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A googolplex is the large number 10^googol, or equivalently, 10^10¹⁰⁰ or 10^{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}. Written out in ordinary decimal notation, it is 1 followed by 10¹⁰⁰ zeroes; that is, a 1 followed by a googol of zeroes. Its prime factorization is 2^googol ×5^googol.

History

In 1920, Edward Kasner's nine-year-old nephew, Milton Sirotta, coined the term googol, which is 10¹⁰⁰, and then proposed the further term googolplex to be "one, followed by writing zeroes until you get tired".^[1] Kasner decided to adopt a more formal definition because "different people get tired at different times and it would never do to have Carnera [be] a better mathematician than Dr. Einstein, simply because he had more endurance and could write for longer".^[2] It thus became standardized to 10^(10¹⁰⁰) = 10^10¹⁰⁰, due to the right-associativity of exponentiation.^[3]

Size

A typical book can be printed with 10⁶ zeros (around 400 pages with 50 lines per page and 50 zeros per line). Therefore, it requires 10⁹⁴ such books to print all the zeros of a googolplex (that is, printing a googol zeros).^[4] If each book had a mass of 100 grams, all of them would have a total mass of 10⁹³ kilograms. In comparison, Earth's mass is 5.97 × 10²⁴ kilograms^[5], the mass of the Milky Way galaxy is estimated at 1.8 × 10⁴² kilograms^[6], and the total mass of all the stars in the observable universe is estimated at 2 × 10⁵² kg.^[7]

To put this in perspective, the mass of all such books required to write out a googolplex would be vastly greater than the mass of the observable universe by a factor of roughly 5 × 10⁴⁰.

In pure mathematics

In pure mathematics, there are several notational methods for representing large numbers by which the magnitude of a googolplex could be represented, such as tetration, hyperoperation, Knuth's up-arrow notation, Steinhaus–Moser notation, or Conway chained arrow notation.

In the physical universe

In the PBS science program Cosmos: A Personal Voyage, Episode 9: "The Lives of the Stars", astronomer and television personality Carl Sagan estimated that writing a googolplex in full decimal form (i.e., "10,000,000,000...") would be physically impossible, since doing so would require more space than is available in the known universe. Sagan gave an example that if the entire volume of the observable universe is filled with fine dust particles roughly 1.5 micrometers in size (0.0015 millimeters), then the number of different combinations in which the particles could be arranged and numbered would be about one googolplex.^[8]^[9]

10⁹⁷ is a high estimate of the elementary particles existing in the visible universe (not including dark matter), mostly photons and other massless force carriers.^[10]

Mod n

The residues (mod n) of a googolplex, starting with mod 1, are:

0, 0, 1, 0, 0, 4, 4, 0, 1, 0, 1, 4, 3, 4, 10, 0, 1, 10, 9, 0, 4, 12, 13, 16, 0, 16, 10, 4, 24, 10, 5, 0, 1, 18, 25, 28, 10, 28, 16, 0, 1, 4, 24, 12, 10, 36, 9, 16, 4, 0, ... (sequence A067007 in the OEIS)

This sequence is the same as the sequence of residues (mod n) of a googol up until the 17th position.

