A clever way to estimate enormous numbers - Michael Mitchell

譯者: Jesse Chen 陳鉦翰 審譯者: 盧 曉天

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

儘管你不喜歡 每天還是得用到數字

Some numbers, such as the speed of sound,

一些小而容易掌握的數字 如音速

are small and easy to work with.

Other numbers, such as the speed of light,

另一些大又麻煩的數字 如光速

are much larger and cumbersome to work with.

We can use scientific notation to express these large numbers

我們能用科學記號 使這些大數字更容易辨識

in a much more manageable format.

So we can write 299,792,458 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

as 3.0 times 10 to the eighth meters per second.

Correct scientific notation

把第一項數值改成科學記號的規則 比1大但比10小

requires that the first term range in value

so that it is greater than one but less than 10,

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

and the second term represents the power of 10 or order of magnitude

by which we multiply the first term.

運用十的次方能迅速估算出 我們只需了解其大約數值的數字

We can use the power of 10 as a tool in making quick estimations

when we do not need or care for the exact value of a number.

For example, the diameter of an atom

舉例來說 原子的直徑約10的負12次方公尺

is approximately 10 to the power of negative 12 meters.

The height of a tree is approximately 10 to the power of one meter.

樹的高度約10的1次方公尺

The diameter of the Earth is approximately 10 to the power of seven meters.

而地球的直徑約10的7次方公尺

把十次方數當作估算工具 有時能輕易估計數字

The ability to use the power of 10 as an estimation tool

can come in handy every now and again,

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's

而這也是數學和科學的必要技巧 尤其處理「西瓜式的費米問題」

but is also an essential skill in math and science,

especially when dealing with what are known as Fermi problems.

「費米問題」以物理學家 恩里科．費米 的名字命名 且他因能利用一些看似極少的數據

Fermi problems are named after the physicist Enrico Fermi,

who's famous for making rapid order-of-magnitude estimations,

迅速估算數量級、數字而聞名

or rapid estimations, with seemingly little available data.

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,

1954年，進行三位一體核試時 費米在核爆途中扔下一些紙張

Fermi dropped a few pieces of paper during the blast

利用紙張往後落下的距離 測量爆炸的威力

and used the distance they traveled backwards as they fell

to estimate the strength of the explosion as 10 kilotons of TNT,

約一萬噸的黃色炸藥 相當於兩萬噸的數量級

which is on the same order of magnitude as the actual value of 20 kilotons.

One example of the classic Fermi estimation problems

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

is to determine how many piano tuners there are

in the city of Chicago, Illinois.

At first, there seem to be so many unknowns

剛開始 會出現很多看似無法解決的問題

that the problem appears to be unsolvable.

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

That is the perfect application for a power-of-10 estimation,

as we don't need an exact answer -

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,

芝加哥是一個很大的城市 我們不太會知道確切的人口數

but we may be unsure about exactly how many people live in the city.

Are the one million people? Five million people?

一百萬人嗎？還是五百萬人？

This is the point in the problem

問題的重點在於 很多人對無法預估數字感到苦惱

where many people become frustrated with the uncertainty,

而我們可以藉由運用十的次方輕易做到

but we can easily get through this by using the power of 10.

估計芝加哥城人口約是10的6次方

We can estimate the magnitude of the population of Chicago

as 10 to the power of six.

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

還是能了解其實際人數不會超過三百萬人

of just under three million people.

So if there are approximately 10 to the sixth people in Chicago,

如果芝加哥城人口約有10的6次方 那鋼琴呢？

how many pianos are there?

If we want to continue dealing with orders of magnitude,

要是我們還是想用數量級來處理 就可以估測

we can either say that one out of 10

每十人或每百人就有一人擁有鋼琴

or one out of one hundred people own a piano.

Given that our estimate of the population includes children and adults,

先前的人口預算包括大人和小孩 現在只算小孩的部分

we'll go with the latter estimate,

芝加哥的鋼琴數約有10的4次方，即約一萬

which estimates that there are approximately 10 to the fourth,

or 10,000 pianos, in Chicago.

有這麼多部鋼琴 那調音師到底有幾位？

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

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

一天調幾部鋼琴 調音師工作幾天

how many pianos are tuned in one day,

or how many days a piano tuner works,

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,

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

and say that a piano tuner tunes roughly 10 to the second pianos in a given year,

which is approximately a few hundred pianos.

約一百部鋼琴

Given our previous estimate of 10 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 10 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

現在我們就可以說 芝加哥城的調音師人數約有10的2次方

we can say that there are approximately 10 to the second piano tuners in Chicago.

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.

再簡單不過，每個費米問題 都假想高估和低估會令其平衡

In any Fermi problem, it is assumed

that the overestimates and underestimates balance each other out,

而其估計誤差 通常只與其實際數值相差一個數量級

and produce an estimation

that is usually within one order of magnitude of the actual answer.

我們用黃頁來確認這個例子 芝加哥到底有幾位調音師

In our case we can confirm this by looking in the phone book

for the number of piano tuners listed in Chicago.

有幾位呢？ 答案：81。

What do we find? 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 10.

看，這就是十的力量 -Translated by Jesse Chen 陳鉦翰