Googolplex: Difference between revisions
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{{About||the headquarters of Google Inc.|Googleplex|the Canadian construction toy|Googolplex (toy)}} |
{{About||the headquarters of Google Inc.|Googleplex|the Canadian construction toy|Googolplex (toy)}} |
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A '''googolplex''' is the number 10{{sup|[[googol]]}}, i.e. 10{{sup|10{{sup|100}}}}. The [[Multiplicative inverse|reciprocal]] of the googolplex is called [[googolminex]].<ref>https://sites.google.com/site/largenumbers/home/a-1/small_numbers</ref> |
A '''googolplex''' is the number 10{{sup|[[googol]]}}, i.e. 10{{sup|(10{{sup|100}})}}. The [[Multiplicative inverse|reciprocal]] of the googolplex is called [[googolminex]].<ref>https://sites.google.com/site/largenumbers/home/a-1/small_numbers</ref> |
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==History== |
==History== |
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In the [[Public Broadcasting Service|PBS]] science program ''[[Cosmos: A Personal Voyage]]'', [[Cosmos: A Personal Voyage#Episode 9: "The Lives of the Stars"|Episode 9: "The Lives of the Stars"]], [[astronomer]] and television personality [[Carl Sagan]] estimated that writing a googolplex in standard 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. |
In the [[Public Broadcasting Service|PBS]] science program ''[[Cosmos: A Personal Voyage]]'', [[Cosmos: A Personal Voyage#Episode 9: "The Lives of the Stars"|Episode 9: "The Lives of the Stars"]], [[astronomer]] and television personality [[Carl Sagan]] estimated that writing a googolplex in standard 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. |
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A typical book can be printed with 10{{sup|6}} zeros (around 400 pages with 50 lines per page and 50 zeros per line.<ref>{{cite book|title=Googolplex Written Out|year=2013|isbn=978-0-9900072-1-0|url=http://www.GoogolplexWrittenOut.com}}</ref> Therefore it requires 10{{sup|94}} such books to print all zeros of googolplex. If each book has a size of 210 mm × 297 mm × 13 mm, the total volume of all the books is 8.1{{e|90}} m{{sup|3}}, which is many orders of magnitude larger than the [[visible universe]], which has a volume of only 4{{e|80}} m{{sup|3}} <ref name=wolframvolume>{{cite web|url=http://www.wolframalpha.com/input/?i=volume+of+the+observable+universe|title=Volume of the observable universe|publisher=Wolfram Alpha}}</ref>. |
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An average book of 60 cubic inches can be printed with 5{{e|5}} zeroes (5 characters per word, 10 words per line, 25 lines per page, 400 pages), or 8.3{{e|3}} zeros per cubic inch. The ''observable'' (i.e. past-facing light-cone) universe contains 6{{e|83}} cubic inches (<sup>4</sup>/<sub>3</sub> × ''π'' × (14{{e|9}} light years in inches)<sup>3</sup>). Therefore if the universe were stuffed with paper that were printed with zeros, then the universe could contain only 5.3{{e|87}} zeros — far short of a googol of zeros. With only about 2.5{{e|89}} [[elementary particle]]s in the [[observable universe]], even if an elementary particle represented each digit, far too few particles would be needed to represent a googolplex. |
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With only about 2.5{{e|89}} {{citation needed|date=December 2013}} [[elementary particle]]s in the [[observable universe]], even if only one elementary particle is used to represented each digit, there are not enough particles to represent a googolplex. |
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| ⚫ | Printing digits of a googolplex would be |
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| ⚫ | Printing digits of a googolplex in one long line would be unreasonable, even in one-[[point (typography)|point]] font (0.353 mm per digit). Writing a googolplex in that font would take about 3.53{{e|97}} meters. The observable universe is estimated to be 8.80{{e|26}} metres, or 93 billion [[light-years]], in [[diameter]],<ref name=ly93>{{cite web | last = Lineweaver | first = Charles | coauthors = Tamara M. Davis | year = 2005 | url = http://www.sciam.com/article.cfm?id=misconceptions-about-the-2005-03&page=5 | title = Misconceptions about the Big Bang | publisher = [[Scientific American]] | accessdate = 2008-11-06}}</ref>; the required line to write the necessary zeroes is therefore 4.0{{e|70}} times as long as the observable universe. |
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Writing the number takes too long: if a person can write two digits per second, then writing a googolplex would take around about 1.51{{e|92}} years, which is about 1.1{{e|82}} times the [[age of the universe]].<ref name="fpx.de">[http://www.fpx.de/fp/Fun/Googolplex/GetAGoogol.html Page, Don, "How to Get a Googolplex"], 3 June 2001.</ref> |
Writing the number takes too long: if a person can write two digits per second, then writing a googolplex would take around about 1.51{{e|92}} years, which is about 1.1{{e|82}} times the [[age of the universe]].<ref name="fpx.de">[http://www.fpx.de/fp/Fun/Googolplex/GetAGoogol.html Page, Don, "How to Get a Googolplex"], 3 June 2001.</ref> |