References

^ Bialik, Carl (14 June 2004). "There Could Be No Google Without Edward Kasner". The Wall Street Journal Online. Archived from the original on 30 November 2016. (retrieved 17 March 2015)
^ Edward Kasner & James R. Newman (1940) Mathematics and the Imagination, page 23, NY: Simon & Schuster
^ Anthony J. Dos Reis (2012). Compiler Construction Using Java, JavaCC, and Yacc. John Wiley & Sons. p. 91. ISBN 978-1-118-11277-9. Extract of page 91
^ Nitsche, Wolfgang (August 2013). Googolplex Written Out (PDF). Stanford, CA, USA. ISBN 978-0-9900072-1-0. Archived from the original on 2 July 2017.{{cite book}}: CS1 maint: location missing publisher (link)
^ Williams, David (2024), Earth Fact Sheet, Greenbelt, MD, USA: NASA, archived from the original on 12 January 2024
^ Letzter, Rafi (2019), Our Large Adult Galaxy Is As Massive As 890 Billion Suns, New York, NY, USA, archived from the original on 21 October 2021{{citation}}: CS1 maint: location missing publisher (link)
^ Alessandro Domenico De Angelis; Mário João Martins Pimenta; Ruben Conceição (2021). Particle and Astroparticle Physics: Problems and Solutions. Springer Nature. p. 10. ISBN 978-3-030-73116-8. Extract of page 10
^ "Googol, Googolplex - & Google" - LiveScience.com Archived 26 July 2020 at the Wayback Machine 8 August 2020.
^ "Large Numbers That Define the Universe" - Space.com Archived 2 November 2019 at the Wayback Machine 8 August 2020.
^ Robert Munafo (24 July 2013). "Notable Properties of Specific Numbers". Archived from the original on 6 October 2020. Retrieved 28 August 2013.

External links

The dictionary definition of googolplex at Wiktionary
Weisstein, Eric W. "Googolplex". MathWorld.
googolplex at PlanetMath.
Padilla, Tony; Symonds, Ria. "Googol and Googolplex". Numberphile. Brady Haran. Archived from the original on 29 March 2014. Retrieved 6 April 2013.

[1] Bialik, Carl (14 June 2004). "There Could Be No Google Without Edward Kasner". The Wall Street Journal Online. Archived from the original on 30 November 2016. (retrieved 17 March 2015)

[2] Edward Kasner & James R. Newman (1940) Mathematics and the Imagination, page 23, NY: Simon & Schuster

[3] Anthony J. Dos Reis (2012). Compiler Construction Using Java, JavaCC, and Yacc. John Wiley & Sons. p. 91. ISBN 978-1-118-11277-9. Extract of page 91

[4] Nitsche, Wolfgang (August 2013). Googolplex Written Out (PDF). Stanford, CA, USA. ISBN 978-0-9900072-1-0. Archived from the original on 2 July 2017.{{cite book}}: CS1 maint: location missing publisher (link)

[5] Williams, David (2024), Earth Fact Sheet, Greenbelt, MD, USA: NASA, archived from the original on 12 January 2024

[6] Letzter, Rafi (2019), Our Large Adult Galaxy Is As Massive As 890 Billion Suns, New York, NY, USA, archived from the original on 21 October 2021{{citation}}: CS1 maint: location missing publisher (link)

[7] Alessandro Domenico De Angelis; Mário João Martins Pimenta; Ruben Conceição (2021). Particle and Astroparticle Physics: Problems and Solutions. Springer Nature. p. 10. ISBN 978-3-030-73116-8. Extract of page 10

[8] "Googol, Googolplex - & Google" - LiveScience.com Archived 26 July 2020 at the Wayback Machine 8 August 2020.

[9] "Large Numbers That Define the Universe" - Space.com Archived 2 November 2019 at the Wayback Machine 8 August 2020.

[10] Robert Munafo (24 July 2013). "Notable Properties of Specific Numbers". Archived from the original on 6 October 2020. Retrieved 28 August 2013.

