Back-of-the-envelope calculations: When close is good enough

Some engineers, professional and amateur, share something with Enrico Fermi: the ability to take rough data, pencil whip it through a few calculations, and decide if a project is possible or what it might take to get something done. Stefan Funk, a physicist at Stanford University is particularly good at tackling these types of problems, also known as Fermi problems, and teaches course on them, “Science on the Back of an Envelope.” The course is even mandatory for physics grad students at the university.

To make the course more interesting, he takes problems from popular culture. For example, taking a cue from the 1998 movie Armageddon, he had students determine if it would be possible to blow up an asteroid heading toward Earth if you knew the size, speed, and current position of the asteroid. Another problem came from James and the Giant Peach: How may seagulls would it take to carry a house-sized peach from England to New York. And you know peaches float, so they are less dense than water. (To see handwritten back-of-the-envelope calculations for these two problems, go to Asteroid or Peach.)

Taking seemingly nonsensical problems from movies and books lets students focus on solving the problem rather than getting the correct number to three decimal places. And they are more interesting.

“It’s about being able to step back and look at your results. Does it make sense? If you’re only off by a factor of two or three, that’s no big deal. If you’re off by, say 1020, that’s a big problem,” says Funk in an interview with Symmetry magazine.

I think Prof. Funk’s course would be great for all sorts of people, including politicians, screenwriters, , business folks off all stripes, and magazine editors.

Discuss this Blog Entry 5

Michael Mahon (not verified)
on Jul 7, 2014

Since the calculator displaced the slide rule, the art of estimation has declined.

In order to use a slide rule, you must be able to estimate the result of a calculation to the correct order of magnitude and one significant digit. The slide rule then provides the next two significant digits.

No estimation is thought to be necessary when using a digital calculator; however, experiments have shown that many calculator users happily accept results that are incorrect by orders of magnitude!

Estimation is a skill that must be taught if we need designers who can quickly traverse a design decision tree, pruning impossible or impractical branches. The alternative is much wasted time and effort calculating or simulating useless alternative solutions--or never even arriving at a practical solution!

Estimation is a critical tool to a designer, and it has almost no educational support.

Michael Mahon (not verified)
on Jul 7, 2014

And I should have added, citizens' inability to think quantitatively makes them highly susceptible to snake oil salesmen. ;-)

Anonymous Poster (not verified)
on Jul 15, 2014

I saw an interesting video about an easy method of estimation. It's called "zequals" and is presented by mathematics writer Rob Eastaway.

TayL (not verified)
on Jul 20, 2014

I still have a slide rule, somewhere.
An interesting adjunct to estimation is consistent units.
Or rational proportions.
So many bloggers have no sense of scale.
Or basic facts. See pols quote numbers inconsistent with US Census data.
Or marketing people coming up with numbers out of thin air.
Of course Fermi shut down his nuclear pile as soon as he confirmed it went critical. Limited facts, estimated construction. He wasn't that sure that it wouldn't blow up.

on Jan 3, 2015

The idea of not automatically providing water in restaurants is reasonable, both for real water savings and as a symbolic gesture, and I’ve seen it done in places suffering drought before. But is it really plausible that the average American eats at a sit-down restaurant with waiters five times a week? That would be an impossible budget-buster for most of us. I would guess that the typical American who eats out five days a week is having most of those meals at the sort of place where you carry your own prepaid, paper-wrapped food away from the counter in a tray, and only grudgingly do you get a cup for water instead of $oda.

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A questioning, sometimes humorous look at technologies, engineering, and the world.


Stephen Mraz

Steve serves as Senior Editor of Machine Design.  He has 23 years of service and has a B.S. Biomedical Engineering from CWRU. Steve was a E-2C Hawkeye Naval Flight Officer in the U.S. Navy. He...
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