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A Planck space has a volume of a [[Planck length]] cubed, which is the smallest measurable volume, at approximately 4. |
A Planck space has a volume of a [[Planck length]] cubed, which is the smallest measurable volume, at approximately 4.22{{e|-105}} m<sup>3</sup> = 4.22{{e|-99}} cm<sup>3</sup> <ref>{{cite web|title=Planck volume|url=http://www.wolframalpha.com/input/?i=planck+volume|publisher=Wolfram Alpha}}</ref>. Therefore 2.4 cm<sup>3</sup> contain about a [[googol]] Planck spaces. Only about 4{{e|80}} cubic metres exist in the observable universe <ref name=wolframvolume />, giving about 9.5{{e|184}} Planck spaces in the entire observable universe; therefore, a googolplex is far larger than even the number of the smallest measurable spaces in the observable universe. |
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==Scale== |
==Scale== |
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===In the physical universe=== |
===In the physical universe=== |
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One [[googol]] is presumed to be greater than the number of [[hydrogen]] atoms in the [[observable universe]], which has been variously estimated to be between 10<sup>79</sup> and 10<sup>81</sup>.<ref>[http://www.cs.umass.edu/~immerman/stanford/universe.html Mass, Size, and Density of the Universe] Article from National Solar Observatory, 21 May 2001.</ref> A googol is also greater than the number of [[Planck time]]s elapsed since the [[Big Bang]], which is estimated at about 8{{e|60}}.<ref name="wolframalpha.com">[http://www.wolframalpha.com/input/?i=convert+age+of+the+universe+to+Planck+times convert age of the universe to Planck times – Wolfram|Alpha], 8 August 2011</ref> Thus in the physical world it is difficult to give examples of numbers that compare to the vastly greater googolplex. In analyzing [[quantum state]]s and [[black hole]]s, physicist [[Don Page (physicist)|Don Page]] writes that "determining experimentally whether or not information is lost down black holes of solar mass ... would require more than 10{{sup|76.96}} measurements to give a rough determination of the final density matrix after a [[Black hole evaporation#Black hole evaporation|black hole evaporates]]".<ref>[http://arxiv.org/pdf/hep-th/9411193 Page, Don N., "Information Loss in Black Holes and/or Conscious Beings?"], 25 Nov. 1994, for publication in ''Heat Kernel Techniques and Quantum Gravity'', S. A. Fulling, ed. (Discourses in Mathematics and Its Applications, No. 4, Texas A&M University, Department of Mathematics, College Station, Texas, 1995)</ref> The end of the Universe via [[Big Freeze]] without [[proton decay]] is expected to be around 10{{sup|10{{sup|75}}}} years into the future, which is still short of a googolplex. |
One [[googol]] is presumed to be greater than the number of [[hydrogen]] atoms in the [[observable universe]], which has been variously estimated to be between 10<sup>79</sup> and 10<sup>81</sup>.<ref>[http://www.cs.umass.edu/~immerman/stanford/universe.html Mass, Size, and Density of the Universe] Article from National Solar Observatory, 21 May 2001.</ref> A googol is also greater than the number of [[Planck time]]s elapsed since the [[Big Bang]], which is estimated at about 8{{e|60}}.<ref name="wolframalpha.com">[http://www.wolframalpha.com/input/?i=convert+age+of+the+universe+to+Planck+times convert age of the universe to Planck times – Wolfram|Alpha], 8 August 2011</ref> Thus in the physical world it is difficult to give examples of numbers that compare to the vastly greater googolplex. In analyzing [[quantum state]]s and [[black hole]]s, physicist [[Don Page (physicist)|Don Page]] writes that "determining experimentally whether or not information is lost down black holes of solar mass ... would require more than 10{{sup|76.96}} measurements to give a rough determination of the final density matrix after a [[Black hole evaporation#Black hole evaporation|black hole evaporates]]".<ref>[http://arxiv.org/pdf/hep-th/9411193 Page, Don N., "Information Loss in Black Holes and/or Conscious Beings?"], 25 Nov. 1994, for publication in ''Heat Kernel Techniques and Quantum Gravity'', S. A. Fulling, ed. (Discourses in Mathematics and Its Applications, No. 4, Texas A&M University, Department of Mathematics, College Station, Texas, 1995)</ref> The end of the Universe via [[Big Freeze]] without [[proton decay]] is expected to be around 10{{sup|(10{{sup|75}})}} years into the future, which is still short of a googolplex. |
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In a separate article, Page shows that the number of [[State function|states]] in a black hole with a mass roughly equivalent to the [[Andromeda Galaxy]] is in the range of a googolplex.<ref name="fpx.de"/> |
In a separate article, Page shows that the number of [[State function|states]] in a black hole with a mass roughly equivalent to the [[Andromeda Galaxy]] is in the range of a googolplex.<ref name="fpx.de"/> |
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Revision as of 06:00, 14 December 2013
A googolplex is the number 10googol, i.e. 10(10100). The reciprocal of the googolplex is called googolminex.[1]
History
In 1938, 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". 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).
