Nathan's Googolisms - Nathan's Notations

latex:
a = b\to c\to \\ \left[ a,d \right] = b\to c\to d \\ \left[ a \right] = b\to c\to b \\ \left[ a,c,d \right] = b\to c\to c\to d \\ [a,,d] = b\to c\to b\to d \\ [a,a\overset{\text{limit}}{\longrightarrow} n,d] = b\to c\text{ }\underset{\times\text{n}} {\underbrace{\to a}}\to d \\ [a,a,] = b\to c\text{ }\underset{\small \times\normalsize \infty }{\underbrace{\to a}}\to b \\ [a,a,d] = b\to c\text{ }\underset{\small \times\normalsize \infty }{\underbrace{\to a}}\to d \\ \\\text{Example:} \\ x=5\to 4\to \\ [x,x\overset{\text{limit}}{\longrightarrow} 2,5] = 5\to 4\to (5\to 4\to (5\to 4\to 5)) \\ = \\ 5\underset{\underset{\underset{5}{\underbrace{5\uparrow \cdots \uparrow 5}}}{\underbrace{5\uparrow \cdots \uparrow 4}}}{\underbrace{\uparrow \cdots \uparrow}} 4 \\\text{Chain Array Recursion Notation (CARN):}\\ \text{Adds a couple new factors!} \\ [a,a\overset{\text{limit}}{\longrightarrow} n,\overset{\small \text{recursion limit}} {a\rightrightarrows k},j_{z},d] = \\ \text{It's hard to explain in LaTeX, so I'll explain in words. }\\ \text{The beginning works the exact same, but the special part is k and j. }\\ \text{k is the recursion limit of the limit.} \\ \text{This means this is how many times the expression is nested inside of the limit, and the limit's limit, etc.} \\ \text{j is the level of recursion.} \\ \text{If j = 1, the recursion level is 1, which means the recursion limit will nest the expression in the limit.} \\ \text{If j = 2, the recursion level is 2, which means recursion limit will nest the expression in the first recursion limit.} \\ \text{If j = 3, the recursion level is 3, which means recursion limit will nest the expression in the second recursion limit (which is j = 2)} \\ \text{and so on...} \\ \text{The subscript z on j represents the first j. If you have }j_{1}\text{, that means there is a nested expression inside of the j and that is its value.}\\\text{ The value for j originally will now be the j in the inner-most nested-expression} \\ \text{Finally, if you have a recursion limit, the original limit value acts as the limit in the inner-most nested-expression.} \\\\\text{Example:} \\ x=10\to 10\to \\ [x,x\overset{\text{limit}}{\longrightarrow} 10,\overset{\small \text{recursion limit}} {x\rightrightarrows 2},1_{0},10] \\ = \\ [x,x\overset{\text{limit}}{\longrightarrow} [x,x\overset{\text{limit}}{\longrightarrow} [x,x\overset{\text{limit}}{\longrightarrow} 10,10],10],10]

Sigma Ligma Notation

uses image to display defintion as its all in latex

latex:

\small \textbf{Sigma Ligma Notation} \\ \large \left\{ \underset{\Large \Sigma}{\underline{a,b,c,d,e}} \right\} \\\\ \text{This represents a sequence of } b \text{ hyperoperations on } a, \text{ starting from hyperoperation level } c, \\ \text{increasing the operation level } d \text{ times, with a step size of } e. \\ \text{Let the result be:} \\\\\Large a \mathbin{[c_1]} a \mathbin{[c_2]} a \dots \mathbin{[c_{b-1}]} a \\\\ \text{Where } [n] \text{ is the } n\text{-th hyperoperation (e.g., } [0] = -,\ [1] = \times,\ [2] = \uparrow,\ \text{etc.), and each } c_i \text{ is defined as:} \\ \textbf{If } d \le b: \\\\\Large c_i = c + e \cdot \left\lfloor \frac{d \cdot (b - i)}{b} \right\rfloor \\\\ \textbf{If } d > b,\ \text{let } q = \left\lfloor \frac{d}{b} \right\rfloor,\ r = d \bmod b.\ \text{Then:} \\ \\\Large c_i = c + e \cdot \begin{cases} q + 1 & \text{if } i \le r \\ q & \text{otherwise} \end{cases} \\\\ \textbf{Example:} \\ \left\{ \underset{\Large \Sigma}{\underline{a,6,1,3,1}} \right\} = a [1] a [1] a [1] a [2] a [2] a [3] a = ((((a \times a) \times a) \uparrow a) \uparrow a) \uparrow \uparrow a

