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.”
