View source for Googol

{{pp-vandalism|small=yes}}
{{Short description|Large number defined as ten to the 100th power}}
{{distinguish|text=[[Google]] or [[Nikolai Gogol]]}}
A '''googol''' is the [[large number]] 10<sup>100</sup> or ten to the power of one hundred.  In decimal notation, it is written as the [[numerical digit|digit]] 1 followed by one hundred [[0 (number)|zeroe]]s: <!-- Do not use [[Template:Gaps]], it breaks mobile -->'''10,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000,{{wbr}}000'''. Its [[Names of large numbers|systematic name]] is '''ten duotrigintillion''' ([[short scale]]) or '''ten sexdecilliard''' ([[long scale]]). Its [[prime factorization]] is <math>2^{100}\times 5^{100}.</math>

==Etymology==
The term was coined in 1920 by 9-year-old Milton Sirotta (1911–1981), nephew of American mathematician [[Edward Kasner]].<ref>{{cite journal|last=Bialik |first=Carl |url=https://www.wsj.com/articles/SB108575924921724042 |title=There Could Be No Google Without Edward Kasner |journal=The Wall Street Journal Online |date=June 14, 2004 |url-status=live |archive-url=https://web.archive.org/web/20161130145858/http://www.wsj.com/articles/SB108575924921724042 |archive-date=November 30, 2016}}</ref> He may have been inspired by the contemporary comic strip character [[Barney Google and Snuffy Smith|Barney Google]].<ref>{{cite book |title=The Hidden History of Coined Words |author1=Ralph Keyes |edition= |publisher=Oxford University Press |year=2021 |isbn=978-0-19-046677-0 |page=120 |url=https://books.google.com/books?id=JYskEAAAQBAJ}} [https://books.google.com/books?id=JYskEAAAQBAJ&pg=PA120 Extract of page 120]</ref> Kasner popularized the concept in his 1940 book ''[[Mathematics and the Imagination]]''.<ref>{{cite book| author1 = Kasner, Edward| author2 = Newman, James R.| title = Mathematics and the Imagination| url = https://books.google.com/books?id=Ad8hAx-6m9oC&q=Mathematics%20and%20the%20Imagination&pg=PP1| year = 1940| publisher = Simon and Schuster, New York| isbn = 0-486-41703-4| url-status = live| archive-url = https://web.archive.org/web/20140703073029/http://books.google.com/books?id=Ad8hAx-6m9oC&lpg=PP1&dq=Mathematics%20and%20the%20Imagination&pg=PP1| archive-date = 2014-07-03}} The relevant passage about the googol and googolplex, attributing both of these names to Kasner's nine-year-old nephew, is available in {{cite book|editor=James R. Newman |title=The world of mathematics |volume=3 |year=2000 |publisher=Dover Publications |location=Mineola, New York |orig-year=1956 |isbn=978-0-486-41151-4 |pages=2007–2010}}</ref> Other [[Names of large numbers|names]] for this quantity include ''ten duotrigintillion'' on the [[short scale]] (commonly used in English speaking countries),<ref>{{cite book |last1=Bromham |first1=Lindell |title=An Introduction to Molecular Evolution and Phylogenetics |date=2016 |publisher=Oxford University Press |location=New York, NY |isbn=978-0-19-873636-3 |page=494 |edition=2nd |url=https://books.google.com/books?id=Jf9NCwAAQBAJ |access-date=April 15, 2022}}</ref> ''ten thousand sexdecillion'' on the [[long scale]], or ''ten sexdecilliard'' on the [[Peletier long scale]].

== Size ==
A googol has no special significance in mathematics. However, it is useful when comparing with other very large quantities, such as the number of [[subatomic particles]] in the visible universe or the number of hypothetical possibilities in a [[chess]] game. Kasner used it to illustrate the difference between an unimaginably large number and [[infinity]], and in this role it is sometimes used in teaching mathematics. To put in perspective the size of a googol, the mass of an electron, just under {{val|e=-30|u=kg}}, can be compared to the mass of the visible universe, estimated at between {{val|e=50}} and {{val|e=60|u=kg}}.<ref>{{cite web|url=https://hypertextbook.com/facts/2006/KristineMcPherson.shtml |title=Mass of the universe |first=Kristine |last=McPherson |year=2006 |website=The Physics Factbook |editor-last=Elert |editor-first=Glenn |access-date=2019-08-24}}</ref> It is a ratio in the order of about {{val|e=80}} to {{val|e=90}}, or at most one ten-billionth of a googol (0.00000001% of a googol).

