A googolplex is the large number 10googol, or equivalently, 1010100 or 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 2googol ×5googol.

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) = 1010100, 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). 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 5.972 × 1024 kilograms, the mass of the Milky Way galaxy is estimated at 2.5 × 1042 kilograms, and the total mass of all the stars in the observable universe is estimated at 2 × 1052 kg.[4]

To put this in perspective, the mass of all such books required to write out a googolplex would be vastly greater than the masses of the Milky Way and the Andromeda galaxies combined (by a factor of roughly 2.0 × 1050), and greater than the mass of the observable universe by a factor of roughly 7 × 1039.

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.[5][6]

Writing the number would ultimately hasten the heat death of the universe: if a person can write two digits per second, then writing a googolplex would take about 1.58×1092 years, which is about 1.1×1082 times the accepted age of the universe, and each digit written would result in an increase of entropy by the second law of thermodynamics.[7]

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.[8]

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.

See also

References

  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. ↑ 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
  5. ↑ "Googol, Googolplex - & Google" - LiveScience.com Archived 26 July 2020 at the Wayback Machine 8 August 2020.
  6. ↑ "Large Numbers That Define the Universe" - Space.com Archived 2 November 2019 at the Wayback Machine 8 August 2020.
  7. ↑ Page, Don, "How to Get a Googolplex" Archived 6 November 2006 at the Wayback Machine, 3 June 2001.
  8. ↑ 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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