Googolplex

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This article is about the large number. For the headquarters of Google Inc., see Googleplex. For the Canadian construction toy, see Googolplex (toy).

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

History[edit]

In 1938, 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". 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".[1] It thus became standardized to 10(10100).

Size[edit]

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 standard 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. While a googolplex has 1 + 10100 digits, the number of atoms in the known universe is only approximately 1078.[2]

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 zeros of googolplex.[3]

Writing the number takes too long: if a person can write two digits per second, then writing a googolplex would take around about 1.51×1092 years, which is about 1.1×1082 times the age of the universe.[4]

Mod n[edit]

The residue of googolplex (mod n) 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 OEIS)

Scale[edit]

In pure mathematics[edit]

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[edit]

One googol is presumed to be greater than the number of atoms in the observable universe, which has been estimated to be approximately 1078.[2] 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".[5] And 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.[4]

See also[edit]

References[edit]

  1. ^ Edward Kasner & James R. Newman (1940) Mathematics and the Imagination, page 23, NY: Simon & Schuster
  2. ^ a b Silk, Joseph (2005), On the Shores of the Unknown: A Short History of the Universe, Cambridge University Press, p. 10, ISBN 9780521836272 .
  3. ^ Googolplex Written Out. 2013. ISBN 978-0-9900072-1-0. 
  4. ^ a b Page, Don, "How to Get a Googolplex", 3 June 2001.
  5. ^ 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)

External links[edit]