Evil Matrix (Notation Included)

uses image to display defintion as its all in latex

latex: too lazy to reformat the latex gahhh

Penumbral Termination

Staged Array Notation

image is so big that it automatically lowers the quality and you cant read it.

here is latex:

\textbf{Staged Array Notation} \\[-5pt]\small\color{darkgray} \mathcal{NathanTroyAdams154} \\ \textbf{Names:} \\ \{\#\}-\text{Mono Array} \\ /\#/-\text{Dua Array} \\ [\#]-\text{Tria Array} \\ \hat{(}\#\hat{)}-\text{Quadra Array} \\ \boxed{\#}-\text{Penta Array} \\ '^{b}a - a\text{ to the }b_{th}\text{ caliber.} \\ "a - \text{Quantifier }a \\ `a - \text{Geared }a \\ a_{\star^{b}_{c}}-b_{th}\text{ star }c, a \\ a_{\ast^{b}_{c}}-b_{th}\text{ star }2c, a \\ a_{\#^{b}_{c}}-a \text{ hashed to } b, c \\ 〈\#〉-\text{Angled #} \\ a?-\text{Questionable }a \\ a\&^{c}b-a\ c_{th}\text{ ampersanded to } b \\ a\circ b-a\text{ circled to }b \\ a\bullet b-a\text{ shot to }b \\ \bowtie-\ \text{Twist} \\ \Omega\ -\ \text{Finalized} \\ \textbf{Mono} \\ a=\{a\} \\ a\uparrow^{b} a=\{a,b\} \\ \underset{b}{\underbrace{a\to a\to \cdots \to a}}=\{a,'b\} \\ a\to _{b}a=\{a,''b\} \\ a\to ^{b}a=\{a,'''b\} \\ a\to ^{c}_{b}a=\{a,''b,'''c\}=\{a,'''c,''b\} \\ \underset{b}{\underbrace{a\to_{c} a\to_{c} \cdots \to_{c} a}}=\{a,'b,''c\} \\ \underset{b}{\underbrace{a\to^{c} a\to^{c} \cdots \to^{c} a}}=\{a,'b,'''c\} \\ \underset{b}{\underbrace{a\to^{d}_{c} a\to^{d}_{c} \cdots \to^{d}_{c} a}}=\{a,'b,''c,'''d\} \\ \{a,'^{1d}a,'^{2d}a,\cdots ,'^{cd}a\} =\{a,\overset{d}{\overbrace{'\overset{\cdots }{}'}}b,`c\}=\{a,b:\;"d,`c\} \\ (\text{Note that b's value is now unimportant}) \\ \{a,b:\;"1,`4\}=\{a,'a,''a,'''a,''''a\} \\=\\ \{a,\underset{a}{\underbrace{'a,''a,'''a\},'a,''a,'''a\}\cdots ,'a,''a,'''a\}}} \\ \{a,b:\;"1,`5\}=\{a,'a,''a,'''a,''''a,'''''a\} \\=\\ \{a,\underset{a}{\underbrace{'a,''a,'''a,''''a\},'a,''a,'''a,''''a\}\cdots ,'a,''a,'''a,''''a\}}} \\ \{a,b:\;"2,`3\}=\{a,''a,''''a,''''''a\} \\=\\ \underset{\underset{\underset{\underset{\underset{a}{\vdots }}{a}}{\vdots }}{a}}{\underbrace{a\to _{a}a\to _{a}\cdots \to _{a}a}} \\ \{10,1:"1,`10\}=\{10,2:"1,`10\} \\ \textbf{Dua} \\ /a/=\{a,a:"a,a`\} \\ /a,b/=\overset{b}{\overbrace{/\cdots/}}a\overset{b}{\overbrace{/\cdots/}} \\ /^{1}a,b/^{1}=/a,b/ \\ /^{2}a,b/^{2}=//a,b/,/a,b// \\ /^{3}a,b/^{3}=///a,b/,/a,b//,//a,b/,/a,b/// \\ /^{b}a/^{b}=/^{1}a,b/^{1} \\ /a,b,c/=/^{c}a,b/^{c} \\ /a,'b/=/\overset{b}{\overbrace{a,a,\cdots a}}/ \\ //a,'b/,'/a,'b//=/a,''b/ \\ //a,''b/,''/a,''b//=/a,'''b/ \\ /a,'^{c}b/=/a,\overset{c}{\overbrace{'\cdots'}}b/ \\ //a,'^{c}b/,'^{c}/a,'^{c}b//=/a,'^{c+1}b/ \\ \textbf{Tria} \\ [a]=/a,'^{a}a/ \\ [a,b]=[^{b}a]^{b}=\overset{b}{\overbrace{[\cdots[}}a\overset{b}{\overbrace{]\cdots]}} \\ [a,b,c]=[^{b}_{c}a]^{b}_{c}=\overset{c}{\overbrace{[^{b}\cdots[^{b}}}a\overset{c}{\overbrace{]^{b}\cdots]^{b}}} \\ [a,b,c,d]=\overset{d}{\overbrace{[^{b}_{c}\cdots[^{b}_{c}}}a\overset{d}{\overbrace{]^{b}_{c}\cdots]^{b}_{c}}} \\ [a,'b]=[\overset{[a,b]}{\overbrace{a,a,\cdots,a}}] \\ [a,'^{c}b]=[\overset{[a,'^{(c-1)}b)]}{\overbrace{a,a,\cdots,a}}] \\ ['^{a}a]=[\overset{[a,'^{(a-1)}a]}{\overbrace{a,a,\cdots,a}}] \\ [a_{\star ^{1}}]=['^{a}a] \\ [a_{\star ^{2}}]=['^{a}['^{a}a]] \\ [a_{\star ^{b}}]=\overset{b}{\overbrace{['^{a}\cdots ['^{a}['^{a}}}a\overset{b}{\overbrace{]]\cdots ]}} \\ [a_{\ast ^{1}}]=[a_{\star ^{a}}] \\ [a_{\ast ^{2}}]=[a_{\star ^{a_{\star ^{a}}}}] \\ [a_{\ast ^{3}}]=[a_{\star ^{a_{\star ^{a_{\star ^{a}}}}}}] \\ [a_{\ast ^{b}}]=[\overset{b}{\overbrace{a_{\star ^{a_{\star ^{a_{\star ^{a\cdots a_{\star ^{a}}}}}}}}}}] \\ [a_{\star_{1} ^{b}}]=[a_{\star ^{b}}] \\ [a_{\star_{2} ^{b}}]=[a_{\ast ^{b}}] \\ [a_{\star_{3} ^{b}}]=[\overset{b}{\overbrace{a_{\ast ^{a_{\ast ^{a_{\ast ^{a\cdots a_{\ast ^{a}}}}}}}}}}] \\ [a_{\star_{a} ^{1}}]=[a_{\#_{1} ^{1}}] \\ [a_{\#_{1} ^{a}}]=[a_{\#_{2} ^{1}}] \\ [a_{\#_{a} ^{1}}]=[a_{'\#_{1} ^{1}}] \\ [a_{\overset{a}{\overbrace{'\cdots'}}\#_{1} ^{1}}]=[a_{'^{a}\#_{1} ^{1}}]=[a_{(\#_{1} ^{1})_{\star^{1}_{1}}}] \\ [a_{\underset{a}{\underbrace{(\#_{1} ^{1})_{(\#_{1} ^{1})_{\ddots _{(\#_{1} ^{1})_{}}}}}}}]=[^{〈}\ {'} \ ^{〉}a] \\ [^{〈}\ {'} \ ^{〉} \ ^{〈}\ {'} \ ^{〉}a]=[[^{〈}\ {'} \ ^{〉}a]_{\underset{[^{〈}\ {'} \ ^{〉}a]}{\underbrace{(\#_{1} ^{1})_{(\#_{1} ^{1})_{\ddots _{(\#_{1} ^{1})_{}}}}}}}] \\ [^{〈}\ {'} \ ^{〉^{a}}a]=\overset{a}{\overbrace{[\cdots[}}\ ^{〈}\ {'} \ ^{〉^{(a-1)}}a]_{\underset{[^{〈}\ {'} \ ^{〉^{(a-1)}}a]}{\underbrace{(\#_{1} ^{1})_{(\#_{1} ^{1})_{\ddots _{(\#_{1} ^{1})_{}}}}}}}\overset{(a-1)}{\overbrace{]\cdots]}}=[a_{〈\star^{1}_{1}〉}] \\ [a_{\underset{a}{\underbrace{(〈\#_{1} ^{1}〉)_{(〈\#_{1} ^{1}〉)_{\ddots _{(〈\#_{1} ^{1}〉)_{}}}}}}}]=[^{〈〈}\ {'} \ ^{〉〉}a] \\ [^{\overset{a}{\overbrace{〈\cdots〈}}}\ {'} \ ^{\overset{a}{\overbrace{〉\cdots〉}}}a]=[a?] \\ \textbf{Quadra} \\ \hat{(}a\&1\hat{)}=[a?] \\ \hat{(} a\&2 \hat{)}=[^{\overset{[a?]