Editing Back-of-the-envelope calculation

{{Short description|Metaphor for a rough calculation}}
A '''back-of-the-envelope calculation''' is a rough calculation, typically jotted down on any available scrap of paper such as 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 in the U.S. is "back of a [[napkin]]", also used in the business world to describe sketching out a quick, rough idea of a business or product.<ref name="brown2011">{{Cite web|last=Brown|first=Bob|date=2011-07-19|title=Napkins: Where Ethernet, Compaq and Facebook's cool data center got their starts|url=https://www.networkworld.com/article/2220218/napkins--where-ethernet--compaq-and-facebook-s-cool-data-center-got-their-starts.html|access-date=2020-10-06|archive-url=https://web.archive.org/web/20190815005607/https://www.networkworld.com/article/2220218/napkins--where-ethernet--compaq-and-facebook-s-cool-data-center-got-their-starts.html|archive-date=15 August 2019|url-status=live|publisher=[[International Data Group|Network World]]|language=en|quote=[[Robert Metcalfe]]'s early [[Ethernet]] diagrams from his days at Xerox PARC back in the early 1970s might be the most famous napkin sketches in the technology industry.}}</ref> In British English, a similar [[idiom]] is "back of a [[cigarette pack|fag packet]]".

== History ==
In the natural sciences, ''back-of-the-envelope calculation'' is often associated with physicist [[Enrico Fermi]],<ref>[https://web.archive.org/web/20080219101037/http://www.encyclopedia.com/doc/1G1-78334537.html Where Fermi stood. - Bulletin of the Atomic Scientists | Encyclopedia.com (Archived)<!-- Bot generated title -->]</ref> 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 problem]]s.<ref>[http://serc.carleton.edu/quantskills/teaching_methods/boe/index.html  Back of the Envelope Calculations<!-- Bot generated title -->]</ref><ref>[http://www.nap.edu/html/hs_math/be.html High School Mathematics at Work: Essays and Examples for the Education of All Students<!-- Bot generated title -->]</ref>

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.<ref name="Rhodes 1986 p. 674">{{cite book | last=Rhodes | first=Richard | title=The Making of the Atomic Bomb | publisher=Simon & Schuster | publication-place=New York | year=1986 | isbn=978-0-671-44133-3 | oclc=13793436 | page=674}}</ref><ref>{{Cite web |url=http://www.lanl.gov/science/weapons_journal/wj_pubs/11nwj2-05.pdf |title=Nuclear Weapons Journal, Los Alamos National Laboratory, Issue 2 2005. |access-date=2014-09-07 |archive-date=2018-12-29 |archive-url=https://web.archive.org/web/20181229223636/https://www.lanl.gov/science/weapons_journal/wj_pubs/11nwj2-05.pdf |url-status=dead }}</ref>

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.<ref>{{cite journal |first=B.A. |last=Austin |title=Precursors To Radar — The Watson-Watt Memorandum And The Daventry Experiment |url=http://www.bawdseyradar.org.uk/wp-content/uploads/2012/12/Wilkins-Calculations.pdf |journal=International Journal of Electrical Engineering & Education |volume=36 |year=1999 |issue=4 |pages=365–372 |doi=10.7227/IJEEE.36.4.10 |s2cid=111153288 |access-date=2016-07-08 |archive-url=https://web.archive.org/web/20150525040134/http://www.bawdseyradar.org.uk/wp-content/uploads/2012/12/Wilkins-Calculations.pdf |archive-date=2015-05-25 |url-status=dead }}</ref>

Another example is [[Victor Weisskopf]]'s pamphlet ''Modern Physics from an Elementary Point of View''.<ref>[http://cdsweb.cern.ch/record/274976/ Lectures given in the 1969 Summer Lecture Programme, CERN (European Organization for  Nuclear Research), CERN 70-8, 17 March 1970.]</ref> 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.

== Examples ==
In a video interview for the [[University of California, Berkeley]] on the 50th anniversary of the laser, Nobel laureate [[Charles Townes]] described how he pulled an envelope from his pocket while sitting in a park and wrote down calculations during his initial insight into lasers.<ref>[http://laserfest.org/lasers/video-history.cfm Video of interview with Charles Townes; envelope mention comes about halfway in]</ref>

During lunch with NFL commissioner [[Pete Rozelle]] in 1966, [[Tiffany & Co.]] vice president Oscar Riedner made a sketch on a cocktail napkin of what would become the [[Vince Lombardi Trophy]], awarded annually to the winner of the [[Super Bowl]].<ref>{{cite news|title=Vince Lombardi Trophy|url=https://www.profootballhof.com/news/vince-lombardi-trophy/|publisher=NFL Enterprises, LLC|website=ProFootballHOF.com|access-date=November 21, 2019}}</ref>

An important Internet protocol, the [[Border Gateway Protocol]], was sketched out in 1989 by engineers on the back of "three ketchup-stained napkins", and is still known as the three-napkin protocol.<ref>{{Cite news|last=Timberg|first=Craig|title=Net of Insecurity; Quick fix for an early Internet problem lives on a quarter-century later|date=31 May 2015|url=http://www.washingtonpost.com/sf/business/2015/05/31/net-of-insecurity-part-2/|access-date=4 January 2021|newspaper=[[The Washington Post]]|language=en-US|quote=As the prospect of system meltdown loomed, the men began scribbling ideas for a solution onto the back of a ketchup-stained napkin. Then a second. Then a third. The “three-napkins protocol,” as its inventors jokingly dubbed it, would soon revolutionize the Internet. And though there were lingering issues, the engineers saw their creation as a “hack” or “kludge,” slang for a short-term fix to be replaced as soon as a better alternative arrived.|archive-url=https://web.archive.org/web/20150601035758/http://www.washingtonpost.com/sf/business/2015/05/31/net-of-insecurity-part-2/|archive-date=1 June 2015|url-status=live}}</ref>

