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