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A clever way to estimate enormous numbers - Michael Mitchell - Video học tiếng Anh
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A clever way to estimate enormous numbers - Michael Mitchell
A clever way to estimate enormous numbers - Michael Mitchell
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0:15
Whether you like it or not, we use numbers every day.
0:18
Some numbers, such as the speed of sound,
0:20
are small and easy to work with.
0:22
Other numbers, such as the speed of light,
0:24
are much larger and cumbersome to work with.
0:26
We can use scientific notation to express these large numbers
0:29
in a much more manageable format.
0:31
So we can write 299,792,458 meters per second
0:37
as 3.0 times 10 to the eighth meters per second.
0:41
Correct scientific notation
0:43
requires that the first term range in value
0:45
so that it is greater than one but less than 10,
0:47
and the second term represents the power of 10 or order of magnitude
0:50
by which we multiply the first term.
0:53
We can use the power of 10 as a tool in making quick estimations
0:56
when we do not need or care for the exact value of a number.
0:59
For example, the diameter of an atom
1:01
is approximately 10 to the power of negative 12 meters.
1:04
The height of a tree is approximately 10 to the power of one meter.
1:07
The diameter of the Earth is approximately 10 to the power of seven meters.
1:11
The ability to use the power of 10 as an estimation tool
1:13
can come in handy every now and again,
1:15
like when you're trying to guess the number of M&M's in a jar,
1:18
but is also an essential skill in math and science,
1:21
especially when dealing with what are known as Fermi problems.
1:24
Fermi problems are named after the physicist Enrico Fermi,
1:26
who's famous for making rapid order-of-magnitude estimations,
1:29
or rapid estimations, with seemingly little available data.
1:32
Fermi worked on the Manhattan Project in developing the atomic bomb,
1:35
and when it was tested at the Trinity site in 1945,
1:38
Fermi dropped a few pieces of paper during the blast
1:41
and used the distance they traveled backwards as they fell
1:43
to estimate the strength of the explosion as 10 kilotons of TNT,
1:47
which is on the same order of magnitude as the actual value of 20 kilotons.
1:51
One example of the classic Fermi estimation problems
1:54
is to determine how many piano tuners there are
1:56
in the city of Chicago, Illinois.
1:58
At first, there seem to be so many unknowns
2:01
that the problem appears to be unsolvable.
2:03
That is the perfect application for a power-of-10 estimation,
2:06
as we don't need an exact answer -
2:07
an estimation will work.
2:09
We can start by determining how many people live in the city of Chicago.
2:12
We know that it is a large city,
2:14
but we may be unsure about exactly how many people live in the city.
2:17
Are the one million people? Five million people?
2:20
This is the point in the problem
2:22
where many people become frustrated with the uncertainty,
2:25
but we can easily get through this by using the power of 10.
2:28
We can estimate the magnitude of the population of Chicago
2:30
as 10 to the power of six.
2:32
While this doesn't tell us exactly how many people live there,
2:35
it serves an accurate estimation for the actual population
2:38
of just under three million people.
2:40
So if there are approximately 10 to the sixth people in Chicago,
2:43
how many pianos are there?
2:44
If we want to continue dealing with orders of magnitude,
2:47
we can either say that one out of 10
2:49
or one out of one hundred people own a piano.
2:51
Given that our estimate of the population includes children and adults,
2:55
we'll go with the latter estimate,
2:57
which estimates that there are approximately 10 to the fourth,
3:00
or 10,000 pianos, in Chicago.
3:02
With this many pianos, how many piano tuners are there?
3:05
We could begin the process of thinking about how often the pianos are tuned,
3:09
how many pianos are tuned in one day,
3:11
or how many days a piano tuner works,
3:13
but that's not the point of rapid estimation.
3:15
We instead think in orders of magnitude,
3:17
and say that a piano tuner tunes roughly 10 to the second pianos in a given year,
3:21
which is approximately a few hundred pianos.
3:23
Given our previous estimate of 10 to the fourth pianos in Chicago,
3:26
and the estimate that each piano tuner can tune 10 to the second pianos each year,
3:31
we can say that there are approximately 10 to the second piano tuners in Chicago.
3:34
Now, I know what you must be thinking:
3:36
How can all of these estimates produce a reasonable answer?
3:39
Well, it's rather simple.
3:41
In any Fermi problem, it is assumed
3:42
that the overestimates and underestimates balance each other out,
3:46
and produce an estimation
3:47
that is usually within one order of magnitude of the actual answer.
3:50
In our case we can confirm this by looking in the phone book
3:53
for the number of piano tuners listed in Chicago.
3:55
What do we find? 81.
3:57
Pretty incredible, given our order-of-magnitude estimation.
4:00
But, hey - that's the power of 10.
