The Nobel-prize-winning physicist, Enrico Fermi, came up with this simple and intuitive way to
deduce the circumference of the earth.
1. How many time zones do you pass through when you fly from New York to Los Angeles?
Answer, 3
2. How many miles is it, about, over that same distance?
Answer, about 3000.
3. How many miles per time zone, on average?
Answer, about 1000
4. How many time zones must there be around the world?
Answer, 24 because there are 24 hours in a day
5. How many miles around the world?
Answer, 24 time zones x 1000 miles per time zone = 24000 miles
Yes, it is about 24000 miles around the world.
So how do I get the diameter of the Earth
1. the formula for a circle is 2 pi r, right? where r is the radius and pi is about 3
2. so 24000 mi = 2 x 3 x r = 6 x r
3. therefore 24000 mi / 6 = r = 4000 mi
4. the diameter of the earth is 2 x r = 8000 mi, where the diameter is 2 times the radius
5. 1 mi = 1.6 km so 8000 mi x 1.6 km/mi = 12800 km
(the correct answer is ~12742 km)
Estimating – Order Of Magnitude – Estimate
The order of a magnitude is it’s nearest power of ten (10):
10 = 1×101 -> Order is 1
100 = 1×102 -> Order is 2
4,000,000 -> 4×106 -> Order is 6
- Do as little math as possible.
- Total the order of magnitude of terms multiplied.
- For numbers beginning with digits 1 – 4 use the lower magnitude,
i.e. 20 has magnitude 1 and 80 has magnitude 2.
Example: Estimate the order of magnitude of the number of breaths of a human in a lifetime.
(10/min) * (60 min) * (24 hrs) * (365 days) * (75 yrs)
>1+2+1+2+2=8
i.e. 1×108 = 100,000,000
Example: Estimate U.S. annual consumption of gasoline in gallons
1. (200,000,000 cars) * (15 gallons / week) * (52 weeks)
> 8 + 1 + 2 = 11
i.e. 1×1011 = 100,000,000,000
2. (400,000,000 cars) * (40 gallons / week) * (52 weeks)
> 8 + 1 + 2 = 11
i.e. 1×1011
The skill of estimating scale and order of magnitude is necessary for quality problem solving. It is
one basis of the reality check.
My answer to the total length of string in the worlds largest ball of string is an excellent example
of this process. While my guesstimate was less than the width of the known universe, it was of
such absurdity as to provide enough material to – oh, I don’t know – knit a cozy little pouch to
tuck the sun into to stem global warming.
I discerned my answer was complete nonsense by comparing it to the magnitude of the earth’s
diameter. While I may have gotten the wrong answer, I knew and understood it was the wrong
answer.
Update 2010/10/17
I saw somewhere where the rounding was done NOT as normal by rounding up if the last digit is
5 or more and rounding down if the last digit is 4 or below, the point where an order of
magnitude changed was centered around the log of the quantity in base ten.
e.g. 400 has a
magnitude of 3 not 2 because log10 400 (the logarithm of 400 in base 10) is 2.6, which means it
has a magnitude of 2.6, and when 2.6 is rounded it is rounded up, which results in a magnitude
estimate of, drum roll, 3. Emphasis mine. It makes enough sense to me.