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@@ Line 1: / Line 1: @@
-{{two other uses|the number|the headquarters of [[Google Inc.]]|Googleplex||googol (disambiguation)}}
+{{short description|Number ten to the power of a googol}}{{Distinguish|Googleplex}}
+{{Use dmy dates|date=September 2020}}
-A '''googolplex''' is the number 10<sup>[[googol]]</sup>, which can also be written as the number 1 followed by a [[googol]] of [[0 (number)|zeros]] (i.e., 10<sup>100</sup> zeros).
+{{pp|small=yes}}
+<!--IMPORTANT: Please do not try to write out a googolplex in standard form in the article.-->
+A '''googolplex''' is the [[large number]] '''10{{sup|[[googol]]}}''', or equivalently, '''10{{sup|10{{sup|100}}}}''' or '''{{not a typo|10<sup>10,000,000,000,&ZeroWidthSpace;000,000,000,&ZeroWidthSpace;000,000,000,&ZeroWidthSpace;000,000,000,&ZeroWidthSpace;000,000,000,&ZeroWidthSpace;000,000,000,&ZeroWidthSpace;000,000,000,&ZeroWidthSpace;000,000,000,&ZeroWidthSpace;000,000,000,&ZeroWidthSpace;000,000,000,&ZeroWidthSpace;000,000,000</sup>}}'''. Written out in ordinary [[decimal notation]], it is 1 followed by 10<sup>100</sup> zeroes; that is, a 1 followed by a [[googol]] of zeroes. Its prime factorization is 2{{sup|googol}}&nbsp;×5{{sup|googol}}.
+==History==
-:1 googolplex
+In 1920, [[Edward Kasner]]'s nine-year-old nephew, Milton Sirotta, coined the term ''[[googol]]'', which is 10{{sup|100}}, and then proposed the further term ''googolplex'' to be "one, followed by writing zeroes until you get tired".<ref>{{cite journal  | title = There Could Be No Google Without Edward Kasner  | first = Carl  | last = Bialik  | journal = The Wall Street Journal Online  | date = 14 June 2004  | url = https://www.wsj.com/articles/SB108575924921724042  | url-status = live  | archive-url = https://web.archive.org/web/20161130145858/http://www.wsj.com/articles/SB108575924921724042  | archive-date = 30 November 2016 }} (retrieved 17 March 2015)</ref> Kasner decided to adopt a more formal definition because "different people get tired at different times and it would never do to have [[Primo Carnera|Carnera]] [be] a better mathematician than [[Albert Einstein|Dr. Einstein]], simply because he had more endurance and could write for longer".<ref>Edward Kasner & James R. Newman (1940) [[Mathematics and the Imagination]], page 23, NY: [[Simon & Schuster]]</ref> It thus became standardized to 10<sup>(10<sup>100</sup>)</sup> = 10<sup>10<sup>100</sup></sup>, due to the [[associativity|right-associativity]] of [[exponentiation]].<ref>{{cite book |title=Compiler Construction Using Java, JavaCC, and Yacc |author1=Anthony J. Dos Reis |edition= |publisher=John Wiley & Sons |year=2012 |isbn=978-1-118-11277-9 |page=91 |url=https://books.google.com/books?id=FFcTpMi3aKQC}} [https://books.google.com/books?id=FFcTpMi3aKQC&pg=PA91 Extract of page 91]</ref>
-:= 10<sup>[[googol]]</sup>
-:= 10<sup>(10<sup>100</sup>)</sup>
-:= 10<sup>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</sup>
-:= 1.e+100
-==History==
-In 1938, [[Edward Kasner]]'s nine-year-old nephew Milton Sirotta coined the term ''[[googol]]''; Milton then proposed the further term ''googolplex'' to be "one, followed by writing zeroes until you get tired". Kasner decided to adopt a more formal definition "because different people get tired at different times and it would never do to have [[Primo Carnera|Carnera]] be a better mathematician than [[Albert Einstein|Dr. Einstein]], simply because he had more endurance and could write for longer.".<ref>{{cite book | last = Kasner| first = Edward | authorlink = Edward Kasner| year = 2001 | title = Mathematics and the imagination | publisher = Dover Publications | location = Mineola, NY}}</ref>
-It thus became standardized to  10<sup>googol</sup>