[[UTF-8]], the dominant character encoding for the [[World Wide Web]],<ref>{{Cite web|url=https://w3techs.com/technologies/cross/character_encoding/ranking|title=Usage Survey of Character Encodings broken down by Ranking|website=w3techs.com|language=en|access-date=2018-11-01}}</ref> was designed by [[Ken Thompson]] and [[Rob Pike]] on a placemat.<ref name=":0">[https://www.cl.cam.ac.uk/~mgk25/ucs/utf-8-history.txt Email Subject: UTF-8 history], From: "Rob 'Commander' Pike", Date: Wed, 30 Apr 2003..., ''...UTF-8 was designed, in front of my eyes, on a placemat in a New Jersey diner one night in September or so 1992...So that night Ken wrote packing and unpacking code and I started tearing into the C and graphics libraries.  The next day all the code was done...''</ref>

The [[Bailey bridge]] is a type of portable, pre-fabricated, truss bridge and was extensively used by British, Canadian and US military engineering units. [[Donald Bailey (civil engineer)|Donald Bailey]] drew the original design for the bridge on the back of an envelope.<ref name="bailey">{{Cite news|url=https://www.latimes.com/archives/la-xpm-1985-05-07-mn-11177-story.html|title=Sir Donald Bailey, WW II Engineer, Dies|last=Services|first=Times Wire|date=1985-05-07|work=Los Angeles Times|access-date=2018-11-01|language=en-US|issn=0458-3035|quote="He sketched the original design for the Bailey Bridge on the back of an envelope as he was being driven to a meeting of Royal Engineers to debate the failure of existing portable bridges"}}</ref>

The [[Laffer Curve]], which claims to show the relationship between tax cuts and government income, was drawn by [[Arthur Laffer]] in 1974 on a bar napkin to show an aide to President [[Gerald R. Ford]] why the federal government should cut taxes.<ref>[https://www.nytimes.com/2017/10/13/us/politics/arthur-laffer-napkin-tax-curve.html?module=WatchingPortal&region=c-column-middle-span-region&pgType=Homepage&action=click&mediaId=thumb_square&state=standard&contentPlacement=2&version=internal&contentCollection=www.nytimes.com&contentId=https%3A%2F%2Fwww.nytimes.com%2F2017%2F10%2F13%2Fus%2Fpolitics%2Farthur-laffer-napkin-tax-curve.html&eventName=Watching-article-click&_r=0 "This is not Arthur Laffer's famous napin" NY Times 13 Oct. 2017]</ref>

Upon hearing that the [[Saturn I#S-IV stage|S-IV 2nd Stage]] of the [[Saturn I]] would need transport from California to Florida for launch as part of the [[Apollo program]], [[John M. Conroy|Jack Conroy]] sketched the cavernous cargo airplane, the [[Aero Spacelines Pregnant Guppy|Pregnant Guppy]].<ref>{{Cite journal|last=Bloom|first=Margy|date=15 September 2011|title=PilotMag Aviation Magazine {{!}} The Pregnant Guppy {{!}} The Problem: Logistics|url=http://pilotmag.com/guppy|url-status=dead|journal=Pilot Magazine|volume=|pages=|archive-url=https://web.archive.org/web/20110715083759/http://pilotmag.com/guppy|archive-date=15 July 2011|access-date=25 April 2012|quote=[Conroy] listened to the conversations around him, then picked up a cocktail napkin and a ballpoint pen. And with the precision he’d learned during the brief months he’d attended engineering school many years before, he drew an airplane that had never been built, to carry a rocket that had never been launched, to take man to a place nobody had ever been before.  [[John M. Conroy|Jack Conroy]] had just sketched the airplane that would become the Pregnant Guppy.|via=}}</ref>

The [[Video Toaster]] was designed on placemats in a Topeka pizza restaurant.<ref>{{Cite news|last=Reimer|first=Jeremy|date=18 March 2016|title=A history of the Amiga, part 9: The Video Toaster|url=https://arstechnica.com/gadgets/2016/03/a-history-of-the-amiga-part-9-the-video-toaster/|access-date=4 January 2021|publisher=[[Ars Technica]]|language=en-us|url-status=live|archive-url=https://web.archive.org/web/20160318125806/https://arstechnica.com/gadgets/2016/03/a-history-of-the-amiga-part-9-the-video-toaster/|archive-date=18 March 2016|quote=[[Paul Montgomery|Montgomery]] suggested that [[NewTek#Tim Jenison|Jenison]] meet his friend [[Brad Carvey]], who had been working on projects involving robotic vision. The three of them got together in a pizza restaurant in Topeka and started drawing block diagrams on the placemats.}}</ref>

== See also ==
* [[Buckingham pi theorem]], a technique often used in [[fluid mechanics]] to obtain order-of-magnitude estimates
* [[Guesstimate]]
* [[Scientific Wild-Ass Guess]]
* [[Heuristic]]
* [[Order-of-magnitude analysis]]
* [[Rule of thumb]]
* [[Sanity testing]]
* [[Fermi Problem]]

== Notes and references ==
{{Reflist}}

== External links ==
{{Wiktionary|back-of-the-envelope|back-of-an-envelope}}
* [http://www-cse.ucsd.edu/classes/sp97/cse141/botec.html Syllabus at UCSD]

{{Orders of magnitude}}

[[Category:Approximations]]
[[Category:Informal estimation]]
[[Category:Metaphors referring to objects]]