Jump to content

Googol: Difference between revisions

From Wikipedia, the free encyclopedia
Content deleted Content added
No edit summary
No edit summary
 
(643 intermediate revisions by more than 100 users not shown)
Line 1: Line 1:
{{pp-vandalism|small=yes}}
{{distinguish|Google}}
{{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&nbsp;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 2<sup>100</sup>&nbsp;×&nbsp;5<sup>100</sup>.


==Etymology==
A '''googol''' is the [[large number]] 10<sup>100</sup>; that is, the [[numerical digit|digit]] 1 followed by 100 [[0 (number)|zeroes]]:
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 ==
:10,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000,&shy;000
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 10<sup>-30</sup>&nbsp;kg, can be compared to the mass of the visible universe, estimated at between 10<sup>50</sup> and 10<sup>60</sup>&nbsp;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 10<sup>80</sup> to 10<sup>90</sup>, 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 10<sup>80</sup> (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 10<sup>128</sup>. 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&nbsp;light years in diameter), if fully packed with sand, would contain 10<sup>63</sup>&nbsp;grains. If the much larger observable universe of today were filled with sand, it would still only equal {{val|e=95}}&nbsp;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 term was coined in 1938<ref>{{cite book| author = Kasner, Edward and Newman, James R. | title = [http://books.google.com/books?id=Ad8hAx-6m9oC&lpg=PP1&dq=Mathematics%20and%20the%20Imagination&pg=PP1#v=onepage&q&f=false Mathematics and the Imagination]| year = 1940| publisher = Simon and Schuster, New York| isbn = 0-486-41703-4}}</ref> by 9-year-old Milton Sirotta, nephew of American [[mathematician]] [[Edward Kasner]]. Kasner popularized the concept in his 1940 book ''[[Mathematics and the Imagination]]''. Other [[Names of large numbers|names]] for googol include '''ten duotrigintillion''' on the [[Long and short scales|short scale]], '''ten thousand sexdecillion''' on the [[Long and short scales|long scale]], or '''ten sexdecilliard''' on the [[Names of large numbers#Extensions of the standard dictionary numbers|Peletier long scale]].


The decay time for a supermassive [[black hole]] of roughly 1&nbsp;galaxy-mass (10<sup>11</sup>&nbsp;[[Solar mass|solar masses]]) due to [[Hawking radiation]] is on the order of 10<sup>100</sup>&nbsp;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&nbsp;years in the future.
A googol has no special significance in mathematics, but 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. Edward 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 give a sense of how big a googol really is, the mass of an electron, just under 1×10<sup>-30</sup>&nbsp;kg, can be compared to the mass of the visible universe, estimated at between 1×10<sup>50</sup>kg and 1×10<sup>60</sup>&nbsp;kg.<ref>{{cite web|author=Elert, Glenn et al.|title=Mass of the Universe |url=http://hypertextbook.com/facts/2006/KristineMcPherson.shtml}}</ref> It is a ratio in the order of about 10<sup>80</sup> to 10<sup>90</sup>, still much smaller than the value of a googol.


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>
A googol is approximately ''70!'' ([[factorial]] of 70). In the [[binary numeral system]], one would need 333 bits to represent a googol, i.e., 1 googol ≈ 2<sup>332.19281</sup>, or exactly <math>2^{(100/\mathrm{log}_{10}2)}</math>.


== Properties ==
Widespread sounding of the word occurs through its [[namesake]], the famous internet company [[Google]], with the name "Google" being an accidental [[spelling|misspelling]] of "googol" by the company's founders,<ref>[http://www.telegraph.co.uk/culture/qi/5157798/QI-Quite-Interesting-facts-about-100.html QI: Quite Interesting facts about 100], ''telegraph.co.uk''</ref> which was picked to signify that the search engine was intended to provide large quantities of information.<ref>{{cite web |url=http://www.google.com/company.html |title=Google! Beta website |publisher=Google, Inc. |archiveurl=https://web.archive.org/web/19990221202430/www.google.com/company.html |archivedate=February 2, 1999 |accessdate=October 12, 2010}}</ref>
A googol is approximately equal to <math>70!\approx1.1979\times10^{100}</math> ([[factorial]] of 70). Using an [[Integer (computer science)|integral]], [[binary numeral system]], one would need 333&nbsp;bits to represent a googol, i.e., <math>10^{100}=2^{(100/\mathrm{log}_{10}2)}\approx2^{332.19280949}</math>. However, a googol is well within the maximum bounds of an IEEE&nbsp;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&nbsp;''n'') of one&nbsp;googol, starting with mod&nbsp;1, is as follows:
The word is notable for being the subject of the £1 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]] [[Who Wants to Be a Millionaire? (UK game show)#Charles Ingram affair|cheated his way through the show]] with the help of an accomplice. {{Citation needed|date=August 2014}}

: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,&nbsp;... {{OEIS|id=A066298}}

This sequence is the same as that of the [[Googolplex#Mod n|residues (mod&nbsp;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&nbsp;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>


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


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


==External links==
==External links==
{{Wiktionary}}
* {{MathWorld | urlname=Googol | title=Googol}}
* {{PlanetMath | urlname=Googol | title=googol}}
* {{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}}
* {{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}}
{{Large numbers}}

{{num-stub}}
[[Category:Large integers]]
[[Category:Large integers]]
[[Category:Integers]]
[[Category:Integers]]
[[Category:Units of amount]]
[[Category:Units of amount]]
[[Category:1920s neologisms]]

Latest revision as of 19:24, 12 November 2024

A googol is the large number 10100 or ten to the power of one hundred. In decimal notation, it is written as the digit 1 followed by one hundred zeroes: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000. Its systematic name is ten duotrigintillion (short scale) or ten sexdecilliard (long scale). Its prime factorization is 2100 × 5100.

