A clever way to estimate enormous numbers - Michael Mitchell

翻译人员: Minji Seo 校对人员: Yolanda Zhang

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

不管你喜不喜欢， 我们每天都会用到数字。

Some numbers, such as the speed of sound,

有些数字比如音速，

are small and easy to work with.

数值不大，容易计算。

Other numbers, such as the speed of light,

另一些数字，比如光速，

are much larger and cumbersome to work with.

就要大得多，不方便计算。

We can use scientific notation to express these large numbers

我们能用科学计数法来表示它们，

in a much more manageable format.

这样的格式更容易进行操作。

So we can write 299,792,458 meters per second

那么我们可以就把 每秒 299,792,458 米，

as 3.0 times 10 to the eighth meters per second.

写成每秒 3.0 乘 10 的 8 次方米。

Correct scientific notation

把第一项数值

按照科学计数法改写后，

requires that the first term range in value

应该比1大但是比10小，

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

而用来与第一项相乘的第二项的数值

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

by which we multiply the first term.

应该为10的次方数，或者叫数量级。

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

运用10的次方就能迅速估算出

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

我们只需了解其大约数值的数字。

For example, the diameter of an atom

举例来说，原子的直径大约是

is approximately 10 to the power of negative 12 meters.

10的负12次方米。

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

树的高度大约是10的1次方米。

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

而地球的直径大约是10的7次方米。

把10的次方数当作估算工具

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

can come in handy every now and again,

有时能方便我们进行估算，

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

例如，猜广口罐里有几颗M&M豆，

而这也是解决数学和科学问题的 必要技巧，

but is also an essential skill in math and science,

尤其在处理所谓的“费米问题”的时候。

especially when dealing with what are known as Fermi problems.

“费米问题”以物理学家 恩里科费米的名字命名，

Fermi problems are named after the physicist Enrico Fermi,

他因为能利用一些看似极少的数据，

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

迅速估算数量级和数字而闻名于世。

or rapid estimations, with seemingly little available data.

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

费米在曼哈顿计划中 指导制造原子弹，

and when it was tested at the Trinity site in 1945,

1954年，进行三位一体试验时，

Fermi dropped a few pieces of paper during the blast

费米在核爆途中扔下一些纸张，

利用纸张往后落下的距离

and used the distance they traveled backwards as they fell

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

估算爆炸的威力，结论是 相当于一万吨的TNT，

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

跟实际值的两万吨在同一个数量级。

One example of the classic Fermi estimation problems

举一个关于费米问题的经典例子：

is to determine how many piano tuners there are

估算在伊利诺伊州的芝加哥

in the city of Chicago, Illinois.

有多少钢琴调音师。

At first, there seem to be so many unknowns

乍一看存在太多未知的信息，

that the problem appears to be unsolvable.

这个问题个根本无法回答。

这是运用10的次方数极好的例子，

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

因为我们并不需要知道确切的数字——

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.

我们可以从估算芝加哥的人口开始。

We know that it is a large city,

我们都知道芝加哥是一个很大的城市，

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

但并不知道确切的人口数。

Are the one million people? Five million people?

一百万人吗？还是五百万人？

This is the point in the problem

问题的重点在于

很多人对这种不确定性感到棘手，

where many people become frustrated with the uncertainty,

而我们可以通过运用 10的次方数轻易做到。

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

我们估计芝加哥城人口大约是

We can estimate the magnitude of the population of Chicago

as 10 to the power of six.

10的6次方。

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

即使我们不知道确切的人数，

it serves an accurate estimation for the actual population

但还是能了解其实际人数

of just under three million people.

应该不会超过三百万。

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

如果芝加哥人口约有10的6次方，

how many pianos are there?

那会有多少钢琴呢？

If we want to continue dealing with orders of magnitude,

要是我们还想用数量级来处理， 就可以估测，

we can either say that one out of 10

每10人或每100人

or one out of one hundred people own a piano.

就有一人拥有钢琴。

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

先前的人口估算包括大人和小孩，

现在我们只算小孩的部分。

we'll go with the latter estimate,

那么芝加哥的钢琴数约有10的4次方，

which estimates that there are approximately 10 to the fourth,

差不多相当于1万。

or 10,000 pianos, in Chicago.

有这么多架钢琴， 那调音师到底有几位呢？

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

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

可以从一架钢琴多久调一次音，

一天调几架钢琴，

how many pianos are tuned in one day,

or how many days a piano tuner works,

调音师工作几天等等开始着手，

but that's not the point of rapid estimation.

但这不是快速预估的重点。

We instead think in orders of magnitude,

我们在这里用数量级估算，

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

一位调音师一年中， 大约要为10的2次方架钢琴调音，

which is approximately a few hundred pianos.

也就是差不多几百架钢琴。

Given our previous estimate of 10 to the fourth pianos in Chicago,

先前估计出芝加哥的钢琴 约有10的4次方架，

and the estimate that each piano tuner can tune 10 to the second pianos each year,

又估算了每位调音师 一年可以替10的2次方架钢琴调音，

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

现在我们就可以说，芝加哥的 调音师人数约有10的2次方这么多。

Now, I know what you must be thinking:

现在你一定在想：

How can all of these estimates produce a reasonable answer?

为什么这些预估都能算出合理的数字？

Well, it's rather simple.

答案再简单不过。

In any Fermi problem, it is assumed

每个费米问题都会假设

that the overestimates and underestimates balance each other out,

高估和低估会彼此平衡，

而其估计误差

and produce an estimation

通常只与其实际数值相差一个数量级。

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

我们也可以用黄页来确认这个例子中

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

芝加哥到底有几位调音师，

for the number of piano tuners listed in Chicago.

有几位呢？ 答案：81。

What do we find? 81.

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

数量级的估算方法很不可思议吧。

But, hey - that's the power of 10.

看，这就是10的力量。