 ==Size==
+A typical book can be printed with 10{{sup|6}} zeros (around 400 pages with 50 lines per page and 50 zeros per line). Therefore, it requires 10{{sup|94}} such books to print all the zeros of a googolplex (that is, printing a googol zeros).<ref>{{cite book| last=Nitsche| first=Wolfgang | date=August 2013| title=Googolplex Written Out| publication-place =Stanford, CA, USA| isbn=978-0-9900072-1-0| url=http://www.GoogolplexWrittenOut.com| archive-url=https:/upwiki/wikipedia/commons/c/c4/Googolplex_Written_Out.pdf| archive-date=July 2, 2017}}</ref>
-In the [[PBS]] science program ''[[Cosmos: A Personal Voyage]]'', Episode 9: "The Lives of the Stars", [[astronomer]] and television personality [[Carl Sagan]] estimated that writing a googolplex in numerals (i.e., "10,000,000,000...") would be physically impossible, since doing so would require more space than the known universe occupies.
+If each book had a mass of 100 grams, all of them would have a total mass of 10{{sup|93}} kilograms. In comparison, [[Earth]]'s mass is 5.97 × 10{{sup|24}} kilograms<ref>{{Citation| last=Williams| first=David| year=2024| title=Earth Fact Sheet| publisher =NASA | publication-place =Greenbelt, MD, USA| url=https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html| archive-url=https://web.archive.org/web/20240112185637/https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html|archive-date=January 12, 2024}}</ref>, the mass of the [[Milky Way]] galaxy is estimated at 1.8 × 10{{sup|42}} kilograms<ref>{{Citation| last=Letzter| first=Rafi| year=2019| title=Our Large Adult Galaxy Is As Massive As 890 Billion Suns| publication-place =New York, NY, USA| url =https://www.space.com/our-galaxy-is-so-big-good-lord.html| archive-url  =https://web.archive.org/web/20211021183324/https://www.space.com/our-galaxy-is-so-big-good-lord.html| archive-date=October 21, 2021}}</ref>, and the total mass of all the stars in the [[observable universe]] is estimated at 2 × 10<sup>52</sup> kg.<ref>{{cite book |title=Particle and Astroparticle Physics: Problems and Solutions |author1=Alessandro Domenico De Angelis |author2=Mário João Martins Pimenta |author3=Ruben Conceição |edition= |publisher=Springer Nature |year=2021 |isbn=978-3-030-73116-8 |page=10 |url=https://books.google.com/books?id=WXwwEAAAQBAJ}} [https://books.google.com/books?id=WXwwEAAAQBAJ&pg=PA10 Extract of page 10]</ref>
+To put this in perspective, the mass of all such books required to write out a googolplex would be vastly greater than the mass of the observable universe by a factor of roughly 5 × 10<sup>40</sup>.
-An average book of 60 cubic inches can be printed with 5 x 10<sup>5</sup> '0's (5 characters per word, 10 words per line, 25 lines per page, 400 pages), or 8.3&nbsp;×&nbsp;10<sup>3</sup> '0's per cubic inch.  The ''observable'' (i.e. past light cone) universe contains 6&nbsp;×&nbsp;10<sup>83</sup> cubic inches (1.3&nbsp;×&nbsp;''π''&nbsp;×&nbsp;(14&nbsp;×&nbsp;10<sup>9</sup> light year in inches)<sup>3</sup>). This implies that if the universe is stuffed with paper printed with '0's, it could contain only 5.3&nbsp;×&nbsp;10<sup>87</sup> '0's&mdash;far short of a googol of '0's. In fact there are only about 2.5 × 10<sup>89</sup> [[elementary particle]]s in the [[observable universe]] so even if you used an elementary particle to represent each digit you still would have to make the universe's mass about a [[Orders of magnitude (numbers)#1012|trillion]] times larger. Therefore a googolplex can not be written out since a googol of '0's can not fit into the observable universe.
+=== In pure mathematics ===
