK,

IPK不只是表達一公斤的重量。 正確來說，它就是"一公斤"的標準。

and the IPK is the lump of metal that decides what that mass is.

其他所謂的"一公斤"都是完全跟IPK一樣， 而IPK就是一塊決定

A kilogram is defined as being the same mass as the IPK.

質量的金屬。 "一公斤"就是定義與IPK同重的質量。

We made kilograms up just like we made up seconds and weeks and volts and newtons.

我們制定了"公斤"， 就像我們制定"秒"、"週"、"伏特"、"牛頓"。

There's nothing about these things that makes them them.

沒有任何可以用來形容這些東西的東西

Someone just decided one day that that was a kilogram.

不過就是有人某天突發奇想決定:"啊!這就叫一公斤"

Now the fact that I find units fascinating probably says more about me then it does about units,

雖然說我知道我比這些無趣的單位迷人

but I can talk about them all day.

但這些單位還是能讓我說整天說不完

For example, did you know that the International System of Units only includes seven base units

舉例來說， 你知道"國際單位制"(SI制)只有七個標準單位嗎?

and every other unit is derived from those units?

而其他的單位都是由這七個單位所組成的嗎? "速度"是距離除以時間，"加速度"

Speed is length divided by time.

是時間除以秒平方，所以單位是每平方秒多少公尺

Acceleration is speed divided by time again, so meters per second per second.

"力"是加速度乘以質量，因為F=ma，記得嗎? 而"功"的單位"焦耳"

Force is that acceleration multiplied by mass, cause F=ma remember?

是力乘上距離。 "功率"則是功除以時間，

Work done in joules is force multiplied by distance.

也就是單位時間內所做的功。 合情合理!(攤手)

And power is work divided by time, so how much work can be done per unit of time. Makes sense.

看起來很艱深， 而且我可以說實際上有無限多種延伸出來的單位

It goes pretty deep, and it's absolutely correct to say that there are an infinite number of possible derived units,

只是大部分的沒啥路用就是了， 也沒那個必要幫它們命名

just most of them aren't useful enough to name.

豆知識補充! 當我說"瓦特"或"赫茲"(這些單位)時，它們就只是兩個單字

But here's a bit of trivia for you. When I say watts or hertz, those things are just regular words.

英文字首不用大寫，但"赫茲"跟"瓦特"實際上也是出現在歷史上的兩個偉人的姓氏

No special capitalization necessary.

所以字首大寫時就是再說這兩位大大了。

But Hertz and Watt, they were real people with like last names that were capitalized.

這是怎麼回事?OAO 呃...當有單位以你的姓氏命名時，你就知道你已經成為科學界的巨擘了!

So what's up with that? Well, getting a unit named after you is kind of the holy grail of science.

如同Richard Hamming所說:"真正的偉大就是當你的姓跟赫茲還有瓦特一樣-------

To quote Richard Hamming:

姓氏字首被小寫化了。" 當然這些天才們是最先嘗試了解這世界如何運轉的，

"True greatness is when your name - like hertz and watt - is spelled with a lowercase letter."

但是他們並不知道有"基礎運作單位"(或是SI制)的概念存在這世上

Of course when these geniuses were first piecing together how the world works

當時的單位是用那時現有的其他數值來轉換，畢竟，它們哪知道標準重量是多重?

they had no idea that there were fundamental basic units beneath it all.

標準長度又有多長?

They were basing all of their units on arbitrary values because, well,

有趣地是，的確是有一個標準單位的起源是從實際數值轉變而來，儘管不是國際通用

how could there possibly be a fundamental amount of mass or distance.

譬如: "地球自轉一圈所花的時間的86400分之一就是'一秒'。"

Interestingly, one of the standard base units is derived from an actual value though not a universal one.

至少是有提出個所以然，但也點出了另外一個有趣的點

The second is 1/60th of 1/60th of 1/24th of the time it takes for the Earth to rotate a single time.

儘管看起來有依據，但是當你仔細探究，

That's something, at least but it also illustrates an interesting point.

會發現有許多瑕疵存在這制定的方式裡面。

As fundamental as that seems, when you get down to the dirty details things start to get kind of cloudy.

