Grangol

Grangol (short for "grand googol") is a number equal to a power tower of 100 tens topped with 100.Sbiis Saibian, Hyper-E NumbersSbiis Saibian's Ultimate Large Numbers List
It can be defined in Hyper-E notation as E100#100 = EEE...EEE100 (100 E's) = (100 tens). The term was coined by Sbiis Saibian. It is also equal to E2#101.
Grangol is comparable to giggolunex, using hyper-operators.
Aarex Tiaokhiao calls this number powergol.Aarex Tiaokhiao's Large Numbers Site
 coined henodgulus for this number, and it's equal to E100{#,1(1)2}100 = E100{#}100 = E100#100 under a climbing interpretation of Hyper-Hyper-Extended Cascading-E Notation.
== 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
Up-arrow notation 	\(10 \uparrow\uparrow 101\)
Chained arrow notation 	\(10 \rightarrow 101 \rightarrow 2\)
BEAF 	\(\{10,101,2\}\)
Cascading-E notation 	E100#^#1
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 ==
*grangolplex
*grangolgong
*grangoldex
*grangoldudex
*grangoltridex
Category:Numbers
Category:Numbers by Sbiis Saibian
Category:Hyper-E numbers
Category:Tetration level
Category:Googol series
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Category:Grangol regiment