View source for Googolplex

{{short description|Number ten to the power of a googol}}{{Distinguish|Googleplex}}
{{Use dmy dates|date=September 2020}}
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<!--IMPORTANT: Please do not try to write out a googolplex in standard form in the article.-->
A '''googolplex''' is the [[large number]] {{math|10{{sup|10{{sup|100}}}}}}, that is, {{math|10}} [[Power of 10|raised to the power]] of a [[googol]]. If written out in ordinary [[decimal notation]], it would be {{math|1}} followed by a googol (10<sup>100</sup>) zeroes – a physically impossible number to write explicitly.

==History==   
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 }} (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>, which is usually written as 10<sup>10<sup>100</sup></sup> using the conventional interpretation for [[Order of operations#Serial exponentiation|serial exponentiation]].<ref>{{cite book |author1=Anthony J. Dos Reis |url=https://books.google.com/books?id=FFcTpMi3aKQC |title=Compiler Construction Using Java, JavaCC, and Yacc |publisher=John Wiley & Sons |year=2012 |isbn=978-1-118-11277-9 |edition= |page=91}} </ref>

==Size==
A typical book can be printed with one million zeros (around 400 pages with 50 lines per page and 50 zeros per line). Therefore, it requires {{10^|94}} such books to print all the zeros of a googolplex (that is, printing a googol zeros).<ref>Stanford student Wolfgang Nitsche put together a website which will, provided a 94-digit volume number, generate a PDF file consisting of {{10^|6}} zero digits (with an initial one digit in volume 1), and registered an ISBN for the set: {{pb}} {{cite book| last=Nitsche |first=Wolfgang |date=August 2013 |title=Googolplex Written Out |place=Stanford, CA |isbn=978-0-9900072-0-3 |url=http://www.googolplexwrittenout.com/ |publisher=Wolfgang Nitsche }}</ref>
If each book had a mass of 100 grams, all of them would have a total mass of {{10^|93}} kilograms. In comparison, [[Earth]]'s mass is {{Val|5.97e24}} kilograms,<ref>{{Citation| last=Williams| first=David| year=2024| title=Earth Fact Sheet |website=NASA Space Science Data Coordinated Archive | publisher= National Aeronautics and Space Administration |url=https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html |access-date=2024-11-15 |archive-url=https://web.archive.org/web/20250821225050/https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html |archive-date= 2025-08-21 }}</ref> the mass of the [[Milky Way]] galaxy is estimated at {{Val|1.8e42}} kilograms,<ref>{{Citation |last=Letzter |first=Rafi |year=2019 |title=Our Large Adult Galaxy Is As Massive As 890 Billion Suns |website=Space.com |url=https://www.space.com/our-galaxy-is-so-big-good-lord.html}}</ref> and the total mass of all the stars in the [[observable universe]] is estimated at {{Val|2e52|u=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 |year=2021 |isbn=978-3-030-73116-8 |page=10 |url=https://books.google.com/books?id=WXwwEAAAQBAJ}}</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 {{val|5e40}}.

===In the physical universe===
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>{{citation |last=Goodrich |first=Ryan |date=2013 |url=https://www.livescience.com/31981-googol.html |title=Googol, Googolplex - & Google |website=LiveScience.com }}</ref><ref>{{citation |url=https://www.space.com/41721-big-numbers-universe-photos/2.html |title=Photos: Large Numbers That Define the Universe |website=Space.com |last=Saplakoglu |first=Yasemin |date=2018 |access-date=2025-09-12}}</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
 |author=Robert Munafo
 |website=Robert Munafo's home pages
 |date=2025
 |title=Notable Properties of Specific Numbers
 |url=http://mrob.com/pub/math/numbers-19.html
 |access-date=2025-09-12
 }}</ref>

==Mod ''n''==
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 ==
{{Portal|Mathematics}}
* [[Graham's number]]
* [[Names of large numbers]]
* [[Orders of magnitude (numbers)]]
* [[Skewes's number]]

==References==
{{Reflist}}

==External links==
{{Commons}}
* {{Wiktionary-inline}}
* {{MathWorld | urlname=Googolplex | title=Googolplex}}
* {{cite web|title=Googol and Googolplex |url=https://www.numberphile.com/videos/googol-and-googolplex |work=Numberphile|last=Haran |first=Brady |author-link=Brady Haran |others=Padilla, Tony; Symonds, Ria}}

{{Large numbers}}

[[Category:Integers]]
[[Category:Large integers]]
[[Category:Units of amount]]
[[Category:Numbers]]
[[Category:Large numbers]]