[[Carl Sagan]] pointed out that the total number of elementary particles in the universe is around {{val|e=80}} (the [[Eddington number]]) and that if the whole universe were packed with [[neutron]]s so that there would be no empty space anywhere, there would be around {{val|e=128}}. He also noted the similarity of the second calculation to that of [[Archimedes]] in ''[[The Sand Reckoner]]''. By Archimedes's calculation, the universe of [[Aristarchus of Samos|Aristarchus]] (roughly 2 light years in diameter), if fully packed with sand, would contain {{val|e=63}} grains. If the much larger observable universe of today were filled with sand, it would still only equal {{val|e=95}} grains. Another 100,000 observable universes filled with sand would be necessary to make a googol.<ref>{{cite book|last=Sagan|first=Carl|author-link=Carl Sagan|title=Cosmos|year=1981|publisher=Book Club Associates|pages=220–221}}</ref>

The decay time for a supermassive [[black hole]] of roughly 1&nbsp;galaxy-mass ({{val|e=11|u=[[solar mass]]es}}) due to [[Hawking radiation]] is on the order of {{val|e=100|u=years}}.<ref name="page">{{cite journal|last=Page |first=Don N. |title=Particle emission rates from a black hole: Massless particles from an uncharged, nonrotating hole |journal=Physical Review D |publisher=American Physical Society (APS) |volume=13 |issue=2 |date=1976-01-15 |issn=0556-2821 |doi=10.1103/physrevd.13.198 |pages=198–206|bibcode=1976PhRvD..13..198P}} See in particular equation (27).</ref> Therefore, the [[heat death of the universe|heat death]] of an [[future of an expanding universe|expanding universe]] is lower-bounded to occur at least one googol years in the future.

A googol is considerably smaller than a [[Names of large numbers|centillion]].<ref>{{cite book |last1=Stewart |first1=Ian |title=Infinity: A Very Short Introduction |date=2017 |publisher=Oxford University Press |location=New York, NY |isbn=978-0-19-875523-4 |page=20 |url=https://books.google.com/books?id=iewwDgAAQBAJ |access-date=April 15, 2022}}</ref>

== Properties ==
A googol is approximately equal to <math>70! \approx 1.1979 \times 10^{100}</math> ([[factorial]] of 70). Using an [[Integer (computer science)|integral]], [[binary numeral system]], one would need 333 bits to represent a googol, i.e., <math>10^{100} = 2^{(100/\mathrm{log}_{10}2)} \approx 2^{332.19280949}</math>. However, a googol is well within the maximum bounds of an IEEE 754 [[Double-precision floating-point format|double-precision floating point]] type  without full precision in the mantissa.

Using [[modular arithmetic]], the series of [[Modular arithmetic#Residue systems|residues]] (mod ''n'') of one googol, starting with mod 1, is as follows:
:0, 0, 1, 0, 0, 4, 4, 0, 1, 0, 1, 4, 3, 4, 10, 0, 4, 10, 9, 0, 4, 12, 13, 16, 0, 16, 10, 4, 16, 10, 5, 0, 1, 4, 25, 28, 10, 28, 16, 0, 1, 4, 31, 12, 10, 36, 27, 16, 11, 0, ... {{OEIS|id=A066298}}

This sequence is the same as that of the [[Googolplex#Mod n|residues (mod n)]] of a [[googolplex]] up until the 17th position.

