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\begin{center}
\vspace*{-0.25cm} %dirty
\LARGE{Large numbers and numbering schemes}
\end{center}

\begin{tabular}{ | l | l | l | l | l | l | }

%table header row
\rowcolor{TableHeaderColor}
\hline
\vtop{\hbox{\strut SK, európske škálovanie,}\hbox{\strut Peletier system (long scale)}} &
US, Canada (short scale) &
DE, Zahlennamen (long scale) &
RU, \begin{otherlanguage*}{russian}Короткая шкала\end{otherlanguage*} &
number &
$10^n$ 
\\
\hline

%table numbers rows
jeden & one & Eins &\begin{otherlanguage*}{russian}один\end{otherlanguage*} & 1 & $10^0$ \\ \hline
desať & ten & Zehn &\begin{otherlanguage*}{russian}десять\end{otherlanguage*} & 10 & $10^1$ \\ \hline
sto & hundred & Hundert &\begin{otherlanguage*}{russian}сто\end{otherlanguage*} & 100 & $10^2$ \\ \hline
tisíc & thousand & Tausend &\begin{otherlanguage*}{russian}тысяча\end{otherlanguage*} & 1 000 & $10^3$ \\ \hline
desaťtisíc & ten thousand & Zehntausend &\begin{otherlanguage*}{russian}десять тысяч\end{otherlanguage*} & 10 000 & $10^4$ \\ \hline
stotisíc & hundred thousand & Hunderttausend &\begin{otherlanguage*}{russian}сто тысяч\end{otherlanguage*} & 100 000 & $10^5$ \\ \hline

milión & million & Million &\begin{otherlanguage*}{russian}один миллион\end{otherlanguage*} & 1 000 000 & $10^6$ \\ \hline
miliarda & \textbf{billion} & Milliarde &\textbf{\begin{otherlanguage*}{russian}миллиард\end{otherlanguage*}} & 1 000 000 000 & $10^9$ \\ \hline
bilión & trillion & Billion &\textbf{\begin{otherlanguage*}{russian}триллион\end{otherlanguage*}} & 1 000 000 000 000 & $10^{12}$ \\ \hline
biliarda & quadrillion & Billiarde &\textbf{\begin{otherlanguage*}{russian}квадриллион\end{otherlanguage*}} & 1 000 000 000 000 000 & $10^{15}$ \\ \hline
trilión & quintillion & Trillion &\textbf{\begin{otherlanguage*}{russian}квинтиллион\end{otherlanguage*}} & 1 000 000 000 000 000 000 & $10^{18}$ \\ \hline
triliarda & sextillion & Trilliarde &\textbf{\begin{otherlanguage*}{russian}секстиллион\end{otherlanguage*}} & 1 000 000 000 000 000 000 000 & $10^{21}$ \\ \hline
kvadrilión & septillion & Quadrillion &\textbf{\begin{otherlanguage*}{russian}септиллион\end{otherlanguage*}} & 1 000 000 000 000 000 000 000 000 & $10^{24}$ \\ \hline
kvadriliarda & octillion & Quadrilliarde &\textbf{\begin{otherlanguage*}{russian}октиллион\end{otherlanguage*}} & 1 000 000 000 000 000 000 000 000 000 & $10^{27}$ \\ \hline
kvintilión & nonillion & Quintillion &\textbf{\begin{otherlanguage*}{russian}нониллион\end{otherlanguage*}} & 1 000 000 000 000 000 000 000 000 000 000 & $10^{30}$ \\ \hline
kvintiliarda & decillion & Quintilliarde &\textbf{\begin{otherlanguage*}{russian}дециллион\end{otherlanguage*}} & 1 000 000 000 000 000 000 000 000 000 000 000 & $10^{33}$ \\ \hline

sextilión & undecillion & Sextillion && 1 000 000 000 000 000 000 000 000 000 000 000 000 & $10^{36}$ \\ \hline
sextiliarda & duodecillion & Sextilliarde && 1 000 000 000 000 000 000 000 000 000 000 000 000 000 & $10^{39}$ \\ \hline
septilión & tredecillion & Septillion && 1 000 000 000 000 000 000 000 000 000 000 000 000 000 000 & $10^{42}$ \\ \hline
septiliarda & quattuordecillion & Septilliarde && 1 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 & $10^{45}$ \\ \hline
oktilión & quinquadecillion & Oktillion && 1 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 & $10^{48}$ \\ \hline
oktiliarda & sedecillion & Oktilliarde && 1 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 & $10^{51}$ \\ \hline
nonilión & septendecillion & Nonillion && 1 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 & $10^{54}$ \\ \hline
noniliarda & octodecillion & Nonilliarde &&  1 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 & $10^{57}$ \\ \hline
decilión & novendecillion & Dezillion && 1 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 & $10^{60}$ \\ \hline
deciliarda & vigintillion & Dezilliarde && 1 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 & $10^{63}$ \\ \hline

