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BORIS´S SEZRATION NOTATION

Hello,my name is Boris and I´ve allways been facinated by large numbers. So I decided to write this article to show you my own invention,that I like to call "Boris´s sezration notation" because I think the name is cool and I like having my name in something.

Let´s start with the simplest form of it:A two-chain

Sopose you have two whole possitive integers (a and b). The simplest sezration you can do with it is this:\([a,b]\) and that is equal to (in chained arrow notation) a→a→a→....→a and the number of a´s is b

Simple,but effective!

Now,let´s spice things up a bit with a three-chain!

[a,b,c]=c→c→c→....→c And there´s [a,b] many c´s. In other words [a,b,c]=[c,[a,b]]

Okay,but what about a four-chain?

[a,b,c,d]=[d,[a,b,c]]=d→d→d→d→...→d once again there´s [a,b,c] many d´s!!!

EXAMPLES:

[3,2]=3→3=27

[3,3]=3→3→3=3↑↑↑3=tritri

[3,4]=3→3→3→3 ~ G(G(27)) (using Graham`s number)

[3,7]=3→3→3→3→3→3→3

[3,7,5]=5→5→5→5→...→5→5([3,7] many fives)

[2,9,3,6]=6→6→6...→6→6 ([2,9,3] many sixes)

THE EXTENDED SEZRATION!

If what we did previously wasn`t crazy enuff,I have something new for you!

[a|b|a]=[a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a...a,a](where there are b many a`s)

EXAMPLES:

[3|4|3]=[3,3,3,3]=[3,[3,[3,[3]]]=[3,[3,[3↑↑↑3]]]=[3,[3→3→3→...(3↑↑↑3)....→3→3]] WOW!!!

We can also extend the extention!

[4|[5|10|5]|4]=[4|[5,5,5,5,5,5,5,5,5,5]|4]=[4,4,4,4,4,4,.....,4,4] and there are [5,5,5,5,5,5,5,5,5,5] many 4`s!

SECOND LEVEL SEZRATION!

This one was sugested by AarexWikia04

Sopose you have this [a||b||a],that equals [a|[a|[a|[a|[a|....a|[b]|a]|a]|a]|a....]|a] , with b many pairs of a`s on the sides!

EXAMPLE:

[3||4||3]=[3|[3|[3|[3|[4]|3]|3]|3]|3]=[3|[3|[3|[3,3,3,3]|3]|3]|3]=[3|[3|[3,3,3,3,....([3,3,3,3] three`s)....,3,3,3]|3]|3]=[3|[REALY BIG]|3]=[3,3,3,3,....(REALY BIG 3`s!)...3,3,3]

THIRD LEVEL SEZRATION!

[a|||b|||a]=[a||[a||[....a||[b]||a....]||a]||a] , with b many pairs of a.(No examples for this one)

FOURTH LEVEL SEZRATION!

[a||||b||||a]=[a|||[a|||[...a|||[b]|||a...]|||a]|||a] , with b pairs of a`s!

HIGHER DIMENTIONAL SEZRATION!

§=dimention of (§a)=dimention of a

[a(§c)b(§c)a]=[a||||||||||||||....||b|||...|||||||||||||||||||a] ,with c pairs of lines!!!

If you thought you know,what big means...think again!

TETRATIONAL SEZRATION!

[a(§(§b))c(§(§b))a]=[a(§[a(§b)c(§b)a])c(§[a(§b)c(§b)a])a]

In other words:

If [a(§b)c(§b)a]=x ,then [a(§(§b))c(§(§b))a]=[a(§x)c(§x)a]

PENTATIONAL,HEXATIONAL,HEPTATIONAL... SEZRATION!

Pentational-[a(§(§(§b)))c(§(§(§b)))a]

Hexational-[a(§(§(§(§b))))c(§(§(§(§b))))a]

Heptational-[a(§(§(§(§(§b)))))c(§(§(§(§(§b)))))a]

You can probably guess how powerfull these can be and how they work by yourself!

UPPER SEZRATION!