-The time it would take to write such a number also renders the task implausible: if a person can write two digits per second, it would take around about 1.51&nbsp;×&nbsp;10<sup>92</sup> years, which is 1.1&nbsp;×&nbsp;10<sup>82</sup> times the [[age of the universe]], to write a googolplex. <ref name="fpx.de">[http://www.fpx.de/fp/Fun/Googolplex/GetAGoogol.html Page, Don, "How to Get a Googolplex"], 3 June 2001.</ref>
+In [[pure mathematics]], there are several notational methods for representing [[large numbers]] by which the [[Magnitude (mathematics)|magnitude]] of a googolplex could be represented, such as [[tetration]], [[hyperoperation]], [[Knuth's up-arrow notation]], [[Steinhaus–Moser notation]], or [[Conway chained arrow notation]].
+===In the physical universe===
-Thinking of this another way, consider printing the digits of a googolplex in unreadable, [[point (typography)|one-point font]].  [[TeX]] one-point font is 0.35145989&nbsp;mm per digit,<ref>[http://www.cl.cam.ac.uk/~mgk25/metric-typo/ Metric typographic units] 23 February 2003.</ref> so it would take about 3.5&nbsp;×&nbsp;10<sup>96</sup> meters to write a googolplex in one-point font. The observable universe is estimated to be 8.80&nbsp;×&nbsp;10<sup>26</sup> meters, or 93 billion [[light-years]], in [[diameter]],<ref name=ly93>{{cite web | last = Lineweaver | first = Charles | coauthors = Tamara M. Davis | year = 2005 | url = http://www.sciam.com/article.cfm?id=misconceptions-about-the-2005-03&page=5 | title = Misconceptions about the Big Bang | publisher = [[Scientific American]] | accessdate = 2008-11-06}}</ref> so the distance required to write the necessary zeroes is longer than the estimated universe; however, text wrapping would make this task possible.
+In the [[Public Broadcasting Service|PBS]] science program ''[[Cosmos: A Personal Voyage]]'', [[Cosmos: A Personal Voyage#Episode 9: "The Lives of the Stars"|Episode 9: "The Lives of the Stars"]], [[astronomer]] and television personality [[Carl Sagan]] estimated that writing a googolplex in full decimal form (i.e., "10,000,000,000...") would be physically impossible, since doing so would require more space than is available in the known universe. Sagan gave an example that if the entire volume of the [[observable universe]] is filled with fine [[Cosmic dust|dust particles]] roughly 1.5 micrometers in size (0.0015 millimeters), then the number of different [[combinations]] in which the particles could be arranged and numbered would be about one googolplex.<ref>[https://www.livescience.com/31981-googol.html "Googol, Googolplex - & Google" - LiveScience.com] {{Webarchive|url=https://web.archive.org/web/20200726041136/https://www.livescience.com/31981-googol.html |date=26 July 2020 }} 8 August 2020.</ref><ref>[https://www.space.com/41721-big-numbers-universe-photos/2.html "Large Numbers That Define the Universe" - Space.com] {{Webarchive|url=https://web.archive.org/web/20191102034608/https://www.space.com/41721-big-numbers-universe-photos/2.html |date=2 November 2019 }} 8 August 2020.</ref>
+{{10^|97}} is a high estimate of the elementary particles existing in the visible universe (not including [[dark matter]]), mostly photons and other massless force carriers.<ref>{{cite web
-One [[googol]] is also presumed to be greater than the number of [[hydrogen]] atoms in the [[observable universe]], which has been variously estimated to be between 10<sup>79 </sup> and 10<sup>81</sup>.<ref>[http://www.cs.umass.edu/~immerman/stanford/universe.html Mass, Size, and Density of the Universe] Article from National Solar Observatory, 21 May 2001.</ref> A googol is also greater than the number of [[Planck time]]s elapsed since the [[Big Bang]], which is estimated at about 8&nbsp;×&nbsp;10<sup>60</sup>.{{Fact|date=December 2008}}
+ |author=Robert Munafo
+ |date=24 July 2013
