Less Wrong is a community blog devoted to refining the art of human rationality. Please visit our About page for more information.

Rough calculations: Fermi and the art of guessing

9 Post author: XiXiDu 08 September 2011 10:39AM

Fermi problem

In science, particularly in physics or engineering education, a Fermi problem, Fermi question, or Fermi estimate is an estimation problem designed to teach dimensional analysis, approximation, and the importance of clearly identifying one's assumptions. Named after physicist Enrico Fermi, such problems typically involve making justified guesses about quantities that seem impossible to compute given limited available information.

Fermi was known for his ability to make good approximate calculations with little or no actual data, hence the name. One example is his estimate of the strength of the atomic bomb detonated at the Trinity test, based on the distance travelled by pieces of paper dropped from his hand during the blast. Fermi's estimate of 10 kilotons of TNT was remarkably close to the now-accepted value of around 20 kilotons, a difference of less than one order of magnitude.


Scientists often look for Fermi estimates of the answer to a problem before turning to more sophisticated methods to calculate a precise answer. This provides a useful check on the results: where the complexity of a precise calculation might obscure a large error, the simplicity of Fermi calculations makes them far less susceptible to such mistakes. (Performing the Fermi calculation first is preferable because the intermediate estimates might otherwise be biased by knowledge of the calculated answer.)

Fermi estimates are also useful in approaching problems where the optimal choice of calculation method depends on the expected size of the answer. For instance, a Fermi estimate might indicate whether the internal stresses of a structure are low enough that it can be accurately described by linear elasticity; or if the estimate already bears significant relationship in scale relative to some other value, for example, if a structure will be over-engineered to withstand loads several times greater than the estimate.

Although Fermi calculations are often not accurate, as there may be many problems with their assumptions, this sort of analysis does tell us what to look for to get a better answer.

Link: en.wikipedia.org/wiki/Fermi_problem

Fermi Problem: Power developed at the eruption of the Puyehue-Cordón Caulle volcanic system in June 2011

Enrico Fermi was renowned for his ability to make reliable estimates. But how well can you do on a modern estimation problem?


Hernan Asory and Arturo Lopez Davalos at the Comision Nacional De Energia Atomica in Argentina, have set themselves (and their students) a similar estimation task. The problem is to estimate the energy release as well as the volume and mass of sand ejected during the eruption of the Puyehue-Cordon Caulle volcano in Chile on 4 July.

You can look up the calculations and the assumption they make in the paper. You might want to try the estimate yourself.

Link: technologyreview.com/blog/arxiv/27140/

If you want to get better at doing rough mental calulcations, the following books might provide some valuable heuristics:

Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving

Time for some quick arithmetic: Is 3600 x 4.4 x 104 x 32 larger or smaller than 3 x 109?

Finding the right answer, says Sanjoy Mahajan, associate director for teaching initiatives at MIT’s Teaching and Learning Laboratory, does not require crafting a long, tedious calculation. Instead, the key to solving this problem — and many others — lies in having informal tools on hand that let us attack the problem. Though the result may not be perfectly precise, he believes, intuitive mathematical reasoning is often sufficient for our needs.

“That’s not to say exact answers aren’t useful,” says Mahajan, “but if looking for them is your only approach, you may never get any answer at all. Sometimes it’s better to start with something rough.”


Mahajan believes we should learn practical math tools and understand why they work.


Mahajan’s unconventional teaching practices stem from his focus, as a physicist, on finding quick, practical answers. Then again, perhaps rolling up one’s sleeves and hacking through problems is how everyone works. “There is a culture in pure mathematics that emphasizes rigor and careful proofs,” says Strogatz. “Yet all practicing mathematicians know we also use our intuitions, then we clean our answers up.”


So let’s get back to the initial question (the numbers relate to the storage capacity of a data CD-ROM). The key to solving it, says Mahajan, is to recognize that the components of the first, messy-looking number can be broken into powers of 10. Then we can temporarily set aside these powers of 10 — Mahajan calls this “taking out the big part,” one of his tenets of problem-solving — while handling the smaller, simpler multiplication problem.

Okay: Picture the number as (3.6 x 103) x (4.4 x 104) x (3.2 x 101). To multiply powers of 10 in practice, we add them, here producing 108. Leave that aside momentarily and multiply 3.6 x 4.4 x 3.2. The answer is about 50, or 5.0 x 101. Combine that with 108, and we have our answer: Roughly 5.0 x 109, which is bigger than 3 x 109. Street-fighting math, and we barely got a scratch. 

Link: web.mit.edu/newsoffice/2010/street-fight-0329.html

Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks

Yes, even you can learn to do seemingly complex equations in your head; all you need to learn are a few tricks. You’ll be able to quickly multiply and divide triple digits, compute with fractions, and determine squares, cubes, and roots without blinking an eye. No matter what your age or current math ability, Secrets of Mental Math will allow you to perform fantastic feats of the mind effortlessly. This is the math they never taught you in school.

Comments (9)

Comment author: gjm 08 September 2011 01:55:45PM *  5 points [-]

The term "dark art" is usually used in these parts to mean something very different from this...

[EDITED some time after posting, to fix a typo.]

[EDITED again to add: when I wrote the above, the title was something about "the dark art of estimation"; it's been changed now. This also explains Dr_Manhattan's otherwise confusing question.]

Comment author: jimmy 09 September 2011 06:41:53PM 3 points [-]

I strongly recommend the street fighting mathematics book. It is short and full of good stuff.

I end up helping otherwise smart people with techniques from this book all the time.

Here it is in PDF. http://mitpress.mit.edu/books/fullpdfs/Street-FightingMathematics.pdf

Comment author: oooo 06 July 2013 04:17:14AM 0 points [-]

The link is broken. Seems that MIT has an e-book storefront and understandably wants to charge for the electronic version.

Comment author: RichardKennaway 09 July 2013 12:56:14PM 1 point [-]

Current link. The book is free.

Comment author: [deleted] 08 September 2011 03:26:48PM 2 points [-]

This is a good post, but you should change the name or people will downvote without reading.

Comment author: Dr_Manhattan 08 September 2011 02:56:33PM 1 point [-]

What's dark about estimation?

Comment author: lessdazed 08 September 2011 01:33:47PM 1 point [-]

Was this inspired by the sequence rerun?

Comment author: RichardJActon 14 December 2014 07:50:10PM 0 points [-]

This is a Link to a resource I came across for people wishing to teach/learn Fermi calculation it contains a problem set, a potentially useful asset especially for meetup planners.


Comment author: fiddlemath 06 July 2012 03:46:45AM 0 points [-]

Has anyone had experiences in which similar reasoning was useful?

I suspect it is, and often, for someone with this mental habit. I lack this habit, so I don't easily know what opportunities I've missed. I'd like to teach the habit to myself and others, but I will need a handful of motivating cases.