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一种估算巨大数字的聪明方法

A clever way to estimate enormous numbers - Michael Mitchell

无论你是否喜欢,我们每天都在使用数字
Whether you like it or not, we use numbers every day.
有些数字,比如音速
Some numbers, such as the speed of sound,
比较少容易操作
are small and easy to work with.
其他数字,比如光速
Other numbers, such as the speed of light,
大得多而且运算时非常麻烦
are much larger and cumbersome to work with.
我们可以使用科学计数法以比较简洁的方式
We can use scientific notation to express these large numbers
来表达这些大数字
in a much more manageable format.
因此我们可以将299792458米每秒
So we can write 299,792,458 meters per second
看成3乘以10的8次方米每秒
as 3.0 times 10 to the eighth meters per second.
正确的科学计数法
Correct scientific notation
要求第一项范围有效
requires that the first term range in value
所以正确的科学计数法第一项应该大于1,小于10
so that it is greater than one but less than 10,
并且第二项是10或者10的数量级
and the second term represents the power of 10 or order of magnitude
两项相乘就是就是正确的科学计数法
by which we multiply the first term.
我们可以用10的数量级来做快速的估量
We can use the power of 10 as a tool in making quick estimations
当我们不需要或者不在意准确的数值的时候
when we do not need or care for the exact value of a number.
比如,原子的直径
For example, the diameter of an atom
接近10的 -12 次方米
is approximately 10 to the power of negative 12 meters.
一棵树的高度接近10米
The height of a tree is approximately 10 to the power of one meter.
地球的直径大约10的7次方米
The diameter of the Earth is approximately 10 to the power of seven meters.
把10的数量级作为估量工具的能力
The ability to use the power of 10 as an estimation tool
有时是有用的
can come in handy every now and again,
就像你尝试猜测米的数量的时候
like when you’re trying to guess the number of M&M’s in a jar,
科学计数法在数学和科学方面也是一个必要的技能
but is also an essential skill in math and science,
尤其是在处理著名的费米问题的时候
especially when dealing with what are known as Fermi problems.
费米问题得名于物理学家费米恩里科
Fermi problems are named after the physicist Enrico Fermi,
他因做快速的数量级估算而闻名
who’s famous for making rapid order-of-magnitude estimations,
或者是快速的估计看来似乎很小的可获得的数据而闻名
or rapid estimations, with seemingly little available data.
费米在曼哈顿计划中从事研发原子弹的工作
Fermi worked on the Manhattan Project in developing the atomic bomb,
在1945年,研发的原子弹在三倍的位置上进行测试
and when it was tested at the Trinity site in 1945,
在爆炸时,费米扔下一些的纸张
Fermi dropped a few pieces of paper during the blast
利用它们向后掉落的距离
and used the distance they traveled backwards as they fell
估算10吨TNT爆照的威力
to estimate the strength of the explosion as 10 kilotons of TNT,
这相当于20吨的威力
which is on the same order of magnitude as the actual value of 20 kilotons.
一个关于费米问题的例子
One example of the classic Fermi estimation problems
如何确定在芝加哥
is to determine how many piano tuners there are
有多少钢琴调谐器
in the city of Chicago, Illinois.
首先,似乎有很多未知数
At first, there seem to be so many unknowns
这个问题看起来无法解决
that the problem appears to be unsolvable.
它是一个10的数量级的完美应用例子
That is the perfect application for a power-of-10 estimation,
当我们不需要一个精确的答案时
as we don’t need an exact answer –
估算就变得很有用
an estimation will work.
我们可以首先估计有多少人住在芝加哥
We can start by determining how many people live in the city of Chicago.
我们都知道这个城市很大
We know that it is a large city,
我们可能无法确定居住在这个城市举具体人数
but we may be unsure about exactly how many people live in the city.
是一百万人?五百万人?
Are the one million people? Five million people?
这是这个问题的一个要点
This is the point in the problem
许多人因为这个不确定性变得沮丧
where many people become frustrated with the uncertainty,
但是我们可以通过10的数量级轻易解决这个问题
but we can easily get through this by using the power of 10.
我们可以估算芝加哥人口的量级
We can estimate the magnitude of the population of Chicago
大致是10的6 次方
as 10 to the power of six.
这个数目并没有告诉我们芝加哥人口的具体数目
While this doesn’t tell us exactly how many people live there,
它是对于人口数目的准确估计
it serves an accurate estimation for the actual population
不到三百万人
of just under three million people.
所以,如果芝加哥大致有10的6次方个人
So if there are approximately 10 to the sixth people in Chicago,
那么这里有多少钢琴呢
how many pianos are there?
如果我们想继续处理这个数量级
If we want to continue dealing with orders of magnitude,
我们也可以说是每十个人就有一个人拥有一架钢琴
we can either say that one out of 10
或者每100人就有一个人拥有一架钢琴
or one out of one hundred people own a piano.
假设我们的人口估算包括孩子和成年人
Given that our estimate of the population includes children and adults,
我们只考虑成年人
we’ll go with the latter estimate,
那么钢琴数目大致是10的4次方
which estimates that there are approximately 10 to the fourth,
或者说芝加哥有10000架钢琴
or 10,000 pianos, in Chicago.
知道了钢琴数目,那么这里有多少钢琴调谐器呢
With this many pianos, how many piano tuners are there?
我们可以首先思考钢琴多久做一次校准
We could begin the process of thinking about how often the pianos are tuned,
在同一天有多少刚才在被校准
how many pianos are tuned in one day,
或者说经过多少天钢琴调谐器才工作一次
or how many days a piano tuner works,
但是这些都不是快速估计的重点
but that’s not the point of rapid estimation.
我们认为在数量级上
We instead think in orders of magnitude,
说一个钢琴调音师大概每年会调10到10的二次方架钢琴
and say that a piano tuner tunes roughly 10 to the second pianos in a given year,
大概就是100架
which is approximately a few hundred pianos.
假设在芝加哥要10的4次方架钢琴
Given our previous estimate of 10 to the fourth pianos in Chicago,
每个钢琴调音师每年会调100架钢琴
and the estimate that each piano tuner can tune 10 to the second pianos each year,
我们可以说在芝加哥大致有100个钢琴调音师
we can say that there are approximately 10 to the second piano tuners in Chicago.
现在,我知道你们肯定会想
Now, I know what you must be thinking:
这些估算如何产生合理的答案
How can all of these estimates produce a reasonable answer?
很简单
Well, it’s rather simple.
在任何费米问题上,
In any Fermi problem, it is assumed
高估的部分和低估的部分相互抵消
that the overestimates and underestimates balance each other out,
并且产生一种估计
and produce an estimation
那就是不超过一个数量级的正确答案
that is usually within one order of magnitude of the actual answer.
在这个例子中,我们可以通过查看电话簿来证明
In our case we can confirm this by looking in the phone book
在芝加哥的钢琴调音师的数目
for the number of piano tuners listed in Chicago.
我们发现什么呢?其数目是81
What do we find? 81.
多么不可思议,符合我们的估算。那就是10的力量
Pretty incredible, given our order-of-magnitude estimation.But, hey – that’s the power of 10.

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