# Googol

A googol is the large number 10100 or ten to the power of one hundred. In decimal notation, it is written as the digit 1 followed by one hundred zeroes: 10,​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. Its systematic name is ten duotrigintillion (short scale) or ten sexdecilliard (long scale). Its prime factorization is ${\displaystyle 2^{100}\times 5^{100}.}$

## Etymology

The term was coined in 1920 by 9-year-old Milton Sirotta (1911–1981), nephew of American mathematician Edward Kasner.[1] He may have been inspired by the contemporary comic strip character Barney Google.[2] Kasner popularized the concept in his 1940 book Mathematics and the Imagination.[3] Other names for this quantity include ten duotrigintillion on the short scale (commonly used in English speaking countries),[4] ten thousand sexdecillion on the long scale, or ten sexdecilliard on the Peletier long scale.

## Size

A googol has no special significance in mathematics. However, it is useful when comparing with other very large quantities, such as the number of subatomic particles in the visible universe or the number of hypothetical possibilities in a chess game. Kasner used it to illustrate the difference between an unimaginably large number and infinity, and in this role it is sometimes used in teaching mathematics. To put in perspective the size of a googol, the mass of an electron, just under 10−30 kg, can be compared to the mass of the visible universe, estimated at between 1050 and 1060 kg.[5] It is a ratio in the order of about 1080 to 1090, or at most one ten-billionth of a googol (0.00000001% of a googol).

Another way of illustrating the immense size of a googol is to picture the Frontier supercomputer, which as of 2022 is the most powerful supercomputer in the world and measures 680 m2 (7,300 sq ft), almost exactly the same size of a basketball court with run-offs and sidelines.[6] The Frontier is capable of making 1,102,000 TFLOPs (1.1 quintillion calculations per second). If the supercomputer was shrunk down to the size of an atom (for reference, a typical grain of sand might have 37 quintillion atoms),[7] and if every atom in the observable universe (~1080 atoms total[8]) was as powerful as a Frontier supercomputer, it would take approximately 100 seconds of parallel computing to manually add up all the digits [clarification needed] like an adding machine (instead of using shorthand calculations).[dubiousdiscuss]

Carl Sagan pointed out that the total number of elementary particles in the universe is around 1080 (the Eddington number) and that if the whole universe were packed with neutrons so that there would be no empty space anywhere, there would be around 10128. He also noted the similarity of the second calculation to that of Archimedes in The Sand Reckoner. By Archimedes's calculation, the universe of Aristarchus (roughly 2 light years in diameter), if fully packed with sand, would contain 1063 grains. If the much larger observable universe of today were filled with sand, it would still only equal 1095 grains. Another 100,000 observable universes filled with sand would be necessary to make a googol.[9]

The decay time for a supermassive black hole of roughly 1 galaxy-mass (1011 solar masses) due to Hawking radiation is on the order of 10100 years.[10] Therefore, the heat death of an expanding universe is lower-bounded to occur at least one googol years in the future.

A googol is considerably smaller than a centillion.[11]

## Properties

A googol is approximately equal to ${\displaystyle 70!\approx 1.1979\times 10^{100}}$ (factorial of 70). Using an integral, binary numeral system, one would need 333 bits to represent a googol, i.e., ${\displaystyle 10^{100}=2^{(100/\mathrm {log} _{10}2)}\approx 2^{332.19280949}}$. However, a googol is well within the maximum bounds of an IEEE 754 double-precision floating point type without full precision in the mantissa.

Using modular arithmetic, the series of residues (mod n) of one googol, starting with mod 1, is as follows:

0, 0, 1, 0, 0, 4, 4, 0, 1, 0, 1, 4, 3, 4, 10, 0, 4, 10, 9, 0, 4, 12, 13, 16, 0, 16, 10, 4, 16, 10, 5, 0, 1, 4, 25, 28, 10, 28, 16, 0, 1, 4, 31, 12, 10, 36, 27, 16, 11, 0, ... (sequence A066298 in the OEIS)

This sequence is the same as that of the residues (mod n) of a googolplex up until the 17th position.

