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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: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000. Its systematic name is ten duotrigintillion ( short scale) or ten sexdecilliard ( long scale). Its
prime factorization In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a comp ...
is 2100 × 5100.


Etymology

The term was coined in 1920 by 9-year-old Milton Sirotta (1911–1981), nephew of American mathematician Edward Kasner. He may have been inspired by the contemporary comic strip character Barney Google. Kasner popularized the concept in his 1940 book '' Mathematics and the Imagination''. Other names for this quantity include ''ten duotrigintillion'' on the short scale (commonly used in English speaking countries), ''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 Chess is a board game for two players. It is an abstract strategy game that involves Perfect information, no hidden information and no elements of game of chance, chance. It is played on a square chessboard, board consisting of 64 squares arran ...
game. Kasner used it to illustrate the difference between an unimaginably large number and
infinity Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
, 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. 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 universe is around 1080 (the Eddington number) and that if the whole universe were packed with
neutron The neutron is a subatomic particle, symbol or , that has no electric charge, and a mass slightly greater than that of a proton. The Discovery of the neutron, neutron was discovered by James Chadwick in 1932, leading to the discovery of nucle ...
s 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 Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
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  grains. The number would have to be 100,000 times bigger to make a googol. The decay time for a supermassive
black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
of roughly 1 galaxy-mass (1011  solar masses) due to Hawking radiation is on the order of 10100 years. See in particular equation (27). 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.


Properties

A googol is approximately equal to 70!\approx1.1979\times10^ (
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
of 70). Using an
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
, binary numeral system, one would need 333 bits to represent a googol, i.e., 10^=2^\approx2^. 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 In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to mo ...
, 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, ... 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 Google LLC (, ) is an American multinational corporation and technology company focusing on online advertising, search engine technology, cloud computing, computer software, quantum computing, e-commerce, consumer electronics, and artificial ...
, with the name "Google" being an accidental misspelling of "googol" by the company's founders, which was picked to signify that the search engine was intended to provide large quantities of information. 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"; however, no suit was ever filed. 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. 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..


See also

* Googolplex * Graham's number * Skewes' number *
Infinity Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
* Names of large numbers


References


External links

* * * {{Large numbers Large integers Integers Units of amount 1920s neologisms