Googol

Googol [ ˈguːgɔl ] is an English-language name for the number . This number corresponds to a 1 with 100 zeros, written out: ${\ displaystyle 10 ^ {100}}$

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

The systematic German name of this number is ten sex decillion (or ten sedecillion ). The corresponding number name according to the short scale used in the USA is English ten duotrigintillion (also dotrigintillion ). The practical use of these theoretical designations is small, so that only a few applications can be proven.

Word creation or origin

The term googol was established in 1938 by the American mathematician Edward Kasner . He had previously asked his nine-year-old nephew to come up with a word for the number . Kasner published his work on this number in his book "Mathematics and Presentation". ${\ displaystyle 10 ^ {100}}$

properties

One googol is smaller than (read seventy faculty ): ${\ displaystyle 70!}$

70! = 1 x 2 x 3 x 4 ... 67 x 68 x 69 x 70 =

11,978,571,669,969,891,796,072,783,721,689,098,736,458,938,142,546,425,857,555,362,864,628,009,582,789,845,319,680,000,000,000,000,000

(1.19785717 × 10 ¹⁰⁰⁾

This means that there are more than one Googol way of organizing 70 objects differently.

The number of protons in the visible universe is estimated to be around , significantly less than a googol. According to the inflation theory , however, the observable universe represents only a tiny part of the entire universe, which extends many orders of magnitude beyond the observation limit . So the total number of protons is possibly much larger than a googol. ${\ displaystyle 10 ^ {80}}$

The only prime factors of a Googol represented canonical, 2 and 5 . Its predecessor and successor are not prime numbers either. ${\ displaystyle \ mathrm {Googol} = 2 ^ {100} \ cdot 5 ^ {100}}$

Calculation limit of calculators

Scientific pocket calculators , which appeared around 1975, could for the first time display 10 digits mantissa and a (tens) exponent in the range -99 ... +99. With a maximum of 9.999999999 (E) 99, 1 Googol was barely displayable, but a number that came close to the Googol up to the 100 billionth part. Internally, the calculation was typically 3 digits more accurate and thus close to Googol up to its 100th billiardth. Accordingly, these computers could use the faculty 69! calculate within a few seconds, but issued an error message with the calculation command "70 factorial".

Googolplex

The number is called Googolplex . So a googolplex is a 1 with zeros. As mentioned above, a googol is already significantly larger than the number of protons in the visible universe. Compared to a googolplex, however, a googol appears ridiculously small, as the following example illustrates: ${\ displaystyle 10 ^ {\ mathrm {Googol}} = 10 ^ {(10 ^ {100})}}$ ${\ displaystyle 10 ^ {100}}$

A computer needs 42 bytes to store numbers up to one googol in integer arithmetic . Home computers from the 1980s are reaching their limits, the main memory of typical PCs from the 2010s is at , and hard drives are at . All of humanity will have memory of around 40 zettabytes by 2020, which is enough for numbers up to . ${\ displaystyle 10 ^ {(10 ^ {2})}}$ ${\ displaystyle 10 ^ {(10 ^ {5})}}$ ${\ displaystyle 10 ^ {(10 ^ {10})}}$ ${\ displaystyle 10 ^ {(10 ^ {13})}}$ ${\ displaystyle 10 ^ {(10 ^ {23})}}$

A computer that can store up to numbers would have, at the temperature of the cosmic background radiation, a theoretical minimum mass in the order of the earth's mass, and in the order of the estimated mass of the visible universe. The representation of numbers up to a googolplex is not a current problem for mankind, but in theory exceeds the resources available in our universe by several orders of magnitude. ${\ displaystyle 10 ^ {(10 ^ {64})}}$ ${\ displaystyle 10 ^ {(10 ^ {92})}}$ ${\ displaystyle 10 ^ {(10 ^ {100})}}$

Other numbers are:

Googolplexplex (also known as Googolplexian) ${\ displaystyle \ left (10 ^ {\ mathrm {Googolplex}} \ right)}$
Googolplexplexplex ${\ displaystyle \ left (10 ^ {\ mathrm {Googolplexplex}} \ right)}$
Googolplexplexplexplex ${\ displaystyle \ left (10 ^ {\ mathrm {Googolplexplexplex}} \ right)}$
...

It is known that the successor number (Googolplex + 1) any prime number is. One of the prime factors of this number is 316,912,650,057,057,350,374,175,801,344,000,001.

Other number systems

In general, for all number systems , a googol is a one with the number of zeros corresponding to the square of the base .

{\ displaystyle \ mathrm {Googol} _ {(n)} = n ^ {(n ^ {2})}}

{\ displaystyle \ mathrm {Googol} _ {(2)} = 2 ^ {(2 ^ {2})} = 2 ^ {4} = 1 {\ mathpunct {.}} 00 {\ mathpunct {.}} 00_ {(2)} = 16 _ {(10)}}

{\ displaystyle \ mathrm {Googol} _ {(3)} = 3 ^ {(3 ^ {2})} = 3 ^ {9} = 1 {\ mathpunct {.}} 000 {\ mathpunct {.}} 000 {\ mathpunct {.}} 000 _ {(3)} = 19 {\ mathpunct {.}} 683 _ {(10)}}

{\ displaystyle \ mathrm {Googol} _ {(4)} = 4 ^ {(4 ^ {2})} = 4 ^ {16} = 1 {\ mathpunct {.}} 0000 {\ mathpunct {.}} 0000 {\ mathpunct {.}} 0000 {\ mathpunct {.}} 0000 _ {(4)} = 4 {\ mathpunct {.}} 294 {\ mathpunct {.}} 967 {\ mathpunct {.}} 296 _ {(10 )}}

{\ displaystyle \ mathrm {Googol} _ {(10)} = 10 ^ {(10 ^ {2})} = 10 ^ {100}}

A Googol in the two-part system corresponds to only 2 ⁴ = 16, a Googolplex only 2 ¹⁶ = 65,536, only the Googolplexplex 2 ^65,536 ≈ 2.003530 · 10 ^19,728 is significantly larger than the Googol in the ^decimal system.

