Googol

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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 zeros: 10,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000,Template:Wbr000. Its systematic name is ten duotrigintillion (short scale) or ten sexdecilliard (long scale). Its prime factorization is 2100 × 5100.

Etymology

The term was coined in 1920 by nine-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).

Carl Sagan pointed out that the total number of elementary particles in the observable 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. The number would have to be 100,000 times bigger to make a googol.[6]

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.[7] 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.[8]

Properties

A googol is approximately equal to 70!≈1.1979×10100 (factorial of 70). Using an integral, binary numeral system, one would need 333 bits to represent a googol, i.e., 10100=2(100/log102)≈2332.19280949. 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

Googol is a homophone of the company name Google, an accidental misspelling of "googol" by the company's founders;[9] it suggests that the search engine provides large quantities of information.[10] In 2004, Kasner's heirs considered suing Google over their use of "googol";[11] however, no suit was ever filed.[12]

Since October 2009, Google has used the domain "1e100.net", "1e100" being E notation for 1 googol, to identify servers across its network.[13][14]

"Googol" was the £1 million answer in a 2001 episode of the British Who Wants to Be a Millionaire?, which the contestant allegedly won by cheating.[15]

See also

References

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  1. ^ Page Module:Citation/CS1/styles.css has no content.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. ^ Page Module:Citation/CS1/styles.css has no content.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. ^ Page Module:Citation/CS1/styles.css has no content.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. {{cite book}}: ISBN / Date incompatibility (help) The relevant passage about the googol and googolplex, attributing both of these names to Kasner's nine-year-old nephew, is available in Page Module:Citation/CS1/styles.css has no content.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. ^ Page Module:Citation/CS1/styles.css has no content.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. ^ Page Module:Citation/CS1/styles.css has no content.McPherson, Kristine (2006). Elert, Glenn (ed.). "Mass of the universe". The Physics Factbook. Retrieved 2019-08-24.
  6. ^ Page Module:Citation/CS1/styles.css has no content.Sagan, Carl (1981). Cosmos. Book Club Associates. pp. 220–221.
  7. ^ Page Module:Citation/CS1/styles.css has no content.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).
  8. ^ Page Module:Citation/CS1/styles.css has no content.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.
  9. ^ Page Module:Citation/CS1/styles.css has no content.Koller, David (January 2004). "Origin of the name "Google"". Stanford University. Archived from the original on June 27, 2012. Retrieved July 4, 2012.
  10. ^ Page Module:Citation/CS1/styles.css has no content."Google! Beta website". Google, Inc. Archived from the original on February 21, 1999. Retrieved October 12, 2010.
  11. ^ Page Module:Citation/CS1/styles.css has no content."Have your Google people talk to my 'googol' people". 16 May 2004. Archived from the original on 2014-09-04.
  12. ^ Page Module:Citation/CS1/styles.css has no content.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.
  13. ^ Page Module:Citation/CS1/styles.css has no content.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.
  14. ^ Page Module:Citation/CS1/styles.css has no content."What is 1e100.net?". Google Inc. Archived from the original on 9 January 2016. Retrieved 30 December 2015.
  15. ^ Page Module:Citation/CS1/styles.css has no content.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.

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