Googolplex
Template:Short descriptionScript error: No such module "Distinguish". Template:Use dmy dates Template:Pp A googolplex is the large number 1010100Script error: No such module "Check for unknown parameters"., that is, 10Script error: No such module "Check for unknown parameters". raised to the power of a googol. If written out in ordinary decimal notation, it would be 1Script error: No such module "Check for unknown parameters". followed by a googol (10100) zeroes – a physically impossible number to write explicitly.
History
In 1920, Edward Kasner's nine-year-old nephew, Milton Sirotta, coined the term googol, which is 10100, and 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 [be] a better mathematician than Dr. Einstein, simply because he had more endurance and could write for longer".[2] It thus became standardized to 10(10100), which is usually written as 1010100 using the conventional interpretation for serial exponentiation.[3]
Size
A typical book can be printed with one million zeros (around 400 pages with 50 lines per page and 50 zeros per line). Therefore, it requires Template:10^ such books to print all the zeros of a googolplex (that is, printing a googol zeros).[4] If each book had a mass of 100 grams, all of them would have a total mass of Template:10^ kilograms. In comparison, Earth's mass is Script error: No such module "val". kilograms,[5] the mass of the Milky Way galaxy is estimated at Script error: No such module "val". kilograms,[6] and the total mass of all the stars in the observable universe is estimated at Script error: No such module "val"..[7]
To put this in perspective, the mass of all such books required to write out a googolplex would be vastly greater than the mass of the observable universe by a factor of roughly Script error: No such module "val"..
In the physical universe
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. Sagan gave an example that if the entire volume of the observable universe is filled with fine dust particles roughly 1.5 micrometers in size (0.0015 millimeters), then the number of different combinations in which the particles could be arranged and numbered would be about one googolplex.[8][9]
Template:10^ is a high estimate of the elementary particles existing in the visible universe (not including dark matter), mostly photons and other massless force carriers.[10]
Mod n
The residues (mod n) of a googolplex, starting with mod 1, 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 the OEIS)
This sequence is the same as the sequence of residues (mod n) of a googol up until the 17th position.
See also
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References
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- ↑ Script error: No such module "Citation/CS1". (retrieved 17 March 2015)
- ↑ Edward Kasner & James R. Newman (1940) Mathematics and the Imagination, page 23, NY: Simon & Schuster
- ↑ Script error: No such module "citation/CS1".
- ↑ Stanford student Wolfgang Nitsche put together a website which will, provided a 94-digit volume number, generate a PDF file consisting of Template:10^ zero digits (with an initial one digit in volume 1), and registered an ISBN for the set: Template:Pb Script error: No such module "citation/CS1".
- ↑ Script error: No such module "citation/CS1".
- ↑ Script error: No such module "citation/CS1".
- ↑ Script error: No such module "citation/CS1".
- ↑ Script error: No such module "citation/CS1".
- ↑ Script error: No such module "citation/CS1".
- ↑ Script error: No such module "citation/CS1".
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External links
- Template:Sister-inline
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- Script error: No such module "citation/CS1".