地球自轉的速度假如越來越慢，難道一秒鐘也要跟著變短嗎?

The Earth's rotation for example is slowing down.

謀摳林!!!!! 這樣會讓一切的計算的亂掉，所以，以現實狀況來制定秒的方式就漸漸地

Does that mean that seconds should also slow down?

被淘汰了! 但是別擔心，假如要察覺到地球真的越轉越慢的話，

No. That would mess up every calculation ever.

也是要等到海枯石爛才有可能，就算等到了，在加上幾秒去回到原本的平衡就行啦!

So seconds are slowly becoming less and less based on reality.

可是，單位在化學和科學裡是佔有舉足輕重的地位DER

Now don't worry. It's gonna take forever for the Earth to slow down noticeably.

假如單位混淆， 環繞火星的人工衛星可是會正面KISS火星啊!!!!!

And when it does we'll just keep adding leap seconds to keep things balanced.

當你下次拿到B而不是A時，別急著回家哭爸，誰叫你就是不鳥單位不單位的。

But units are extremely important in chemistry and in sciences in general,

至少你不是毀掉一個300萬美元的火星計畫

as we learned when the Mars Climate Orbiter crashed into Mars

什麼叫在意這些單位哩?

because instructions were inputted in the wrong units.

就是叫你注意看好它們! 別讓它們做任何你沒叫它們做的事，因為它們

Next time you get a B instead of an A because you didn't keep track of your units,

超狡猾(腳滑)的

just remember at least you didn't destroy a 300 million dollar mission to Mars.

而大部分的化學就只是這些單位們變來變去而已。 假設你現在在一輛車裡，

But what do I mean when I say keep track of your units?

時速六十英哩，而現在一群不住在美國的人在你後面，他們非常的火

Well. I mean watch them. Do not let them do anything you didn't tell them to do because they're sneaky.

"英哩是什麼啦?能吃嗎? 換成時速幾公里不是好多了嗎?"

And a lot of chemistry is just converting between units.

好啦幫你平反一下， 其實對科學來說公里也是糟糕透頂的單位

So say you are in a car, and the car is going 60 miles per hour.

他們都是相對地而已，我們應該找個更全球化的單位.....譬如: 光 年

Now right now everyone who doesn't live in America is like:

指的是光一年所可以走的距離， 而說到"小時"，小時根本超無聊Der

"Boo, miles are terrible. Convert to kilometers Hank!"

所以我們來把時速60英哩轉換成 光年/秒 當你說時速60英哩，聽起來像只有一個單位

Well I'll do you one better. From a scientific perspective, kilometers are terrible too.

但事實它是個分數，"每小時"60"英哩"

They're just as arbitrary. We should use something more universal.

我們先從甘丹的開始! 小時轉換成秒

Like lightyears. The amount of distance light can travel in a year. And hours, hours is no fun.

所以首先我們換成分鐘，每小時就是60分鐘(60mins/hr)，每60分鐘就是一小時(1hr/60mins)

So let's convert to lightyears per second. 60 miles per hour.

這個分數的型態可以上下隨意換，但是我們希望

When you say it it sounds like a whole number with a single unit.

"小時"的單位能放在分子，為什麼?

But it's not. It's actually a fraction. 60 miles over 1 hour.

因為我們想要單位相消掉。 我們要消滅小時，我們不要它在計算完後

Let's start with the easy part. Getting to the seconds.

還出現在等號後。 然後用相同的方式換算成秒

So first we've got to get to minutes. So there's 60 minutes per hour. And also 1 hour per 60 minutes.

現在來處理光年的轉換

That fraction once we have it can flip either way.

我問估狗大神，它說一光年是5.9乘以10的12次方英哩，

We want it with the hours on the top, on the numerator. Why?

然後嗒噠~ 小時消掉、分鐘消掉、英哩消掉 剩下我們要的光年/秒囉!

Because we want the units to cancel. We want to destroy the hours.

重點來了，如此一來

We don't want them in our units when we're done.

我們的單位就正確囉 接下來經過精密的計算就可以知道

And then the same thing happens again with 1 minute per 60 seconds. Now we go to lightyears.

