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• Whether you like it or not, we use numbers every day.

無論你喜不喜歡，你每天還是得用到數字

• Some numbers, such as the speed of sound, are small and easy to work with.

一些小而容易掌握的數字，如音速，很容易掌握

• Other numbers, such as the speed of light, are much larger and cumbersome to work with.

另一些大又麻煩的數字，如光速，就比較難掌握

• We can use scientific notation to express these larger numbers in a much more manageable format.

我們能用科學記號使這些大數字更容易辨識

• So we can write 299,792,458 meters per second as 3.0 times ten to the eighth meters per second.

所以可以把每秒 299,792,458 公尺 寫成每秒 3.0 乘 10 的 8 次方公尺

• Correct scientific notation requires that the first term range in value set as greater than one but less than ten,

第一項數值改成科學記號的規則是它要比 1 大但比 10 小

• and the second term represents the power of ten, or order of magnitude, by which we multiply the first term

而用來乘上第一項的第二項數值為 10 的次方數或稱數量級

• We can use the power of ten as a tool in making quick estimations when we do not need or care for the exact value of a number.

運用十的次方就能迅速估算出我們只需了解其大約數值的數字

• For example, the diameter of an atom is approximately ten to the power of negative twelve meters.

舉例來說，原子的直徑約為 10 的負 12 次方公尺

• The height of a tree is approximately ten to the power of one meters.

樹的高度約 10 的 1 次方公尺

• And the diameter of the Earth is approximately ten to the power of seven meters.

而地球的直徑約 10 的 7 次方公尺

• The ability to use the power of ten as an estimation tool can come in handy every now and again,

把十次方數當作估算工具有時能輕易估計數字

• like when you're trying to guess the number of M&M's in a jar.

例如猜廣口罐裡有幾顆 M&M 的時候

• But is also an essential skill in math and science, especially when dealing with what are known as Fermi problems.

而這也是數學和科學的必要技巧，尤其處理當你在「費米問題」

• Fermi problems are named after the physicist Enrico Fermi, who's famous for making rapid order-of-magnitude estimations,

「費米問題」以物理學家恩里科．費米的名字命名。因能利用一些看似極少的數據

• or rapid estimations, with seemingly little available data.

迅速估算數字的數量級而聞名

• Fermi worked on the Manhattan Project in developing the atomic bomb,

費米在曼哈頓計畫中指導製造原子彈

• and when it was tested at the Trinity site in 1945, Fermi dropped a few pieces of paper during the blast

1954年，進行三位一體核試時，費米在核爆途中扔下一些紙張

• and used the distance they travelled backwards as they fell to estimate the strength of the explosion

利用紙張往後落下的距離測量爆炸的威力

• as 10 kilotons of TNT, which is on the same order of magnitude as the actual value of 20 kilotons.

約一萬噸的黃色炸藥，與實際數字 20 萬頓的數量級一樣

• One example of the classic Fermi estimation problems is to determine how many piano tuners there are in the city of Chicago, Illinois.

舉一個經典的費米問題： 估算在伊州芝加哥城有多少鋼琴調音師

• At first, there seem to be so many unknowns that the problem appears to be unsolvable.

剛開始會出現很多看似無法解決的問題

• That is the perfect application for a power-of-ten estimation, as we don't need an exact answer.

這是運用十的次方極好的例子，因為我們不需要知道確切的數字

• An estimation will work.

只要估算即可

• We can start by determining how many people live in the city of Chicago.

我們可以從估算芝加哥城的人數開始

• We know that it is a large city, we may be unsure about exactly how many people live in the city.

芝加哥是一個很大的城市，我們不太會知道確切的人口數

• Are there one million people? Five million people?

一百萬人嗎？還是五百萬人？

• This is the point in the problem where many people become frustrated with the uncertainty,

問題的重點在於很多人對無法預估數字感到苦惱

• but we can easily get through this by using the power of ten.

而我們可以藉由運用十的次方輕易做到

• We can estimate the magnitude of the population of Chicago as ten to the power of six.

估計芝加哥城人口約是 10 的 6 次方

• While this doesn't tell us exactly how many people live there,

即使我們不知道確切的人數

• it serves an accurate estimation for the actual population of just under three million people.

這是一個夠準確的方式來估計實際上不到三百萬的人口

• So, if there are approximately ten to the sixth people in Chicago, how many pianos are there?

如果芝加哥城人口約有 10 的 6 次方，那鋼琴呢？

• If we want to continue dealing with orders of magnitude we can either say that

要是我們還是想用數量級來處理，就可以估測

• one out of ten or one out of one hundred people own a piano.

每十人或每百人就有一人擁有鋼琴

• Given that our estimate of the population includes children and adults, we'll go with the latter estimate,

由於人口包括大人與小孩，我們就預估每一百人就有一個人擁有鋼琴

• which estimates that there are approximately ten to the fourth, or 10,000 pianos, in Chicago.

這就估計到芝加哥的鋼琴數約有 10 的 4 次方，即約一萬

• With this many pianos, how many piano tuners are there?

有這麼多部鋼琴，那調音師到底有幾位？

• We could begin the process of thinking about how often the pianos are tuned,

可以從一部鋼琴多久調一次音開始著手

• how many pianos are tuned every one day, or how many days a piano tuner works,

一天調幾部鋼琴，調音師工作幾天

• but that's not the point of rapid estimation.

但這不是快速預估的重點

• We instead think in orders of magnitude and say that a piano tuner tunes roughly ten to the second pianos in a given year,

應用數量級預估 一位調音師一年中，約替 10 的 2 次方部鋼琴調音

• which is approximately a few hundred pianos.

約一百部鋼琴

• Given our previous estimate of ten to the fourth pianos in Chicago,

先前預估了芝加哥城的鋼琴約有 10 的 4 次方部

• and the estimate that each piano tuner can tune ten to the second pianos each year,

又預估了每位調音師一年可以替 10 的 2 次方部鋼琴調音

• we can say that there are approximately ten to the second piano tuners in Chicago.

我們就可以說芝加哥城的調音師人數約有 10 的 2 次方

• Now, I know what you must be thinking:

你一定在想：

• How can all of these estimates produce a reasonable answer?

為什麼這些預估都能算出合理的數字？

• Well, it's rather simple: In any Fermi problem, it is assumed that the overestimates and underestimates balance each other out

再簡單不過，每個費米問題都假想高估和低估會互相抵銷

• and produce an estimation that is usually within one order of magnitude of the actual answer.

而其估計誤差通常只與其實際數值相差一個數量級

• In our case we can confirm this by looking in the phone book for the number of piano tuners listed in Chicago.

我們用黃頁來確認芝加哥到底有幾位調音師

• What do we find? 81.

有幾位呢？ 答案：81。

• Pretty incredible, given our order-of-magnitude estimation.

數量級的預估很不可思議吧

• But, hey, that's the power of ten.

看，這就是十的力量

Whether you like it or not, we use numbers every day.

【TED-Ed】用聰明的方法計算大數字 (A clever way to estimate enormous numbers - Michael Mitchell)

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