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


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


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


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


  • 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.


  • 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,


  • 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.


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


  • or rapid estimations, with seemingly little available data.


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


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


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


  • 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.


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


  • An estimation will work.


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


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


  • Are there one million people? Five million people?


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


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


  • 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,


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


  • 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


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


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


  • 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?


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


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


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


  • 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.


  • 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:


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


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


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


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


  • What do we find? 81.

    有幾位呢? 答案:81。

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


  • But, hey, that's the power of ten.


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


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