== Cultural impact ==
Widespread sounding of the word occurs through the name of the company [[Google]], with the name "Google" being an accidental misspelling of "googol" by the company's founders,<ref>{{cite web |url=http://graphics.stanford.edu/~dk/google_name_origin.html |title=Origin of the name "Google" |last=Koller |first=David |date=January 2004 |publisher=Stanford University |access-date=July 4, 2012 |archive-url=https://web.archive.org/web/20120627081942/http://graphics.stanford.edu/~dk/google_name_origin.html |archive-date=June 27, 2012 |url-status=dead }}</ref> which was picked to signify that the search engine was intended to provide large quantities of information.<ref>{{cite web|url=https://www.google.com/company.html |title=Google! Beta website |publisher=Google, Inc. |archive-url=https://web.archive.org/web/19990221202430/http://www.google.com/company.html |archive-date=February 21, 1999 |access-date=October 12, 2010 |url-status=dead }}</ref> In 2004, family members of Kasner, who had inherited the right to his book, were considering suing Google for their use of the term "googol";<ref>{{cite web|title= Have your Google people talk to my 'googol' people|date= 16 May 2004|url= https://www.baltimoresun.com/2004/05/16/have-your-google-people-talk-to-my-googol-people/|url-status= live|archive-url= https://web.archive.org/web/20140904125042/http://articles.baltimoresun.com/2004-05-16/entertainment/0405150243_1_google-googol-internet-search-engine|archive-date= 2014-09-04}}</ref> however, no suit was ever filed.<ref>{{Cite book |last=Nowlan |first=Robert A. |title=Masters of Mathematics: The Problems They Solved, Why These Are Important, and What You Should Know about Them |publisher=Sense Publishers |year=2017 |isbn=978-9463008938 |location=Rotterdam |pages=221 |language=en}}</ref>

Since October 2009, Google has been assigning domain names to its servers under the domain "1e100.net", the scientific notation for 1 googol, in order to provide a single domain to identify servers across the Google network.<ref>{{cite web |url=https://www.theregister.co.uk/2010/02/08/google_mystery_domain/ |title=Google doppelgänger casts riddle over interwebs |date=8 February 2010 |access-date=30 December 2015 |publisher=The Register |author=Cade Metz |url-status=live |archive-url=https://web.archive.org/web/20160303180937/https://www.theregister.co.uk/2010/02/08/google_mystery_domain/ |archive-date=3 March 2016 }}</ref><ref>{{cite web |url=https://support.google.com/faqs/answer/174717?hl=en |title=What is 1e100.net? |access-date=30 December 2015 |publisher=Google Inc. |url-status=live |archive-url=https://web.archive.org/web/20160109065331/https://support.google.com/faqs/answer/174717?hl=en |archive-date=9 January 2016 }}</ref>

The word is notable for being the subject of the £1&nbsp;million question in a 2001 episode of the British quiz show ''[[Who Wants to Be a Millionaire? (UK game show)|Who Wants to Be a Millionaire?]]'', when contestant [[Charles Ingram]] was discovered to have [[Who Wants to Be a Millionaire? (British game show)#Charles Ingram cheating scandal|cheated his way through the show]] with the help of a confederate in the studio audience.<ref>{{citation|title=Television's Strangest Moments: Extraordinary But True Tales from the History of Television|first1=Quentin|last1=Falk|first2=Ben|last2=Falk|publisher=Franz Steiner Verlag|year=2005|isbn=9781861058744|url=https://books.google.com/books?id=4L7pA2tKumsC&pg=PA245|pages=245–246|contribution=A Code and a Cough: Who Wants to Be a Millionaire? (1998–)}}.</ref>

==See also==
* [[Googolplex]]
* [[Graham's number]]
* [[Skewes' number]]
* [[Infinity]]
* [[Names of large numbers]]

==References==
{{Reflist|30em}}

==External links==
{{Wiktionary}}
* {{MathWorld|urlname=Googol|title=Googol}}
* {{PlanetMath|urlname=Googol|title=Googol}}
* {{cite web|title=Googol and Googolplex|url=http://www.numberphile.com/videos/googolplex.html|work=Numberphile|publisher=[[Brady Haran]]|author=Padilla, Tony|author2=Symonds, Ria|access-date=2013-04-06|archive-url=https://web.archive.org/web/20140329024608/http://www.numberphile.com/videos/googolplex.html|archive-date=2014-03-29|url-status=dead}}

{{Large numbers}}

[[Category:Large integers]]
[[Category:Integers]]
[[Category:Units of amount]]
[[Category:1920s neologisms]]