Googolplex

A googolplex is the number 10googol, or equivalently, 10(10100). Written out in ordinary decimal notation, it is 1 followed by 10100 zeroes, that is, a 1 followed by a googol zeroes.

This article is about the large number. For the headquarters of Google LLC, see Googleplex.

History

In 1920, Edward Kasner's nine-year-old nephew, Milton Sirotta, coined the term googol, which is 10100, 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 a better mathematician than Dr. Einstein, simply because he had more endurance and could write for longer".[2] It thus became standardized to 1010100.

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 x 1024 kilograms, the mass of the Milky Way Galaxy is estimated at 2.5 x 1042 kilograms, and the mass of matter in the observable universe is estimated at 1.5 x 1053 kg.

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 x 1050), and greater than the mass of the observable universe by a factor of roughly 7 x 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.

One googol is presumed to be greater than the number of atoms in the observable universe, which has been estimated to be approximately 1078.[3] Thus, in the physical world, it is difficult to give examples of numbers that compare to the vastly greater googolplex. However, in analyzing quantum states and black holes, physicist Don Page writes that "determining experimentally whether or not information is lost down black holes of solar mass ... would require more than 101076.96 measurements to give a rough determination of the final density matrix after a black hole evaporates".[4] The end of the universe via Big Freeze without proton decay is expected to be around 101075 years into the future.

In a separate article, Page shows that the number of states in a black hole with a mass roughly equivalent to the Andromeda Galaxy is in the range of a googolplex.[5]

Writing the number would take an immense amount of time: if a person can write two digits per second, then writing a googolplex would take about 1.51×1092 years, which is about 1.1×1082 times the accepted age of the universe.[5]

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.[6]

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 March 17, 2015)
  2. Edward Kasner & James R. Newman (1940) Mathematics and the Imagination, page 23, NY: Simon & Schuster
  3. Silk, Joseph (2005), On the Shores of the Unknown: A Short History of the Universe, Cambridge University Press, p. 10, ISBN 9780521836272.
  4. 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)
  5. Page, Don, "How to Get a Googolplex" Archived 6 November 2006 at the Wayback Machine, 3 June 2001.
  6. Robert Munafo (24 July 2013). "Notable Properties of Specific Numbers". Retrieved 28 August 2013.

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