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+ | [[File:Grangol.jpg|thumb|Grangol in zeroes]] |
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⚫ | '''Grangol''' (short for "grand [[googol]]") is equal to |
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+ | [[File:Grangoltower.jpg|thumb|Grangol in towers|259x259px]] |
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⚫ | '''Grangol''' (short for "grand [[googol]]") is a number equal to a power tower of 100 tens topped with a 100.<ref>Sbiis Saibian, [http://sites.google.com/site/largenumbers/home/4-3/Hyper-E Hyper-E Numbers]</ref><ref>[http://sites.google.com/site/largenumbers/home/appendix/a/numbers Sbiis Saibian's Ultimate Large Numbers List]</ref> It can be defined in [[Hyper-E Notation]] as E100#100 = EEE...EEE100 (100 E's) = {{^...|10|100}} (100 tens). The term was coined by [[Sbiis Saibian]]. |
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⚫ | Grangol is close to [[Jonathan Bowers]]' [[giggol]] = E1#100 |
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+ | Grangol is comparable to 10 [[Tetration|tetrated]] to 101, using [[hyper-operator]]s. |
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⚫ | |||
+ | This is a 1 followed by E100#99 zeros, or a googol-99-plex |
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+ | |||
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⚫ | |||
+ | |||
+ | == Comparison with giggol == |
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+ | |||
⚫ | |||
+ | |||
+ | [[fzgiggol|\(\text{giggol}^{\text{giggol}}\)]]\( < \text{grangol}\) |
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+ | |||
+ | so the two numbers are arithmetically extremely far apart. This concept is related to the [[power-tower paradox]]. |
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+ | |||
+ | The proof is fairly straightforward. Since giggol = E1#100 and Ed#p = 10<sup>Ed#(p-1)</sup> it follows that: |
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+ | |||
+ | \begin{eqnarray*} |
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+ | \text{giggol}^{\text{giggol}} &=& E1\#100^{E1\#100} = (10^{E1\#99})^{E1\#100} \\ |
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+ | &<& 10^{(E1\#100)^2} = 10^{10^{2E1\#99}} = E(2E1\#99)\#2 \\ |
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+ | &<& E(10E1\#99)\#2 = E(10\times10^{E1\#98})\#2 \\ |
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+ | &=& E(10^{1+E1\#98})\#2 = E(1+E1\#98)\#3 \\ |
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+ | &<& E(1+E1\#97)\#4 < \cdots < E(1+E1\#1)\#100 \\ |
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+ | &=& E(1+E1)\#100 = E11\#100 \\ |
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+ | &<& E100\#100 = \text{grangol} |
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+ | \end{eqnarray*} |
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+ | |||
+ | Therefore, giggol<sup>giggol</sup> can be given the following lower and upper bound: |
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+ | |||
+ | \[E10\#100 < \text{giggol}^{\text{giggol}} < E11\#100\] |
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+ | |||
+ | <br /> |
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+ | |||
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|- |
|- |
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− | ! scope="col"|Notation |
+ | ! scope="col" |Notation |
− | ! scope="col"|Approximation |
+ | ! scope="col" |Approximation |
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|- |
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|[[Up-arrow notation]] |
|[[Up-arrow notation]] |
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− | |\( |
+ | |\(10 \uparrow\uparrow 101\) |
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|- |
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|[[Chained arrow notation]] |
|[[Chained arrow notation]] |
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− | |\( |
+ | |\(10 \rightarrow 101 \rightarrow 2\) |
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|- |
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|[[BEAF]] |
|[[BEAF]] |
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− | |\(\{ |
+ | |\(\{10,101,2\}\) |
+ | |- |
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+ | |[[Hyperfactorial array notation]] |
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+ | |\(104!1\) |
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+ | |- |
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+ | |[[Strong array notation]] |
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+ | |\(s(10,101,2)\) |
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+ | |- |
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+ | |[[Nested factorial notation]] |
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+ | |\(100![2]\) |
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|- |
|- |
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|[[Fast-growing hierarchy]] |
|[[Fast-growing hierarchy]] |
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|- |
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|[[Slow-growing hierarchy]] |
|[[Slow-growing hierarchy]] |
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− | |\(g_{\varepsilon_0}( |
+ | |\(g_{\varepsilon_0}(101)\) |
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|} |
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− | + | == Sources == |
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− | {{numnav|hectalogue|grangolplex}} |
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− | |||
− | === Sources === |
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<references /> |
<references /> |
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− | + | == See also == |
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*[[grangolplex]] |
*[[grangolplex]] |
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− | |||
*[[grangolgong]] |
*[[grangolgong]] |
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− | |||
*[[grangoldex]] |
*[[grangoldex]] |
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+ | *[[grangoldudex]] |
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*[[grangoltridex]] |
*[[grangoltridex]] |
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[[Category:Numbers by Sbiis Saibian]] |
[[Category:Numbers by Sbiis Saibian]] |
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[[Category:Hyper-E numbers]] |
[[Category:Hyper-E numbers]] |
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− | [[Category:Tetration]] |
+ | [[Category:Tetration level]] |
+ | [[Category:Googol series]] |
Revision as of 01:47, 8 April 2021
Grangol (short for "grand googol") is a number equal to a power tower of 100 tens topped with a 100.[1][2] It can be defined in Hyper-E Notation as E100#100 = EEE...EEE100 (100 E's) = 1010...10100 (100 tens). The term was coined by Sbiis Saibian.