## Cultural impact

Widespread sounding of the word occurs through the name of the company Google, with the name "Google" being an accidental misspelling of "googol" by the company's founders,[12] which was picked to signify that the search engine was intended to provide large quantities of information.[13] In 2004, family members of Kasner, who had inherited the right to his book, were considering suing Google for their use of the term "googol";[14] however, no suit was ever filed.[15]

Since October 2009, Google has been assigning domain names to its servers under the domain "1e100.net", the scientific notation for 1 googol, in order to provide a single domain to identify servers across the Google network.[16][17]

The word is notable for being the subject of the £1 million question in a 2001 episode of the British quiz show Who Wants to Be a Millionaire?, when contestant Charles Ingram was discovered to have cheated his way through the show with the help of a confederate in the studio audience.[18]

## References

1. ^ Bialik, Carl (June 14, 2004). "There Could Be No Google Without Edward Kasner". The Wall Street Journal Online. Archived from the original on November 30, 2016.
2. ^ Ralph Keyes (2021). The Hidden History of Coined Words. Oxford University Press. p. 120. ISBN 978-0-19-046677-0. Extract of page 120
3. ^ Kasner, Edward; Newman, James R. (1940). Mathematics and the Imagination. Simon and Schuster, New York. ISBN 0-486-41703-4. Archived from the original on 2014-07-03. The relevant passage about the googol and googolplex, attributing both of these names to Kasner's nine-year-old nephew, is available in James R. Newman, ed. (2000) [1956]. The world of mathematics. Vol. 3. Mineola, New York: Dover Publications. pp. 2007–2010. ISBN 978-0-486-41151-4.
4. ^ Bromham, Lindell (2016). An Introduction to Molecular Evolution and Phylogenetics (2nd ed.). New York, NY: Oxford University Press. p. 494. ISBN 978-0-19-873636-3. Retrieved April 15, 2022.
5. ^ McPherson, Kristine (2006). Elert, Glenn (ed.). "Mass of the universe". The Physics Factbook. Retrieved 2019-08-24.
6. ^ "Basketball Court Dimensions & Markings | Harrod Sport". www.harrodsport.com. Retrieved 2022-09-14.
7. ^ Yongsheng, Zhong (2016-07-31). Chinese Classic Economics. Paths International. ISBN 978-1-84464-467-4.
8. ^ Villanueva, John Carl (2009-07-31). "How Many Atoms Are There in the Universe?". Universe Today. Retrieved 2022-09-14.
9. ^ Sagan, Carl (1981). Cosmos. Book Club Associates. pp. 220–221.
10. ^ Page, Don N. (1976-01-15). "Particle emission rates from a black hole: Massless particles from an uncharged, nonrotating hole". Physical Review D. 13 (2). American Physical Society (APS): 198–206. Bibcode:1976PhRvD..13..198P. doi:10.1103/physrevd.13.198. ISSN 0556-2821. See in particular equation (27).
11. ^ Stewart, Ian (2017). Infinity: A Very Short Introduction. New York, NY: Oxford University Press. p. 20. ISBN 978-0-19-875523-4. Retrieved April 15, 2022.
12. ^ Koller, David (January 2004). "Origin of the name "Google"". Stanford University. Archived from the original on June 27, 2012. Retrieved July 4, 2012.
13. ^ "Google! Beta website". Google, Inc. Archived from the original on February 21, 1999. Retrieved October 12, 2010.
14. ^ "Have your Google people talk to my 'googol' people". 16 May 2004. Archived from the original on 2014-09-04.
15. ^ Nowlan, Robert A. (2017). Masters of Mathematics: The Problems They Solved, Why These Are Important, and What You Should Know about Them. Rotterdam: Sense Publishers. p. 221. ISBN 978-9463008938.
16. ^ Cade Metz (8 February 2010). "Google doppelgänger casts riddle over interwebs". The Register. Archived from the original on 3 March 2016. Retrieved 30 December 2015.
17. ^ "What is 1e100.net?". Google Inc. Archived from the original on 9 January 2016. Retrieved 30 December 2015.
18. ^ Falk, Quentin; Falk, Ben (2005), "A Code and a Cough: Who Wants to Be a Millionaire? (1998–)", Television's Strangest Moments: Extraordinary But True Tales from the History of Television, Franz Steiner Verlag, pp. 245–246, ISBN 9781861058744.