Googolplex

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Not to be confused with Googleplex.

A googolplex is the large number 1010100, that is, 10 raised to the power of a googol. If written out in ordinary decimal notation, it would be 1 followed by a googol (10100) zeroes – a physically impossible number to write explicitly.

History

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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), which is usually written as 1010100 using the conventional interpretation for serial exponentiation.[3]

Size

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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 the physical universe

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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

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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.

See also

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References

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  1. ^ Bialik, Carl (14 June 2004). "There Could Be No Google Without Edward Kasner". The Wall Street Journal Online. (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.
  4. ^ Stanford student Wolfgang Nitsche put together a website which will, provided a 94-digit volume number, generate a PDF file consisting of 106 zero digits (with an initial one digit in volume 1), and registered an ISBN for the set:
    Nitsche, Wolfgang (August 2013). Googolplex Written Out. Stanford, CA: Wolfgang Nitsche. ISBN 978-0-9900072-0-3.
  5. ^ Williams, David (2024), "Earth Fact Sheet", NASA Space Science Data Coordinated Archive, National Aeronautics and Space Administration, archived from the original on 21 August 2025, retrieved 15 November 2024
  6. ^ Letzter, Rafi (2019), "Our Large Adult Galaxy Is As Massive As 890 Billion Suns", Space.com
  7. ^ Alessandro Domenico De Angelis; Mário João Martins Pimenta; Ruben Conceição (2021). Particle and Astroparticle Physics: Problems and Solutions. Springer. p. 10. ISBN 978-3-030-73116-8.
  8. ^ Goodrich, Ryan (2013), "Googol, Googolplex - & Google", LiveScience.com
  9. ^ Saplakoglu, Yasemin (2018), "Photos: Large Numbers That Define the Universe", Space.com, retrieved 12 September 2025
  10. ^ Robert Munafo (2025). "Notable Properties of Specific Numbers". Robert Munafo's home pages. Retrieved 12 September 2025.
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Wikimedia Commons has media related to Googolplex.
  • The dictionary definition of googolplex at Wiktionary
  • Weisstein, Eric W. "Googolplex". MathWorld.
  • Haran, Brady. "Googol and Googolplex". Numberphile. Padilla, Tony; Symonds, Ria.