}{\overbrace{〈\cdots〈}}}\ {'} \ ^{\overset{[a?]}{\overbrace{〉\cdots〉}}}[a?]] \\ \hat{(}a\&a\hat{)}=[^{\overset{\hat{(}a\&(a-1)\hat{)}}{\overbrace{〈\cdots〈}}}\ {'} \ ^{\overset{\hat{(}a\&(a-1)\hat{)}}{\overbrace{〉\cdots〉}}}\hat{(}a\&(a-1)\hat{)}]=\hat{(}a\&\&1\hat{)} \\ \hat{(}a\&\&a\hat{)}=[^{\overset{\hat{(}a\&\&(a-1)\hat{)}}{\overbrace{〈\cdots〈}}}\ {'} \ ^{\overset{\hat{(}a\&\&(a-1)\hat{)}}{\overbrace{〉\cdots〉}}}\hat{(}a\&\&(a-1)\hat{)}]=\hat{(}a\&\&\&1\hat{)} \\ \hat{(}a\overset{b}{\overbrace{\&\cdots\&}}c\hat{)}=\hat{(}a\&^{b}c\hat{)} \\ \hat{(}a\&^{a}a\hat{)}=\hat{(}'^{1}a\hat{)} \\ \hat{(}\hat{(}'^{1}a\hat{)}\&^{\hat{(}'^{1}a\hat{)}}\hat{(}'^{1}a\hat{)}\hat{)}=\hat{(}'^{2}a\hat{)} \\ \hat{(}\hat{(}'^{(a-1)}a\hat{)}\&^{\hat{(}'^{a-1)}a\hat{)}}\hat{(}'^{a-1)}a\hat{)}\hat{)}=\hat{(}'^{a}a\hat{)}=\hat{(}'^{'1}a\hat{)} \\ \hat{(}'^{'2}a\hat{)}=\hat{(}'^{'1}\hat{(}'^{'1}a\hat{)}\hat{)} \\ \hat{(}'^{'3}a\hat{)}=\hat{(}'^{'2}\hat{(}'^{'2}a\hat{)}\hat{)}=\hat{(}'^{'1}\hat{(}'^{'1}\hat{(}'^{'1}\hat{(}'^{'1}a\hat{)}\hat{)}\hat{)}\hat{)} \\ \hat{(}'^{'a}a\hat{)}=\overset{2^{a}}{\overbrace{\hat{(}'^{'1}\cdots'^{'1}\hat{(}}}\ '^{'1}a\overset{2^{a}}{\overbrace{\hat{)}\cdots\hat{)}=\hat{(}'^{'^{'}1}a\hat{)}}} \\ \hat{(}'^{'^{'}a}a\hat{)}=\overset{2^{a}}{\overbrace{\hat{(}'^{'^{'}1}\cdots'^{'^{'}1}\hat{(}}}\ '^{'^{'}1}a\overset{2^{a}}{\overbrace{\hat{)}\cdots\hat{)}=\hat{(}'^{'^{'^{'}}1}a\hat{)}}} \\ \hat{(}a\bigg\{\ '^{'^{\cdot^{\cdot^{\cdot^{'}}}}a}a\hat{)}=\overset{2^{a}}{\overbrace{\hat{(}a\bigg\{\ '^{'^{\cdot^{\cdot^{\cdot^{'}}}}1}\cdots a\bigg\{\ '^{'^{\cdot^{\cdot^{\cdot^{'}}}}1}\hat{(}}}\ a\bigg\{\ '^{'^{\cdot^{\cdot^{\cdot^{'}}}}1}a\overset{2^{a}}{\overbrace{\hat{)}\cdots\hat{)}=\hat{(}(a+1)\bigg\{\ '^{'^{\cdot^{\cdot^{\cdot^{'}}}}1}a\hat{)}}}=\hat{(}a\circ 1\hat{)} \\ \hat{(}a\circ 2\hat{)}=\hat{(}\hat{(}a\circ 1\hat{)}\bigg\{\ '^{'^{\cdot^{\cdot^{\cdot^{'}}}}\hat{(}a\circ 1\hat{)}}\hat{(}a\circ 1\hat{)}\hat{)} \\ \hat{(}a\circ a\hat{)}=\hat{(}\hat{(}a\circ (a-1)\hat{)}\bigg\{\ '^{'^{\cdot^{\cdot^{\cdot^{'}}}}\hat{(}a\circ (a-1)\hat{)}}\hat{(}a\circ (a-1)\hat{)}\hat{)} \\ \textbf{Penta} \\ \boxed{a}=\hat{(}a\circ a\hat{)} \\ \boxed{a,b}=\boxed{\overset{b}{\overbrace{\cdots}}\underset{\large \vdots }{\overset{\large \vdots }{\boxed{a}}}\cdots} \\ \boxed{a,b,c}=\boxed{\overset{c}{\overbrace{\cdots}}\underset{\large \vdots }{\overset{\large \vdots }{\boxed{a,b},\boxed{a,b}}}\cdots,\boxed{a,b}} \\ \boxed{a\bullet ^{1}}=\boxed{\overset{a}{\overbrace{a,a,\cdots,a}}} \\ \boxed{a\bullet ^{2}}=\boxed{\overset{\boxed{a\bullet ^{1}}}{\overbrace{\boxed{a\bullet ^{1}},\boxed{a\bullet ^{1}},\cdots,\boxed{a\bullet ^{1}}}}} \\ \boxed{a\bullet ^{a}}=\boxed{\overset{\boxed{a\bullet ^{(a-1)}}}{\overbrace{\boxed{a\bullet ^{(a-1)}},\boxed{a\bullet ^{(a-1)}},\cdots,\boxed{a\bullet ^{(a-1)}}}}} \\ \boxed{a\bullet^{1}_{1}}=\boxed{a\bullet^{1}}=\boxed{a\bullet} \\ \boxed{a\bullet ^{\boxed{a\bullet}}}=\boxed{a\bullet_{2}^{1}} \\ \boxed{(a\bullet_{2}^{1})^{\boxed{(a\bullet_{2}^{1})}}}=\boxed{a\bullet_{3}^{1}} \\ \boxed{(a\bullet_{(a-1)}^{1})^{\boxed{(a\bullet_{(a-1)}^{1})}}}=\boxed{a\bullet_{a}^{1}}=\boxed{'^{1}a} \\ \boxed{a\bullet_{'^{1}a}^{1}}=\boxed{'^{2}a} \\ \boxed{a\bullet_{'^{(a-1)}a}^{1}}=\boxed{'^{a}a}=\boxed{a_{\bowtie ^{1}}} \\ \boxed{a_{\bowtie ^{1}}}=\boxed{a_{\bowtie ^{1}_{1}}} \\ \boxed{'^{'^{a}a}a}=\boxed{a_{\bowtie ^{2}}} \\ \boxed{'^{'^{'^{a}a}a}a}=\boxed{a_{\bowtie ^{3}}} \\ \boxed{a_{\bowtie ^{a}}}=\boxed{a_{\bowtie ^{1}_{2}}} \\ \boxed{a_{\bowtie ^{a_{\bowtie ^{a}}}}}=\boxed{a_{\bowtie ^{1}_{3}}} \\ \boxed{a_{\overset{a_{\bowtie}}{\overbrace{\bowtie ^{1}_{a_{\bowtie_{\ddots _{a_{\bowtie}}}}}}}}}=\boxed{^{(}\ '\:^{)\omega}a} \\ \boxed{^{(}\ '\:^{)\omega^{^{(}\ '\:^{)\omega}}}a}=\boxed{^{(}\ '\:^{)\omega^{\omega}}a} \\ \boxed{^{(}\ '\:^{)\omega^{^{(}\ '\:^{)\omega^{(}\ '\:^{)\omega}}}}a}=\boxed{^{(}\ '\:^{)\omega^{\omega^{\omega}}}a} \\ \boxed{^{(}\ '\:^{)\overset{\boxed{^{(}\ '\:^{)\omega}a}}{\overbrace{\omega^{\omega^{\omega^{\cdot^{\cdot^{\cdot^{\omega}}}}}}}}}a}=\boxed{a_{\omega}} \\ \boxed{a_{\omega_{\omega}}}=\boxed{\boxed{a_{\omega}}_{\omega}} \\ \boxed{a\overset{a_{\omega}}{\overbrace{_{\omega_{\ddots_{\omega}}}}}}=\boxed{a_{\Gamma_{1}^{1}}} \\ \boxed{a\overset{a_{\Gamma_{1}^{1}}}{\overbrace{_{\omega_{\ddots_{\omega}}}}}}=\boxed{a_{\Gamma_{1}^{2}}} \\ \boxed{a\overset{a_{\Gamma_{1}^{(a-1)}}}{\overbrace{_{\omega_{\ddots_{\omega}}}}}}=\boxed{a_{\Gamma_{1}^{a}}}=\boxed{a_{\Gamma_{2}^{1}}} \\ \boxed{a_{\Gamma_{a}^{1}}}=\boxed{a\overset{a_{\Gamma_{(a-1)}^{(a-1)}}}{\overbrace{_{\omega_{\ddots_{\omega}}}}}} \\ = \\ \boxed{a_{\Omega}}