[a/b]=[a(§(§(...(§a)))...)b(§(§(...(§a)))...)a] where there are b many §`s

[a/b/2]=[[a/b]/[a/b]]

[a/b/3]=[[a/b/2]/[a/b/2]]

[a/b/4]=[[a/b/3]/[a/b/3]]

[a/b/c]=[[a/b/c-1]/[a/b/c-1]]

If [a/b/c]=x,then [a/b/c+1]=[x/x]=z and [a/b/c+2]=[[x/x]/[x/x]]=[z/z]

[a/b/c/2]=[[a/b/c]/[a/b/c]]

[a/b/c/3]=[[a/b/c/2]/[a/b/c/2]]

[a/b/c/d]=[[a/b/c/d-1]/[a/b/c/d-1]]

[a//b]=[a/a/a/a/a/a/..../a],whith b a`s

[a///b]=[a//a//a//a//...//a],whith b a`s

And so on!

ULTRA-SEZRATION!

Things like [n///n] can be expressed like this [n/n]​_{3}(where 3 is a subscript).

Now,you can write things like [n/n]_{100},without having to write them down,but that's not ultra-sezration(yet)!

Ultra-sezration occurs,when things like [a(@)b] appear.

[a(@)b]=[a/b]_{[a/b]_{[a/b]..._{[a/b]}...} with [a/b] subscripts of [a/b]

[a(@+1)b]=[a(@)[a(@)[a(@)[.....[a(@)a]]]]....]],with b+1 a's

[a(@+2)b]=[a(@+1)[a(@+1)[a(@+1)[...[a(@+1)a]...] - b+1 a's

[a(@+@)b]=[a(@+[a(@+[a(@+....[a(@)a])a])a])a...)a] - b+1 a's

[a(@+@+@)b] or [a(3@)b]=[a(@+@+[a(@+@+[a(@+@+....[a(@+@)a])a])a...])a] And so on!

[a(@^2)b]=[a(@*[a(@*[a(@*[...[a(@*b)a])a])a])a.....)a] with [a(@*b)a] a's

[a(@^3)b]=[a(@^2*[a(@^2*[....[a(@^2*b)a]a)]a)....a)])a] - [a(@^2*b)a] a's

[a(@^@)b]=[a(@^[a(@^[a(@^[....[a(@^b)a])a])a....])a] - [a(@^b)a] a's

[a(@^@^@)b]=[a(@^@^[a(@^@^[....[a(@^@^b)a])a])a....])a] - [a(@^@^b)a] a`s

[a(@↑↑@)b]=[a(@↑↑[a(@↑↑[...[a(@↑↑b)a])a])a]....)a])a] - [a(@↑↑b)a] a's

[a(@↑↑↑@)b]=[a(@↑↑↑[a(@↑↑↑[a(@...a(@↑↑↑[a(@↑↑↑b)a])a])a])a...])a] - [a(@↑↑↑b)a] a's

[a(@↑↑↑↑@)b]=[a(@↑↑↑↑[a(@↑↑↑↑[a(@....a(@↑↑↑↑[a(@↑↑↑↑b)a])a])a])a....])a]

[a(@→@→5)b]=[a(@→[a(@→[a(@→...[a(@→b→5)a]→5)a]→5)a]→5)a]...]→5)a] - [a(@→b→5)a] a's

[a(@→@→c)b]=[a(@→[a(@→[...[a(@→[a(@→b→c)a]→c)a]→c)a]→c)a]....]→c)a] - [a(@→b→c)a] a's

[a(@→@→@→2)b]=[a(@→@→[a(@→@→[a(@→@→....[a(@→@→[a(@→@→b→2)a]→2)a]→2)a]...]→2)a] - [a(@→@→b→2)a] a's

• NEW*

(suggested by Chronolegends,Alemagno12,AarexWikia04)

[a(g_@)b]=[a(@→@→@→@→@→@→...(@)...→@→2)b]~[a([@,@])b]

[a(g+1_@)b]=[a(g_@)[a(g_@)...[a(g_@)a]]]...]] - [a(g_@)b] pairs of brackets

[a(g+2_@)b]=[a(g+1_@)[a(g+1_@)...[a(g+1_@)a]]....] - [a(g_@)b] pairs of brackets

[a(g*2_@)b]~[a([@,@,@])b]

[a(g*3_@)b]~[a([@,@,@,@])b]

[a(g*n_@)b]~[a([@|n+1|@])b]

[a(gg_@)b]=[a(g*b_@)[....[a(g*b_@)[a(g*b_@)a]]...]] - [a(g*b_@)a] pairs of brackets