原來時速60英哩就等同於 每秒走9.3乘以10的-12次方光年

I asked Google, and there's 1 light-year in every 5.9 * 10^12 miles.

我們剛表演了完了一題重要的考題 但"這一切到底喜低衝蝦毀,有意義嗎?"

Looking at this we see that the hours cancel and the minutes cancel and the miles cancel.

有滴!! 因為9.3乘上10的-12次方是個非常非常非常小的數字，

Leaving us with lightyears per second. That's really what matters.

所以當你駕駛這台車時， 你每秒只不過是以宇宙超級霹靂無敵世界小

We've come out with the correct units.

的光年比例來前進

The rest is just hammering at the calculator to discover that a car going 60 mph is also going

現在差不多有50到100000人在看這部影片

9.3 * 10^-12 lightyears per second.

我在猜大概頂多七個人跟我一起算剛才那題， 而且不用計算機

Now we perform an important test. The "does this make sense?" test.

不是硬要給你難堪啦，這只是我的猜測啦!

And yes indeed it does because 9.3 * 10^-12 is a very, very, very, very small number.

假如你之後想用計算機來算我的問題也是可以， 應該會蠻有幫助的，

Which makes sense because when you're traveling in a car you're going

至少不會像個人體計算機宅宅。 但是當你用計算機時，你會發現一件有趣的事

a very, very, very, very, very, very, very tiny fraction of a light-year every second.

當我說9.3乘上10的-12次方時， 你的計算可能顯示是

Now there are probably gonna be fifty to a hundred thousand people that watch this video.

9.3487658140029乘以10的-12次方之類的。 為什麼是安捏哩?

And I'm gonna guess that maybe a solid seven of you did the math along with me with your calculator out.

這麼多數字，為什麼我只說兩個? 難道我只是要省時間嗎?

Now I'm not giving you a hard time. That's just my guess.

不是這樣der，只是我會多花時間講這些而已(靠北=皿=) 你以為我記憶這些數字會有困難嗎?

If you want to follow along with your calculator in the future that might be helpful.

當然不，因為我剛才就表演給你看過了啊!(跩屁XD)

It would at very least be very nerdy.

好啦秀秀，我跟你講到底為什麼要這樣做?

But if you have been following along with your calculator, you might maybe have noticed something interesting.

當你做出個實驗性的計算數據時，有兩種數值 一種是"測量值"，一種是"估計值"。

I said 9.3 * 10^-12. When your calculator...

測量值就好比一分鐘內有幾秒，一打雞蛋有多少個

Your calculator probably said something like 9.3487658140029 * 10^-12.

它們是可以被定義(辨識)的， 因此我們精確地知道它的數值直到小數點後無限位數。

So why, when I had so many more numbers to give, did I only give two? Was I trying to save time?

當我說有一打雞蛋時，你就知道是12顆

Well obviously not, because now I appear to be wasting time talking about it.

而不是12.0000000001或是11.9999999。 就是12。

Do you think that it would be too hard for me to remember all those numbers?

但那並不是我的汽車真正行駛的速率，

Well obviously not, because I just did it. So I will tell you why.

它並不是以時速60英哩"整"的時速前進的。 我只知道我的車速介於兩個整數間

When you're doing experimental calculations, there's two kinds of numbers. There's exact and measured.

因為那就是我從儀表板上所能讀到的資訊。 所以汽車可能是以時速59.87390039英哩

Exact numbers are like the number of seconds in a minute or the number of eggs in a dozen.

或是時速60.49321289英哩。 但是不管是哪一個儀錶板還是顯示60

They're defined that way and thus we know them in effect all the way out to an infinite number of decimal places.

儘管我多會測量車速， 我無法像知道一打雞蛋十二個那樣確切地

If I say that there are a dozen eggs you know that that's 12. It's not 12.0000000001

知道現在到底確切的車速是多少， 所以第二種數值這就出現了

or 11.9999999. It's 12.

"估計值" 關於估計值很酷的是，你永遠猜不到

But that's not true for the number of miles per hour my car was going.

它竟然一次告訴你兩個訊息!

That car wasn't going 60.0000-out into infinity mph.

第一，它告訴你已經測量的數據是多少。 第二，它告訴你這個值是測量到多精準。

I only know the speed of my car to two decimal places because that's all I get from the speedometer.

