The term was coined in 1920 by 9-year-old Milton Sirotta (1911–1981), nephew of U.S. mathematician Edward Kasner.[1] Kasner popularized the concept in his 1940 book Mathematics and the Imagination.[2] Other names for googol include ten duotrigintillion on the short scale, ten thousand sexdecillion on the long scale, or ten sexdecilliard on the Peletier long scale.
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 give a sense of how big a googol really is, 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.[3] It is a ratio in the order of about 1080 to 1090, or only about 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 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.[4]
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.[5] Therefore, the heat death of an expanding universe is lower-bounded to occur at least one googol years in the future.
A googol is approximately 70! (factorial of 70). Using an integral, binary numeral system, one would need 333 bits to represent a googol, i.e., 1 googol = 2 ( 100 / l o g 10 2 ) {\displaystyle 2^{(100/\mathrm {log} {10}2)}} 2^{(100/\mathrm{log}{10}2)} ≈ 2332.19280949. However, a googol is well within the maximum bounds of an IEEE 754 double-precision floating point type, but 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.
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,[6] which was picked to signify that the search engine was intended to provide large quantities of information.[7] 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;[8] 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.[9][10]
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 cheated his way through the show with the help of a confederate in the studio audience.[11]
A googolplex is the number 10googol, or equivalently, 10(10100). Written out in ordinary decimal notation, it is 1 followed by 10100 zeroes, that is, a 1 followed by a googol zeroes.
In 1920, 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".[1] Kasner decided to adopt a more formal definition because "different people get tired at different times and it would never do to have Carnera a better mathematician than Dr. Einstein, simply because he had more endurance and could write for longer".[2] It thus became standardized to 1010100.
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 the zeros of a googolplex (that is, printing a googol zeros). If each book had a mass of 100 grams, all of them would have a total mass of 1093 kilograms. In comparison, Earth's mass is 5.972 x 1024 kilograms, the mass of the Milky Way Galaxy is estimated at 2.5 x 1042 kilograms, and the mass of matter in the observable universe is estimated at 1.5 x 1053 kg.[citation needed]
To put this in perspective, the mass of all such books required to write out a googolplex would be vastly greater than the masses of the Milky Way and the Andromeda galaxies combined (by a factor of roughly 2.0 x 1050), and greater than the mass of the observable universe by a factor of roughly 7 x 1039.
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 full decimal 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.
One googol is presumed to be greater than the number of atoms in the observable universe, which has been estimated to be approximately 1078.[3] 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".[4] 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.[5]
Writing the number would take an immense amount of time: if a person can write two digits per second, then writing a googolplex would take about 1.51×1092 years, which is about 1.1×1082 times the accepted age of the universe.[5]
1097 is a high estimate of the elementary particles existing in the visible universe (not including dark matter), mostly photons and other massless force carriers.[6]
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