11,582
pages

Tetralogue is equal to E1#4 = EEEE1 = 10101010 using Hyper-E notation.[1][2] The term was coined by Sbiis Saibian. It is equal to a one followed by a trialogue (ten trillitrestrigintatrecentillitrestrigintatrecentilliduotrigintatrecentillion) zeros. It is also equal to 10^^4 in up-arrow notation.

Writing down the full decimal expansion would take 109,999,999,994 books of 400 pages each, with 2,500 digits on each page.

Username5243 calls this number goodcolduplex or dooquol, and it's equal to 10[2]4 = 10[1]10[1]10[1]10 in Username5243's Array Notation.[3]

BlankEntity calls this number Roentgenic Great Billion.[4]

## Etymology

The name of this number is based on Greek prefix "tetra-" and the suffix "-logue" for "logarithm".

## Names in -illion systems

In Conway-Wechsler system,[5] $$10^{10^{10^{10}}}$$ is expressed in short scale as follows by using Fish's result.[6]

ten trilli(3,333,333,332 times trestrigintatrecentilli)duotrigintatrecentillion

Using Saibian's generalization of Bowers' -illions, it is approximately:

one nanegigatriotriacontetriahectemegatriotriacontetriahectekillatriotriacontetriahectillion

## Approximations in other notations

Notation Approximation
Up-arrow notation $$10 \uparrow\uparrow 4$$ (exact)
Down-arrow notation $$10 \downarrow\downarrow (10^{10}+1)$$ (exact)
Chained arrow notation $$10 \rightarrow 4 \rightarrow 2$$ (exact)
Steinhaus-Moser Notation $$10[3][3][3]$$
Copy notation $$9[9[9[10]]]$$
Notation Array Notation $$(10\{2,2\}4)$$ (exact)
BEAF $$\{10,4,2\}$$ (exact)
Bashicu matrix system $$(0)(1)(0)(0)[427]$$
Hyperfactorial array notation $$((13!0)!0)!0$$
Nested Factorial Notation $$13!^3$$
Fast-growing hierarchy $$f_2(f_2(f_2(28))))$$
Hardy hierarchy $$H_{(\omega^2) 3}(28)$$
Slow-growing hierarchy $$g_{\omega^{\omega^{\omega^\omega}}}(10)$$ (exact)

## Sources

1. Saibian, Sbiis. Hyper-E Numbers. Retrieved 2014-11-16.
2. Saibian, Sbiis. Ultimate Large Numbers List Part I. Retrieved 2014-11-16.
3. Part 1 - My Large Numbers
4. BlankEntity. BlankEntity's Googology - Class 1 and higher. Retrieved 2022-04-24.
5. Conway and Guy (1995) "The book of Numbers" Copernicus
6. User blog:Kyodaisuu/Illion name of 10^10^x in Conway-Wechsler system