Tailed Arrow Notation

Coruscating Defluctuation

Crusading Arrow Notation

Expanding Arrow Notation

Extended Pie Scale

This is based off MiTerSkyArk's Pie Scale Function. ( PSF(n) = π*(10↑n) )

^ is superscript, _ is subscript, ↑ is Knuth's Up Arrow.

Π^(1) (0) = Π(0) = PSF(0) = π

If a less than or equal to 1, it is ignored.

Π(1) = PSF(1) = π*10

Π(n#1) = Π(n) = PSF(n) = π*(10↑n)

Π(n#2) = PSF(PSF(n)) = π*(10↑(π*(10↑n)))

Π(n#k) = PSF(...k PSF('s...PSF(n)...)

Π(n##1) = Π(n#(Π(n#n)))

Π(n#^(b)1) = Π(n#(...b Π(n#('s...Π(n#n)...)

Π^(a) (n#1) = PSF_(a) (n) = π*(10↑^(a)n)

Example:

Π^(2) (3#^(2)1) = Π^(2) (3#(Π^(2) (3#3))) = Π^(2) (3#(PSF_(2) (PSF_(2) (PSF_(2) (3))))) = PSF_(2) (...(PSF_(2) (PSF_(2) (PSF_(2) (3)))) PSF_(2) ('s...PSF_(2) (3)...)

Simpler way to write it:

EPS = Π

Π^(2) (1) = EPS^(2) (1)

DRAGONLADY

DRAGONLADY is inspired by this song.

b and k are positive integers.

on the last line before Large Number Example, k refers to repetition of [n], then finally [b]. n represents #.

On the first page about [b] =, the unshown numbers are also b.

Etymology :

Other suffixes and prefixes can be added, such as Tera-, -illion, Quad-, etc.

Mono will be used a lot, but that just represents the one. You can use any similar prefix interchangeably to represent differences.

Suffixes :

[1] = Monolemna (-lemna)

#[1] = Monophotil (-photil)

#+2[1] = Monobiathriphotil (-mono) (-athri) (-photil)

2#[1] = Monobimultiphotil (-multi) (-bi)

#^(2)[1] = Monobiexpophotil (-expo)

[#][1] = Monolemnaphotil

2¬[1] = Monobimegielphotil (-megiel)

2¬^(3)[1] = Monobitrimegielphotil (-tri)

Factorial Array Notation

Functions

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