googol $=$ 10 sedeciliárd & googol $=$ 10 duotrigintillion & Googol & \textbf{\begin{otherlanguage*}{russian}Гугол $=$ десять дуотригинтиллионов\end{otherlanguage*}} & \scriptsize{10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000} & $10^{100}$ \\ \hline

kvinkvagintiliarda & centillion & Quinquagintilliarde &&&$10^{303}$ \\ \hline
centilión & novenonagintacentillion & Zentillion &&&$10^{600}$ \\ \hline
centiliarda & ducentillion & Zentilliarde &&&$10^{603}$ \\ \hline
kvingentiliarda & millinillion & Quingentilliarde &&&$10^{3 003}$ \\ \hline
milinilión &  & Mi-lli-ni-llion &&&$10^{6 000}$ \\ \hline
milinilinilión &  & Mi-lli-ni-lli-ni-llion &&&$10^{6 000 000}$ \\ \hline
googolplex & googolplex & Googolplex & \textbf{\begin{otherlanguage*}{russian}гуголплекс\end{otherlanguage*}} & $10^{10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000}$ & $10^{10^{100}}$ \\ \hline
\end{tabular}

$\newline$

%Graham
\hspace{-0.32cm} %dirty
\begin{tabular}{l l l l l @{}m{13cm}@{} }
\textbf{Graham's number} = $g_{64}$, growth: $f_{\omega+1}(64)$, last digits: $\dots04 575 627 262 464 195 387$, definition:&
$\begin{aligned}
g_0 &=& 4 \\ g_1 &=& 3 \uparrow\uparrow\uparrow\uparrow 3 \\ g_2 &=& 3 \underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{g_1 \text{ arrows}} 3 \\
g_{k + 1} &=& 3 \underbrace{\uparrow\uparrow\uparrow\cdots\uparrow\uparrow\uparrow}_{g_k \text{ arrows}} 3 \\ g_{64} &=& \text{Graham's number}
\end{aligned}$
& where & $\begin{aligned} a \uparrow^1 b &=& a^b \\ a \uparrow^n 1 &=& a \\ a \uparrow^{n + 1} (b + 1) &=& a \uparrow^n (a \uparrow^{n + 1} b) \\ \end{aligned}$ & $\quad$ &
Graham's number is so large that the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume, possibly the smallest measurable space. But even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Nor even can the number of digits of that number. And so forth, for a number of times far exceeding the total number of Planck volumes in the observable universe. Graham's number cannot even be expressed in this way by power towers of the form $a^{b^{c^{.^{.^{.}}}}}$.
\end{tabular} 

$\newline$

$\text{\textbf{TREE}}(n)=$ the length of the longest sequence of trees $T_1, T_2, \dots$ labelled from $\{1, 2, \dots n\}$ such that, for all $i, T_i$ has at most $i$ vertices, and for all $i, j$ such that $i < j$, there is no label-preserving homeomorphic embedding from $T_i$ into $T_j$, growth:$f_{\vartheta(\Omega^\omega\omega)}(n)$ .

$\newline$

\textbf{Loader's number} is defined as $D^5(99)=D(D(D(D(D(99)))))$ where $D(k)$ is an accumulation of all possible expressions provable within approximately $\log(k)$ inference steps in the calculus of constructions (encoding proofs as binary numbers).

$\newline$

Some other numbers and functions: $\textbf{SSCG}(2)\geq 3 \cdot 2^{3 \cdot 2^{95}}-8 \approx 10^{3.5775 \cdot 10^{28}}$ (simple subcubic graph number), $\text{SSCG}(2)\ll\text{TREE}(3)\ll\text{SSCG}(3)$; $\textbf{SCG}(13)$ (subcubic graph sequence)

$\textbf{BH}(100)$ (Buchholz hydra) (growth rate: $\Pi_1^1-\text{CA}+\text{BI}$), $\textbf{USGDCS}_2(k)$ (greedy clique sequence), $\textbf{BB}(n)$ (busy beaver function), Rayo's function, FOOT (first-order oodle theory) function, BIG FOOT.

\vspace{0.5cm} %dirty

\noindent\rule[0.5ex]{\linewidth}{0.5pt}

%Infinity
\begin{center}
Nekonečno [SK], infinity [EN], die Unendlichkeit [DE], \begin{otherlanguage*}{russian}Бесконечность\end{otherlanguage*} [RU], $\boldsymbol{\infty}=\{\text{that which is \textbf{not} finite}\}$\par
Infinity refers to something without any limit, and is a concept relevant in a number of fields, predominantly mathematics and physics. Infinity is \textbf{not} a number.\par
The English word infinity derives from Latin \textbf{\textit{infinitas}}, which can be translated as "unboundedness", itself derived from the Greek word \textbf{\textit{apeiros}}, meaning "endless".\par
\end{center}

\begin{flushright}
compiled by: Michal Farkaš (last update: April 2018)$\quad \quad \quad$
\end{flushright}
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