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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.

  • Correct scientific notation requires that the first term range in value set as greater than one but less than ten,

  • and the second term represents the power of ten, or order of magnitude, by which we multiply the first term

  • 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.

  • The height of a tree is approximately ten to the power of one meters.

  • And the diameter of the Earth is approximately ten to the power of seven meters.

  • 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.

  • 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.

  • One example of the classic Fermi estimation problems is to determine how many piano tuners there are in the city of Chicago, Illinois.

  • 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.

  • 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?

  • 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.

  • 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,

  • which is approximately a few hundred pianos.

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

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

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

  • 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.

  • 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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B1 TED-Ed estimation fermi piano magnitude chicago

【TED-Ed】A clever way to estimate enormous numbers - Michael Mitchell

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    VoiceTube posted on 2013/03/20
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