-Thus in the physical world it is difficult to give examples of numbers that compare closely to a googolplex. In analyzing [[quantum state]]s and [[black hole]]s, physicist Don Page writes that "determining experimentally whether or not information is lost down black holes of solar mass ... would require more than 10<sup>10<sup>76.96</sup></sup> measurements to give a rough determination of the final density matrix after a black hole evaporates".<ref>[http://arxiv.org/pdf/hep-th/9411193 Page, Don N., "Information Loss in Black Holes and/or Conscious Beings?"], 25 Nov. 1994, for publication in ''Heat Kernel Techniques and Quantum Gravity'', S. A. Fulling, ed. (Discourses in Mathematics and Its Applications, No. 4, Texas A&M University, Department of Mathematics, College Station, Texas, 1995)</ref>
+ |title=Notable Properties of Specific Numbers
+ |url=http://mrob.com/pub/math/numbers-19.html
+ |access-date=2013-08-28
+ |archive-date=6 October 2020
+ |archive-url=https://web.archive.org/web/20201006200300/http://mrob.com/pub/math/numbers-19.html
+ |url-status=live
+ }}</ref>
+==Mod ''n''==
-In a separate article, Page shows that the number of [[State function|states]] in a black hole with a mass roughly equivalent to the [[Andromeda Galaxy]] is in the range of a googolplex.<ref name="fpx.de"/>
+The [[Modular arithmetic|residues (mod ''n'')]] of a googolplex, starting with mod 1, are:
+:0, 0, 1, 0, 0, 4, 4, 0, 1, 0, 1, 4, 3, 4, 10, 0, 1, 10, 9, 0, 4, 12, 13, 16, 0, 16, 10, 4, 24, 10, 5, 0, 1, 18, 25, 28, 10, 28, 16, 0, 1, 4, 24, 12, 10, 36, 9, 16, 4, 0, ... {{OEIS|id=A067007}}
+This sequence is the same as the sequence of [[Googol#Properties|residues (mod ''n'') of a googol]] up until the 17th position.
+== See also ==
-In [[pure mathematics]], the [[Magnitude (mathematics)|magnitude]] of a googolplex could be related to other forms of large number notation such as [[tetration]], [[Knuth's up-arrow notation]], [[Steinhaus-Moser notation]], or [[Conway chained arrow notation]].
+{{Portal|Mathematics}}
+* [[Graham's number]]
+* [[Names of large numbers]]
+* [[Orders of magnitude (numbers)]]
+* [[Skewes's number]]
+==References==
-Some [[sequence]]s grow very quickly; for instance, the first two [[Ackermann function#Ackermann numbers|Ackermann numbers]] are 1 and 2<sup>2</sup>&nbsp;=&nbsp;4; but then the third is  a  [[tetration|power tower]] of threes more than seven trillion high.
+{{Reflist}}
-Yet, much larger still is [[Graham's number]], perhaps the largest [[natural number]] mathematicians actually ever talk about.
-==See also==
-{{portal|Mathematics|Nuvola_apps_edu_mathematics_blue-p.svg}}
-*[[Large numbers]]
-*[[Names of large numbers]]
-== References ==
-{{reflist}}
 ==External links==
+{{Commons}}
+* {{Wiktionary-inline}}
 * {{MathWorld | urlname=Googolplex | title=Googolplex}}
 * {{PlanetMath | urlname=Googolplex | title=googolplex}}
+* {{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=6 April 2013|archive-url=https://web.archive.org/web/20140329024608/http://www.numberphile.com/videos/googolplex.html|archive-date=29 March 2014|url-status=dead}}
 {{Large numbers}}
@@ Line 50: / Line 57: @@
 [[Category:Integers]]
 [[Category:Large integers]]
+[[Category:Units of amount]]
-[[ca:Googolplex]]
+[[Category:Numbers]]
+[[Category:Large numbers]]
-[[cs:Googolplex]]
-[[da:Googolplex]]
-[[de:Googol#Googolplex]]
-[[es:Googolplex]]
-[[eo:Gugloplekso]]
-[[eu:Googolplex]]
-[[fr:Gogolplex]]
-[[ko:구골플렉스]]
-[[id:Googolplex]]
-[[is:Googolplex]]
-[[it:Googolplex]]
-[[lv:Gugolplekss]]
-[[hu:Googolplex]]
-[[nl:Googolplex]]
-[[ja:グーゴルプレックス]]
-[[no:Googolplex]]
-[[nn:Googolplex]]
-[[pl:Googolplex]]
-[[pt:Googolplex]]
-[[ro:Googolplex]]
-[[ru:Гуголплекс]]
-[[simple:Googolplex]]
-[[sk:Googolplex]]
-[[fi:Googolplex]]
-[[sv:Googolplex]]
-[[th:กูกอลเพลกซ์]]
-[[tr:Googolplex]]
-[[zh-yue:Googolplex]]
-[[zh:古戈爾普勒克斯]]