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A back-of-the-envelope calculation[1] is a rough calculation to test a hypothesis. It is not necessarily written on the back of an envelope. It is more trustworthy than a guess, but less definite than a mathematical proof.

The idiom originates in the practice of quickly jotting down calculations on the nearest available piece of paper, such as an envelope. The defining characteristic of back-of-the-envelope calculations is the use of simplified, scaled-down models. A similar phrase is "back of a napkin".



In the hard sciences, back-of-the-envelope calculation is often associated with physicist Enrico Fermi,[2] who was well known for emphasizing ways that complex scientific equations could be approximated within an order of magnitude using simple calculations. He went on to develop a series of sample calculations which are called "Fermi Questions" or "Back-of-the-Envelope Calculations" and used to solve Fermi problems.[3][4]

Fermi was known for getting quick and accurate answers to problems which would stump other people. The most famous instance came during the first atomic bomb test in New Mexico on July 16, 1945. As the blast wave reached him, Fermi dropped bits of paper. By measuring the distance they were blown, he could compare to a previously computed table and thus estimate the bomb energy yield. He estimated 10 kilotons of TNT; the measured result was 18.6.

Another example is Victor Weisskopf's pamphlet Modern Physics from an Elementary Point of View.[5] In these notes Weisskopf used back-of-the-envelope calculations to calculate the size of a hydrogen atom, a star, and a mountain, all using elementary physics.



  • When British engineer James Nasmyth was asked to build a bigger hammer after the workpiece was too big for existing trip hammers, Nasmyth, after a little thought, sketched out on a piece of paper the design for the steam hammer.
  • Legend has it that the original Southwest Airlines "Texas Triangle", connecting its major cities (Dallas, Houston, San Antonio) was created on the back of a napkin.
  • Though admittedly not his own discovery, Arthur Laffer famously illustrated a graph that lower income taxes would mean higher revenue, as with a retailer lowering higher-than-optimum prices. This Laffer Curve drawn on a cocktail napkin became a well-known rebuttal to confiscatory tax rates.

See also

Notes and references

  1. ^ Paritosh, Praveen; Forbus, Ken (2005). "Analysis of Strategic Knowledge in Back of the Envelope Reasoning". 20th National Conference on Artificial Intelligence (AAAI-05). Pittsburgh, PA. 978-1-57735-236-5. Retrieved 2009-03-06.  
  2. ^ Where Fermi stood. - Bulletin of the Atomic Scientists |
  3. ^ Back of the Envelope Calculations
  4. ^ High School Mathematics at Work: Essays and Examples for the Education of All Students
  5. ^ Lectures given in the 1969 Summer Lecture Programme, CERN (European Organization for Nuclear Research), CERN 70-8, 17 March 1970.

External links


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