Grangol is comparable to 10 tetrated to 101, using hyper-operators.
This is a 1 followed by E100#99 zeros, or a googol-99-plex
Aarex Tiaokhiao calls this number powergol.[3]
Comparison with giggol
Grangol is close to and slightly larger than Jonathan Bowers' giggol = E1#100. It is comparable only from a googologist's perspective, however — it can be shown that:
\(\text{giggol}^{\text{giggol}}\)\( < \text{grangol}\)
so the two numbers are arithmetically extremely far apart. This concept is related to the power-tower paradox.
The proof is fairly straightforward. Since giggol = E1#100 and Ed#p = 10Ed#(p-1) it follows that:
\begin{eqnarray*} \text{giggol}^{\text{giggol}} &=& E1\#100^{E1\#100} = (10^{E1\#99})^{E1\#100} \\ &<& 10^{(E1\#100)^2} = 10^{10^{2E1\#99}} = E(2E1\#99)\#2 \\ &<& E(10E1\#99)\#2 = E(10\times10^{E1\#98})\#2 \\ &=& E(10^{1+E1\#98})\#2 = E(1+E1\#98)\#3 \\ &<& E(1+E1\#97)\#4 < \cdots < E(1+E1\#1)\#100 \\ &=& E(1+E1)\#100 = E11\#100 \\ &<& E100\#100 = \text{grangol} \end{eqnarray*}
Therefore, giggolgiggol can be given the following lower and upper bound:
\[E10\#100 < \text{giggol}^{\text{giggol}} < E11\#100\]
Approximations in other notations
Notation | Approximation |
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Up-arrow notation | \(10 \uparrow\uparrow 101\) |
Chained arrow notation | \(10 \rightarrow 101 \rightarrow 2\) |
BEAF | \(\{10,101,2\}\) |
Hyperfactorial array notation | \(104!1\) |
Strong array notation | \(s(10,101,2)\) |
Nested factorial notation | \(100![2]\) |
Fast-growing hierarchy | \(f_3(100)\) |
Hardy hierarchy | \(H_{\omega^3}(100)\) |
Slow-growing hierarchy | \(g_{\varepsilon_0}(101)\) |
Sources
See also
Originals: googol · googolplex
Cockburn's examples: gargoogolplex · fzgoogolplex · (mega)fugagoogolplex
Googol-n-plex: googolduplex · googoltriplex · -quadriplex · -quinplex · -sextiplex · -septiplex · -octiplex · -noniplex · -deciplex · -centiplex
Googol-103n-plex: googolmilliplex · -megaplex · -gigaplex · -teraplex · -petaplex · -exaplex · -zettaplex · -yottaplex · -xennaplex · -vekaplex · -mekaplex
Bowers' extensions: giggol · gaggol · geegol · boogol · biggol · troogol · goobol · more...
Saibian's extensions: googol-minutia · googolchime · googoltoll · googolgong · grangol · greagol · gigangol · gugold · graatagold · gugolthra · throogol · godgahlah · tethrathoth · more...
