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Back-of-the-Envelope Calculations: Order of Magnitude Estimates in Science, Study notes of Piano

The concept of order of magnitude estimates, a simple yet powerful tool used in science to approximate values and determine the reasonableness of calculated results. The importance of everyday guesses, the legacy of enrico fermi and his famous calculations, and the usefulness of order of magnitude calculations in various scientific contexts.

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Module 1, Lesson 6
The Back of the Envelope
Objective: By the end of this lesson you will be able to perform simple back-of-the-envelope
calculations that can be used to approximate values without tedious calculations and to
determine if calculated values are reasonable. You will also recognize the value of these
calculations in doing science and incrementally building models that describe the world around
us.
1. Everyday Guesses
Ordinarily, when we do estimates, we tend to say things like "about a dozen," or "around four
thousand." If we look closely, we can see that not all estimates, even verbal ones, are of the
same accuracy. The vaguest sorts of estimates are essentially how many zeros are in any
measurement. This is known as an order of magnitude estimate - finding the closest power of
ten to the number you're looking for. These calculations are usually simple enough that you can
perform them on very little paper, such as the back on an envelope.
Suppose you're at a crowded hockey stadium. An order of magnitude estimate means that,
instead of saying, "There are 4335 people at this hockey game", you can say, "There are a few
thousand people at this hockey game." You don't have enough information to say with any
precision how many people there are, or even how many thousands of people there are. But
you do know enough to say with certainty that there are too many to count in the hundreds, and
not enough to count in the tens of thousands.
2. Enrico Fermi
Enrico Fermi is one of the great great physicists in history. His work spanned many fields, and
included both experimental and theoretical results. Fermi was a master of back-of-the-envelope
calculations and famously used them to calculate the tonnage of TNT the first atomic blast was
equivalent to. As Fermi recalls in My Observations During the Explosion at Trinity on July 16,
194,
“About 40 seconds after the explosion the air blast reached me. I tried to estimate its
strength by dropping from about six feet small pieces of paper before, during, and after
the passage of the blast wave. Since, at the time, there was no wind I could observe
very distinctly and actually measure the displacement of the pieces of paper that were in
the process of falling while the blast was passing. The shift was about 2 1/2 meters,
which, at the time, I estimated to correspond to the blast that would be produced by ten
thousand tons of T.N.T.”
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Module 1, Lesson 6

The Back of the Envelope

Objective: By the end of this lesson you will be able to perform simple back-of-the-envelope calculations that can be used to approximate values without tedious calculations and to determine if calculated values are reasonable. You will also recognize the value of these calculations in doing science and incrementally building models that describe the world around us.

1. Everyday Guesses

Ordinarily, when we do estimates, we tend to say things like "about a dozen," or "around four thousand." If we look closely, we can see that not all estimates, even verbal ones, are of the same accuracy. The vaguest sorts of estimates are essentially how many zeros are in any measurement. This is known as an order of magnitude estimate - finding the closest power of ten to the number you're looking for. These calculations are usually simple enough that you can perform them on very little paper, such as the back on an envelope. Suppose you're at a crowded hockey stadium. An order of magnitude estimate means that, instead of saying, "There are 4335 people at this hockey game", you can say, "There are a few thousand people at this hockey game." You don't have enough information to say with any precision how many people there are, or even how many thousands of people there are. But you do know enough to say with certainty that there are too many to count in the hundreds, and not enough to count in the tens of thousands.

2. Enrico Fermi

Enrico Fermi is one of the great great physicists in history. His work spanned many fields, and included both experimental and theoretical results. Fermi was a master of back-of-the-envelope calculations and famously used them to calculate the tonnage of TNT the first atomic blast was equivalent to. As Fermi recalls in My Observations During the Explosion at Trinity on July 16, 194 , “About 40 seconds after the explosion the air blast reached me. I tried to estimate its strength by dropping from about six feet small pieces of paper before, during, and after the passage of the blast wave. Since, at the time, there was no wind I could observe very distinctly and actually measure the displacement of the pieces of paper that were in the process of falling while the blast was passing. The shift was about 2 1/2 meters, which, at the time, I estimated to correspond to the blast that would be produced by ten thousand tons of T.N.T.”

Emilio Segrè, who witnessed the event together with Fermi, states in his biography Enrico Fermi, Physicist that Fermi had done the necessary calculations in advance, "having prepared himself a table of numbers, so that he could tell immediately the energy liberated from this crude but simple measurement." Fermi also started asking order of magnitude questions of his students, such that they because to be known as Fermi Problems. These are designed to teach the skill of making justified guesses about incomplete sets of data. The classic Fermi problem is “How many piano tuners are in Vancouver?” How do we start to answer this? Google perhaps? Let’s try a more systematic approach. By making a series of assumptions

  1. There are approximately 2,000,000 people living in the Greater Vancouver area.
  2. On average, there are two people in each household in Vancouver.
  3. Roughly one household in 100 has a piano that is tuned regularly.
  4. Pianos that are tuned regularly are tuned on average about once per year.
  5. It takes a piano tuner about two hours to tune a piano, including travel time.
  6. Each piano tuner works eight hours in a day, five days in a week, and 50 weeks in a year. Each of these assumptions can be disputed, a key aspect of any good physics debate, and refined to get closer to the real answer. From these assumptions we can compute that the number of piano tunings in a single year in Vancouver is ! (2,000,000 persons in Vancouver) / (2 persons/household) × (1 piano/100 households) × ! (1 piano tuning per piano per year) = 10,000 piano tunings per year in Vancouver. We can similarly calculate that the average piano tuner performs ! (50 weeks/year)×(5 days/week)×(8 hours/day)/(2 hours to tune a piano) = 1000 piano ! tunings per year per piano tuner. Dividing gives ! (6,666 piano tunings per year in Vancouver) / (1000 piano tunings per year per piano ! tuner) = 10 piano tuners in Vancouver. We should now think if this is a reasonable number. If we got a number below 1, we know we made a mistake. If we got a number that was around 200,000, then every tenth person you know would be a piano tuner, which isn’t true. I personally don’t know any professional piano tuners, and I know about 1,000 people, so I would expect the number of piano tuners to be less than 2,000,000/1,000 = 1000. A quick Google search leads me to think there are about 20- (feel free to dispute this), so we got very close with our calculation, within an order of magnitude.

• You "cut corners," and aren't taking into account more detailed aspects of whatever you're

studying

• You want to "play it safe" and make a conservative estimate

• You have calculated a more precise result but want to know if your answer is reasonable

DON'T USE ORDER OF MAGNITUDE WHEN

• You're comparing two quantities that are less than around 100 times each other

• You have access to complete information that is very precise