Googolplex
{{short description|Number ten to the power of a googol}}{{Distinguish|Googleplex}}
{{Use dmy dates|date=September 2020}}
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A googolplex is the large number 10{{sup|googol}}, or equivalently, 10{{sup|10{{sup|100}}}} or {{not a typo|1010,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000}}. 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{{sup|googol}} ×5{{sup|googol}}.
==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".{{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) 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".Edward Kasner & James R. Newman (1940) Mathematics and the Imagination, page 23, NY: Simon & Schuster It thus became standardized to 10(10100) = 1010100, due to the right-associativity of exponentiation.{{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]
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).{{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://upload.wikimedia.org/wikipedia/commons/c/c4/Googolplex_Written_Out.pdf| archive-date=July 2, 2017}}
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,{{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}} the mass of the Milky Way galaxy is estimated at 1.8 × 10{{sup|42}} kilograms,{{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}} and the total mass of all the stars in the observable universe is estimated at 2 × 1052 kg.{{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]
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.[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.[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.
{{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.{{cite web
|author=Robert Munafo
|date=24 July 2013
|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
}}
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, ... {{OEIS|id=A067007}}
This sequence is the same as the sequence of residues (mod n) of a googol up until the 17th position.
See also
{{Portal|Mathematics}}
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}}