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  • 2640 lumens. 1 foot. 2.3 kilograms. 9 volts. Aaah!


  • 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?


  • 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.


  • 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.


  • Someone just decided one day that that was a kilogram.


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


  • but I can talk about them all day.


  • 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.


  • 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.


  • No special capitalization necessary.


  • 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."


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


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


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


  • 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.


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


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


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


  • 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.


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


  • 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.


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


  • as we learned when the Mars Climate Orbiter crashed into Mars


  • 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,


  • 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?


  • 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.


  • 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.


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


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


  • 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.


  • That fraction once we have it can flip either way.


  • 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.


  • 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.


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


  • 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.


  • 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.


  • It would at very least be very nerdy.


  • 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...


  • 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?


  • 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?


  • 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.


  • 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.


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


  • 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.


  • First, they tell you the number that was measured.


  • 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.


  • 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:


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


  • With my speedometer there are two: 6 and 0.


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


  • 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.


  • 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.


  • 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,


  • 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.


  • 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


  • 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.


  • 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.


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


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


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


  • 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