With the basis of the value system , however, Googol and Googolplex are growing rapidly. A googolplex based on base 3 is 3 ^19,683 ≈ 1.505416 · 10 ^9,391 already larger than a googol in the ^decimalsystem .

Notoriety

Some objects were named after this number:

Google - the search engine derives its name from Googol, based on the endeavor to index as many Internet pages as possible . The name of the company's headquarters is Googleplex , based on the Googolplex. Google uses a domain 1e100.net based on Googol for its own servers.
Googolgon , a polygon with googol (10 ¹⁰⁰ ) corners
Googolgram , a star inscribed in a polygon with googol (10 ¹⁰⁰ ) corners
gugolplex , an art project in Cologne since 2005, which aims to combine musical and literary elements
One In A Googolplex , German pop band
Googol Plex , former hardcore punk band in Japan

Citations of the term in the literature:

Googol - a novel by HD Klein ; the continuation of the novel is called Googolplex
Dr. Francis Googol - fictional teacher, who through the non-fiction book Dr. Googol's wondrous world of numbers by Clifford A. Pickover (English original title: Wonders of Numbers ) leads
Gugelplex Sterndenker - name of a supercomputer in Douglas Adam's novel The Hitchhiker's Guide to the Galaxy
The American author Jonathan Safran Foer uses the term Googol in his novel Extrem loud and incredibly close, which was published in German in 2005

In a Peanuts comic, Schroeder explains to his admirer Lucy that the chances of her marriage are 1 to Googol

Quotes of the term in the film:

Googolplex is the name of a Springfield movie theater in the animated television series The Simpsons
Googolplex is the name of a mall in Disney's animated series Phineas and Ferb
In the film Back to the Future III , Dr. Emmett L. Brown ("Doc Brown") had the chance to meet his wife, 1 at Googolplex

literature

Edward Kasner: New Names in Mathematics . In: Scripta Mathematica 5 (1938), No. 1, pp. 5-14, here especially pp. 11-14.

Web links

Wiktionary: Googol - explanations of meanings, word origins, synonyms, translations

Edward Kasner (English)
http://www.fpx.de/fp/Fun/Googolplex/
http://whatis.techtarget.com/definition/0,,sid9_gci213798,00.html
Eric W. Weisstein : Googol . In: MathWorld (English).
Thomas Foregger, Pedro Sanchez, John Smith u. a .: Googol . In: PlanetMath . (English)

Individual evidence

^ Original title: Mathematics and the Imagination .
↑ Wolfram Alpha: prime factors of 10 ^ 100-1: http://www.wolframalpha.com/input/?i=prime+factors+of+10^100%2B1
↑ https://www.youtube.com/watch?v=nKL6PFb_e3g Texas Instruments SR-10 & SR-50 Calculators, youtube.com, crusinscamp, Video (04:45), August 9, 2011, accessed July 12, 2016 - English.
^ For the first time with the edition of the pocket calculator SR-50 from Texas Instruments , 1974. - Available in Austria 1975/1976 for 2500 Schilling.
↑ Thomas Jüngling: Zettabytes: data volume doubles every two years. In: welt.de . July 16, 2013, accessed October 7, 2018 .
^ The Anatomy of a Large-Scale Hypertextual Web Search Engine
↑ What is 1e100.net?
^ Douglas Adams The Hitchhiker's Guide to the Galaxy Heyne-Verlag; 3. Edition; Paperback edition 06/2005; Page 162. ISBN 3-453-50016-4 .

[1] Original title: Mathematics and the Imagination .

[2] Wolfram Alpha: prime factors of 10 ^ 100-1: http://www.wolframalpha.com/input/?i=prime+factors+of+10^100%2B1

[3] ttps://www.youtube.com/watch?v=nKL6PFb_e3g Texas Instruments SR-10 & SR-50 Calculators, youtube.com, crusinscamp, Video (04:45), August 9, 2011, accessed July 12, 2016 - English.

[4] For the first time with the edition of the pocket calculator SR-50 from Texas Instruments , 1974. - Available in Austria 1975/1976 for 2500 Schilling.

[welt-118099520-5] Thomas Jüngling: Zettabytes: data volume doubles every two years. In: welt.de . July 16, 2013, accessed October 7, 2018 .

[Google-Paper-6] The Anatomy of a Large-Scale Hypertextual Web Search Engine

[7] What is 1e100.net?

[8] Douglas Adams The Hitchhiker's Guide to the Galaxy Heyne-Verlag; 3. Edition; Paperback edition 06/2005; Page 162. ISBN 3-453-50016-4 .