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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.



【TED-Ed】大きな数を推定する巧妙な方法 — マイケル・ミッチェル (A clever way to estimate enormous numbers - Michael Mitchell)

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