A googolplex is the number 10googol, or equivalently, 1010100.
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". It thus became standardized to 1010100.
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. Although a googolplex is 10^10^100, the number of atoms in the known universe is 10^78.
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.
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.
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)
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
One googol is presumed to be greater than the number of hydrogen atoms in the observable universe, which has been variously estimated to be between 1079 and 1081. 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". And the end of the Universe via Big Freeze without proton decay is expected to be around 101075 years into the future.
- Edward Kasner & James R. Newman (1940) Mathematics and the Imagination, page 23, NY: Simon & Schuster
- Googolplex Written Out. 2013. ISBN 978-0-9900072-1-0.
- Page, Don, "How to Get a Googolplex", 3 June 2001.
- Mass, Size, and Density of the Universe Article from National Solar Observatory, 21 May 2001.
- 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)
- Weisstein, Eric W., "Googolplex", MathWorld.
- googolplex at PlanetMath.org.
- Padilla, Tony; Symonds, Ria. "Googol and Googolplex". Numberphile. Brady Haran.
- Googolplex written out