通常聽到這個一堆人腦袋已經打結了

So the car could have been going 59.87390039 mph or 60.49321289 mph; the speedometer would still say 60.

但是有估計值你只要知道後面還有無限的數字

And no matter how well I measure the car's speed,

一直到無窮盡的位數，只是你不知道到底是多少。 你無法，是癡人說夢!

I will never know it at the same level of precision that I know the number of eggs in a dozen.

當我的秤說我是175磅時，並不代表我重175.000000磅 只能說我是175.多少磅

So that's the second type of number, measured numbers.

阿那些在5後面的數字哩? 我們全部朧母災! 所以是這樣的，

Now the cool thing about measured numbers,

當你不知道測量物的精準度到哪時， 估計值一點屁用都沒有

because you never ever know them exactly, is that they tell you two things at once.

因此，你最好把精準度留下，要不然你可能會

First, they tell you the number that was measured.

因為搞不清楚胰島素或其他藥物的精確劑量而誤殺了誰

And second, they tell you the precision at which that number was measured.

我們因為這樣就設下了規則，就是所謂的 "有效數字"

People often get their heads all tangled up about this,

這些數字就是你實際上知道的數字 我的儀錶板上有兩個有效數字，6跟0

but with a measured number you just have to remember that the actual number goes out to infinite decimal places,

但是0很怪，因為有時只是個佔位的符號 比如說:

you just never know all of them. You can't. It's impossible,.

最快的飛機可以達到時速13,000英哩， 順帶一提，它真的可以!

So when my scale says 175 lbs, that doesn't mean 175.000000 lbs. It means 175.something lbs.

這是一台無人滑翔機在2011的測量。 但這並非精確的數字，所以那些0只是佔位的符號

And all those numbers after the five? We don't know them.

所以當有個數字結尾是一個零，或兩到三個零，

And here's the thing, a measured number can be pretty unhelpful if you don't have knowledge

我們很難去界定說這些零是不是有效的。 但是當用了科學符號後，

of the precision of the measurement.

這一切都變得簡單多了，既然是科學人，當然要用科學符號囉! 所以時速60英哩應該寫成

So you have to conserve the precision through your calculations

6.0x10 我們把0視為有效數字因為是我們寫的

or else you might end up killing someone with an imprecise dose of insulin or something.

要不然直接寫6x10就好了啊! 我們保留那個零因為我們真的確切知道

So we have a set of rules for what are called significant figures:

當你懂了科學記號你就會覺得它超ㄅㄧㄤˋ的

these are the digits in your number that you actually know.

當你有問題時，找估狗或計算機你就明白

With my speedometer there are two: 6 and 0.

到底是在講哪個數字，但是指數的數字會指引你

But 0 is weird, because sometimes it's just used as a placeholder.

到底要怎麼移動小數點

Like if I said that the fastest plane can go 13,000 mph, which it can by the way.

所以乘以10時，你就把小數點往右移一格，就得到60了 如果乘10的-1次方就往左移

An unmanned military test glider did it in 2011.

就得到0.60 乘以十的五次方哩?

That's not an exact number, those zeroes are just placeholders.

1,2,3,4,5 600,000就跑出來啦 當然囉!你的有效數字要記得保留

So when a number ends in a zero, or two or three zeroes, it's hard to tell if those zeroes are significant.

所以2.4590乘以10的-4次方就是 0.00024590而你的有效數字

But this all gets so much simpler when you use scientific notation, which since it's science we should.

依然是一樣的五個

So 60 mph would instead be 6.0 * 10^1. We get that zero is significant because we wrote it.

懂了你的答案該有幾個有效數字後 要記得兩個原則

Otherwise it would just be 6 * 10^1. We keep that zero around because we actually know it.

當在做加減法計算，只要考慮小數點後面的數字有幾個

Scientific notation is awesome by the way, once you get the hang of it.

兩個數值中， 小數點後擁有最少數字的就是答案應有的有效數字

If you're having trouble you can always just type it into Google or your calculator to

舉例來說 1495.2+1.9903

see exactly what number we're talking about,

經過精密的計算

but the number of the exponent just tells you how many places to move the decimal point.