Etymology

The term was coined in 1920 by 9-year-old Milton Sirotta (1911–1981), nephew of American mathematician Edward Kasner.[1] He may have been inspired by the contemporary comic strip character Barney Google.[2] Kasner popularized the concept in his 1940 book Mathematics and the Imagination.[3] Other names for this quantity include ten duotrigintillion on the short scale (commonly used in English speaking countries),[4] 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 10-30 kg, can be compared to the mass of the visible universe, estimated at between 1050 and 1060 kg.[5] It is a ratio in the order of about 1080 to 1090, 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 1080 (the Eddington number) and that if the whole universe were packed with neutrons so that there would be no empty space anywhere, there would be around 10128. 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 (roughly 2 light years in diameter), if fully packed with sand, would contain 1063 grains. If the much larger observable universe of today were filled with sand, it would still only equal 1095 grains. Another 100,000 observable universes filled with sand would be necessary to make a googol.[6]

The decay time for a supermassive black hole of roughly 1 galaxy-mass (1011 solar masses) due to Hawking radiation is on the order of 10100 years.[7] Therefore, the heat death of an expanding universe is lower-bounded to occur at least one googol years in the future.

A googol is considerably smaller than a centillion.[8]

Properties

A googol is approximately equal to (factorial of 70). Using an integral, binary numeral system, one would need 333 bits to represent a googol, i.e., . However, a googol is well within the maximum bounds of an IEEE 754 double-precision floating point type without full precision in the mantissa.

Using modular arithmetic, the series of 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, ... (sequence A066298 in the OEIS)

This sequence is the same as that of the 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,[9] which was picked to signify that the search engine was intended to provide large quantities of information.[10] 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";[11] however, no suit was ever filed.[12]

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.[13][14]

The word is notable for being the subject of the £1 million question in a 2001 episode of the British quiz show Who Wants to Be a Millionaire?, when contestant Charles Ingram was discovered to have cheated his way through the show with the help of a confederate in the studio audience.[15]

See also

References

  1. ^ Bialik, Carl (June 14, 2004). "There Could Be No Google Without Edward Kasner". The Wall Street Journal Online. Archived from the original on November 30, 2016.
  2. ^ Ralph Keyes (2021). The Hidden History of Coined Words. Oxford University Press. p. 120. ISBN 978-0-19-046677-0. Extract of page 120
  3. ^ Kasner, Edward; Newman, James R. (1940). Mathematics and the Imagination. Simon and Schuster, New York. ISBN 0-486-41703-4. Archived from the original on 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 James R. Newman, ed. (2000) [1956]. The world of mathematics. Vol. 3. Mineola, New York: Dover Publications. pp. 2007–2010. ISBN 978-0-486-41151-4.
  4. ^ Bromham, Lindell (2016). An Introduction to Molecular Evolution and Phylogenetics (2nd ed.). New York, NY: Oxford University Press. p. 494. ISBN 978-0-19-873636-3. Retrieved April 15, 2022.
  5. ^ McPherson, Kristine (2006). Elert, Glenn (ed.). "Mass of the universe". The Physics Factbook. Retrieved 2019-08-24.
  6. ^ Sagan, Carl (1981). Cosmos. Book Club Associates. pp. 220–221.
  7. ^ Page, Don N. (1976-01-15). "Particle emission rates from a black hole: Massless particles from an uncharged, nonrotating hole". Physical Review D. 13 (2). American Physical Society (APS): 198–206. Bibcode:1976PhRvD..13..198P. doi:10.1103/physrevd.13.198. ISSN 0556-2821. See in particular equation (27).
  8. ^ Stewart, Ian (2017). Infinity: A Very Short Introduction. New York, NY: Oxford University Press. p. 20. ISBN 978-0-19-875523-4. Retrieved April 15, 2022.
  9. ^ Koller, David (January 2004). "Origin of the name "Google"". Stanford University. Archived from the original on June 27, 2012. Retrieved July 4, 2012.
  10. ^ "Google! Beta website". Google, Inc. Archived from the original on February 21, 1999. Retrieved October 12, 2010.
  11. ^ "Have your Google people talk to my 'googol' people". 16 May 2004. Archived from the original on 2014-09-04.
  12. ^ Nowlan, Robert A. (2017). Masters of Mathematics: The Problems They Solved, Why These Are Important, and What You Should Know about Them. Rotterdam: Sense Publishers. p. 221. ISBN 978-9463008938.
  13. ^ Cade Metz (8 February 2010). "Google doppelgänger casts riddle over interwebs". The Register. Archived from the original on 3 March 2016. Retrieved 30 December 2015.
  14. ^ "What is 1e100.net?". Google Inc. Archived from the original on 9 January 2016. Retrieved 30 December 2015.
  15. ^ Falk, Quentin; Falk, Ben (2005), "A Code and a Cough: Who Wants to Be a Millionaire? (1998–)", Television's Strangest Moments: Extraordinary But True Tales from the History of Television, Franz Steiner Verlag, pp. 245–246, ISBN 9781861058744.