A back-of-the-envelope calculation is a rough calculation, typically jotted down on any available scrap of paper such as the actual back of an envelope. It is more than a guess but less than an accurate calculation or mathematical proof. The defining characteristic of back-of-the-envelope calculations is the use of simplified assumptions. A similar phrase is "back of a napkin", which is also used in the business world to describe sketching out a quick, rough idea of a business or product. In British English, a similar idiom is "back of a fag packet".
In the hard sciences, back-of-the-envelope calculation is often associated with physicist Enrico Fermi, 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.
Fermi was known for getting quick and accurate answers to problems that would stump other people. The most famous instance came during the first atomic bomb test in New Mexico on 16 July 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.
Perhaps the most influential example of such a calculation was carried out over a period of a few hours by Arnold Wilkins after being asked to consider a problem by Robert Watson Watt. Watt had learned that the Germans claimed to have invented a radio-based death ray, but Wilkins' one-page calculations demonstrated that such a thing was almost certainly impossible. When Watt asked what role radio might play, Wilkins replied that it might be useful for detection at long range, a suggestion that led to the rapid development of radar and the Chain Home system.
Another example is Victor Weisskopf's pamphlet Modern Physics from an Elementary Point of View. 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.
Nobel laureate Charles Townes describes in a video interview for the University of California, Berkeley on the 50th anniversary of the laser, how he pulled an envelope from his pocket while sitting in a park and wrote down calculations during his initial insight into lasers.
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