Miscellany: googolbang (10100!) · googolminex (10^-10100) · googolteen · googolty · great googol(plex) · gooprol · little googol/googolbit · zootzootplex
Grangol—greagol series: grangol · grangolplex · grangolgong · googoldex · googoldexiplex · googoldexiduplex · ecetondex · googolplexidex · googolduplexidex · tria-taxis · grangoldex · grangoldexigong · ecetondudex · tetra-taxis · grangoldudex · grangoldudexigong · ecetontridex · penta-taxis · grangoltridex · grangoltridexigong · ecetonquadridex · hexa-taxis · grangolquadridex · grangolquadridexigong · ecetonquintidex · hepta-taxis · grangolquintidex · grangolquintidexigong · octa-taxis · enna-taxis · deka-taxis · hecta-taxis
Gigangol—gugold series: gigangol · gigangolgong · ecetontetrex · tria-exaxis · gigangoltetrex · gigangoltetrexigong · ecetondutetrex · tetra-exaxis · gigangoldutetrex · gigangoldutetrexigong · deka-exaxis · hecta-exaxis · gorgegol · gorgegolgong · ecetonpentex · tria-eptaxis · gorgegolpentex · gorgegolpentexigong · ecetondupentex · tetra-eptaxis · gorgegoldupentex · gorgegoldupentexigong · deka-eptaxis · hecta-eptaxis · gulgol · gulgolgong · ecetonhex · tria-octaxis · gulgolhex · gulgolhexigong · ecetonduhex · tetra-octaxis · gulgolduhex · gulgolduhexigong · deka-octaxis · hecta-octaxis · gaspgol · gaspgolgong · ecetonheptex · tria-ennaxis · gaspgolheptex · gaspgolheptexigong · ecetonduheptex · tetra-ennaxis · gaspgolduheptex · gaspgolduheptexigong · deka-ennaxis · hecta-ennaxis · ginorgol · ginorgolgong · ecetonoctex · tria-dekaxis · ginorgoloctex · ginorgoloctexigong · ecetonduoctex · tetra-dekaxis · ginorgolduoctex · ginorgolduoctexigong · deka-dekaxis · hecta-dekaxis · gargantuul · googondol · googoadol · goograndol · googreadol · googigandol · googorgedol · googuldol · googaspdol · googinordol · googarganduul · googonkosol · googontritol · googonsartol · googonpetol · googonextol · googoneptol · googonogdol · googonentol
Gugolthra—throogol series: gugolthra · great gugold · gugolthra-plexitris · graatagolthra · greegolthra · grinningolthra · golaagolthra · gruelohgolthra · gaspgolthra · ginorgolthra · gargantuulthra · googondolthra · gugoltesla · gugoltesla-plexitetris · graatagoltesla · greegoltesla · grinningoltesla · golaagoltesla · gruelohgoltesla · gaspgoltesla · ginorgoltesla · gargantuultesla · googondoltesla · gugolpeta · graatagolpeta · greegolpeta · grinningolpeta · golaagolpeta · gruelohgolpeta · gaspgolpeta · ginorgolpeta · gargantuulpeta · googondolpeta · gugolhexa · graatagolhexa · greegolhexa · grinningolhexa · golaagolhexa · gruelohgolhexa · gaspgolhexa · ginorgolhexa · gargantuulhexa · googondolhexa · gugolhepta · graatagolhepta · greegolhepta · grinningolhepta · golaagolhepta · gruelohgolhepta · gaspgolhepta · ginorgolhepta · gargantuulhepta · googondolhepta · gugolocta · graatagolocta · greegolocta · grinningolocta · golaagolocta · gruelohgolocta · gaspgolocta · ginorgolocta · gargantuulocta · googondolocta · gugolenna · graatagolenna · greegolenna · grinningolenna · golaagolenna · gruelohgolenna · gaspgolenna · ginorgolenna · gargantuulenna · googondolenna · gugoldeka · graatagoldeka · greegoldeka · grinningoldeka · golaagoldeka · gruelohgoldeka · gaspgoldeka · ginorgoldeka · gargantuuldeka · googondoldeka · gugolendeka · gugoldodeka · gugoltriadeka · gugoltetradeka · gugolpentadeka · gugolhexadeka · gugolicosa · gugoltrianta · gugolsaranta · gugolpeninta · gugolexinta · gugolebdominta · gugologdonta · gugoleneninta
Tetroogol—godgahlah series: tetroogol · grand tetroogol · tetrangol · tetreagol · tetrugold · tetraatagold · tetrugolthra · tetrugoltesla · tetrithroogol · tetrithrootrigol · tetrithrootergol · tetrithroopetol · tetrithroohexol · tetrootrigol · tetratrithroogol · tetratrithrootrigol · tetrootergol · tetraterithroogol · tetroopetol · tetroohexol · tetrooheptgol · tetroogogdol · tetroogentol · tetroodekol · pentoogol · pentrangol · pentugold · pentithroogol · pentitetroogol · pentootrigol · pentootergol · pentoopetol · pentoohexol · hexoogol · hexootrigol · heptoogol · ogdoogol · ogdahepti-hexapenti-tetrithroogol · ogdatri-hexi-pentateri-tetrapetithrinortrigolthra · entoogol · dektoogol · endektoogol · icosolus · triantolus · sarantolus · penintolus · exintolus · ebdomintolus · ogdontolus · enenintolus
Godgahlah - godgahlah-ex-grand godgahlah series: godgahlah · chilliatolus · myriatolus · godgahlahgong · grand godgahlah · grand godgahlahgong · grand grand godgahlah · grand grand grand godgahlah · grand grand grand grand godgahlah · hundred-ex-grand godgahlah · godgahlah-ex-grand godgahlah