你得到1497.1903然後你只保留到小數點後一位 因為1495.2的小數點後只有一位

So to the 1st power you move it one to the right and you get 60.

所以答案是1497.2 至於乘法就只要把有效數字控制在

To the negative 1st power you move the decimal point one place to the left and you get 0.60.

與你所有測量值中擁有最少的有效數字相同即可

To the fifth power, one, two, three, four, five, and you get six with five zeroes or 600,000.

所以60x5.0839=305.304，因為60只有兩個有效數字 所以一切在前兩位數字以後的只能以0填滿

Of course your significant figures get preserved, so 2.4590 * 10^-4 is 0.00024590 and you still

答案就是300囉，可是我們只要顯示出有效數字即可

get the same five sig figs.

因此第二個零是有效的， 第三個零以後的就用科學符號寫出來

Now to the magic of figuring out how many sig figs your answer should have.

即為3.0x10^2

There are two simple rules for this.

因為科學的緣故! 我知道不把你擁有的所有數字寫上去好像很不合常理

If it's addition or subtraction it's only the number of figures after the decimal point that matters.

但是你要知道: 那些數字在有效數字後面的數字是什麼?

The number with the fewest figures after the decimal point

它們都跟珠利輪一樣，謊話連篇! 它們都是騙人數字，你根本就不知道那些數字

decides how many figures you can have after the decimal in your answer.

假如你寫下來的話，別人就會信以為真 就好像新北市民看三環三線的剪綵一樣

So 1,495.2+1.9903 you do the math.

你知道化學界怎麼對待說謊的人嗎? 我們殺了他 或是用選票送他回去當市長

First you get 1,497.1903 and then you round to the first decimal,

謝謝收看這集的Crash Course化學單元 今天你學到了一些

because that first number only had one figure after the decimal. So you get 1,497.2.

如何理解化學中的數學的訣竅，希望你記得這集以免

And for multiplication just make sure the answer has the same sig figs as your least precise measurement.

以後為這些東西煩惱

So 60 x 5.0839 = 305.034, but we only know two sig figs,

如何轉換單位是個就算不是化學也會需要用到的技巧

so everything after those first two numbers is zeroes: 300.

科學符號會讓你看起來像是知道自己在說啥

Of course then we'd have to point out to everyone that the second zero but not the third is significant,

當個懲罰那些用錯有效數字的人， 看到錯誤就ㄌㄧㄚˇㄍㄨㄥˊ

so we'd write it out with scientific notation: 3.0 * 10^2. Because science!

像個KMTer看到689 2.0就要帶風向一樣

Now I know it feels counterintuitive not to show all of the numbers that you have at your fingertips,

所以好好享受我賦予你的新能力吧 我們下次見:)

but you've got to realize: all of those numbers beyond the number of sig figs you have? They're lies.

(亂打中，若有雷同存屬巧合) Crash Course 是由Nick Jenkins所編輯和指導的

They're big lying numbers. You don't know those numbers.

劇本我用麥可筆是寫在黨工們的褲子上的 Michael Aranda是聲音後吱 動畫團隊是Thought Bubble

And if you write them down people will assume that you do know those numbers.

如果你有問題、意見或點子給我們，我們都在五樓下面(誤

And you will have lied to them. And do you know what we do with liars in chemistry? We kill them!

謝謝收看Crash Course化學單元 (翻譯: Charlie Cheng)

Thank you for watching this episode of Crash Course Chemistry.

Today you learned some keys to understanding the mathematics of chemistry,

and you want to remember this episode in case you get caught up later down the road:

How to convert between units is a skill that you'll use even when you're not doing chemistry.

Scientific notation will always make you look like you know what you're talking about.

Being able to chastise people for using the wrong number of significant digits is basically

math's equivalent of being a grammar Nazi.

So enjoy these new powers I have bestowed upon you, and we'll see you next time.

Crash Course Chemistry was filmed, edited, and directed by Nick Jenkins.

This episode was written by me, Michael Aranda is our sound designer, and our graphics team is Thought Bubble.

If you have any questions, comments or ideas for us, we are always down in the comments.

Thank you for watching Crash Course Chemistry.