Size
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 standard 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.
A typical book can be printed with 106 zeros (around 400 pages with 50 lines per page and 50 zeros per line.[3] Therefore it requires 1094 such books to print all zeros of googolplex. If each book has a size of 210 mm × 297 mm × 13 mm, the total volume of all the books is 8.1×1090 m3, which is many orders of magnitude larger than the visible universe, which has a volume of only 4×1080 m3 [4].
With only about 2.5×1089 [citation needed] elementary particles in the observable universe, even if only one elementary particle is used to represented each digit, there are not enough particles to represent a googolplex.
Printing digits of a googolplex in one long line would be unreasonable, even in one-point font (0.353 mm per digit). Writing a googolplex in that font would take about 3.53×1097 meters. The observable universe is estimated to be 8.80×1026 metres, or 93 billion light-years, in diameter,[5]; the required line to write the necessary zeroes is therefore 4.0×1070 times as long as the observable universe.
Writing the number takes too long: if a person can write two digits per second, then writing a googolplex would take around about 1.51×1092 years, which is about 1.1×1082 times the age of the universe.[6]
A Planck space has a volume of a Planck length cubed, which is the smallest measurable volume, at approximately 4.22×10−105 m3 = 4.22×10−99 cm3 [7]. Therefore 2.4 cm3 contain about a googol Planck spaces. Only about 4×1080 cubic metres exist in the observable universe [4], giving about 9.5×10184 Planck spaces in the entire observable universe; therefore, a googolplex is far larger than even the number of the smallest measurable spaces in the observable universe.
Scale
In pure mathematics
In pure mathematics, there are several notational methods for representing large numbers by which the magnitude of a googolplex could be represented, such as tetration, Knuth's up-arrow notation, Steinhaus-Moser notation, or Conway chained arrow notation.
In the physical universe
One googol is presumed to be greater than the number of hydrogen atoms in the observable universe, which has been variously estimated to be between 1079 and 1081.[8] A googol is also greater than the number of Planck times elapsed since the Big Bang, which is estimated at about 8×1060.[9] Thus in the physical world it is difficult to give examples of numbers that compare to the vastly greater googolplex. 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 1076.96 measurements to give a rough determination of the final density matrix after a black hole evaporates".[10] The end of the Universe via Big Freeze without proton decay is expected to be around 10(1075) years into the future, which is still short of a googolplex.
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.[6]
If the entire volume of the observable universe (taken to be 3×1080 m3) were packed solid with fine dust particles about 1.5 micrometres in size, then the number of different ways of ordering these particles (that is, assigning the number 1 to one particle, then the number 2 to another particle, and so on until all particles are numbered) would be approximately one googolplex. [citation needed][clarification needed]
See also
References
- ^ https://sites.google.com/site/largenumbers/home/a-1/small_numbers
- ^ Edward Kasner & James R. Newman (1940) Mathematics and the Imagination, page 23, NY: Simon & Schuster
- ^ Googolplex Written Out. 2013. ISBN 978-0-9900072-1-0.
- ^ a b "Volume of the observable universe". Wolfram Alpha.
- ^ Lineweaver, Charles (2005). "Misconceptions about the Big Bang". Scientific American. Retrieved 6 November 2008.
{{cite web}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ a b Page, Don, "How to Get a Googolplex", 3 June 2001.
- ^ "Planck volume". Wolfram Alpha.
- ^ Mass, Size, and Density of the Universe Article from National Solar Observatory, 21 May 2001.
- ^ convert age of the universe to Planck times – Wolfram|Alpha, 8 August 2011
- ^ Page, Don N., "Information Loss in Black Holes and/or Conscious Beings?", 25 Nov. 1994, for publication in Heat Kernel Techniques and Quantum Gravity, S. A. Fulling, ed. (Discourses in Mathematics and Its Applications, No. 4, Texas A&M University, Department of Mathematics, College Station, Texas, 1995)
External links
- Weisstein, Eric W. "Googolplex". MathWorld.
- googolplex at PlanetMath.
- Padilla, Tony. "Googol and Googolplex". Numberphile. Brady Haran.
{{cite web}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - Googolplex written out