[a(ggg_@)b]=[a(gg_@)[a(gg_@)[...[a(gg_@)a]]...]] - [a(gg_@)b] pairs of brackets

[a(№g_@)b]=[a(ggggg...(b)...ggg_@)a]

[a(g_g_@)b]=[a(№g_@)[a(№g_@)[....[a(№g_@)a]...]] - [a(№g_@)b] pairs of brackets

[a(g_g_g_@)b]=[a(g_g_@)[a(g_g_@)[....[a(g_g_@)a]...]] - [a(g_g_@)b] pairs of brackets

[a(№(g_g)_@)b]=[a(g_g_g_g_g_...(b)..._g_g)a]

[a(g__g_@)b]=[a(№(g_g)_@)[a(№(g_g)_@)[...[a(№(g_g)_@)[a(№(g_g)_@)a])a])a...])a] - [a(№(g_g)_@)b] pairs of brackets

[a(g___g_@)b]=[a(g__g_@)[a(g__g_@)[...[a(g__g_@)[a(g__g_@)a])a])a...])a] - [a(g__g_@)b] pairs of brackets

[a(№(_g_)_g_@)b]=[a(g_________...(b)...___g_@)a]

[a(G_@)b]~[a([@(§b)@(§b)@])a]

[a(G+1_@)b]~[a([@(§(§b))@(§(§b))@])a]

[a(G+2_@)b]~[a([@(§(§(§b)))@(§(§(§b)))@])a]

[n(G+n_@)n]~[n([@/n+1]n)

[n(G*2_@)n]~[n([@/@/n+1])n]

[n(G*n_@)n]~[n([@/@/@/@...(n).../@/n+1])n]~[n([@//n+1])n]

[n(G^2_@)n]~[n([@//[@/n+1]])n]

[n(G^3_@)n]~[n([@///[@/n+1]])n]

[n(G^c_@)n]~[n([@/[@/n+1]]_{[c]})n]

[n(G↑↑2_@)n]~[n([@(@)n])n]

[n(G↑↑3_@)n]~[n([@(@^2)n])n]

[n(G↑↑4_@)n]~[n([@(@^@)n])n]

[n(G↑↑5_@)n]~[n([@(@↑³@)n])n]

[n(G↑↑6_@)n]~[n([@(@↑↑↑↑@)n])n]

[n(G↑↑b_@)n]~[n([@(@↑...(b-2)...↑@)n])n] (for b ≥ 5 )

[n(G↑³2_@)n]~[n([@(@→@→@→2)n])n]

[n(G↑³3_@)n]~[n([@(@→@→@→3)n])n]

[n(G↑³f_@)n]~[n([@(@→@→@→f)n])n]

[n(G↑↑↑↑f_@)n]~[n([@(@→@→@→@→f)n])n]

[n(G→f→v_@)n]~[n([@(@→@→@→...(v)...→@→@→f)n])n]~[n([@([@,v])n])n]~[n([@(g_@)v])n]

[n([G,4]_@)n] > [n([@(g*n_@)n])n]

[n([G,5]_@)n] > [n([@(G_@)n])n]

[n([G,6]_@)n] > [n([@(G+n_@)n])n]

[n([G,7]_@)n] > [n([@(G*n_@)n])n]

[n([G,8]_@)n] > [n([@(G^n_@)n])n]

[n([G,9]_@)n] > [n([@(G↑↑n_@)n])n]

[n([G,10]_@)n] > [n([@(G↑³n_@)n])n]

[n([G,11]_@)n] > [n([@(G↑↑↑↑n_@)n])n] [n([G,t]_@)n] > [n([@(G↑↑...(t-7)....↑↑n_@)n])n] (for t ≥ 8)

[n([G,G]_@)n] > [n([@(G→G→G→2_@)n])n]

[n([G,2,G]_@)n]=[n([G,[G,2]]_@)n] > [n([@([G,G]_@)n])n]

[n([G,3,G]_@)n]=[n([G,[G,3]]_@)n] > [n([@([G,2,G]_@)n])n]

[n([G,r,G]_@)n] > [n([@([G,r-1,G]_@)n])n]

[n([G,G,G]_@)n] > [n([@([G,n,G+n]_@)n])n]

[n([G,G,G,G]_@)n] > [n([@([G,G,n,G*n]_@)n])n]

[n([G|4|G]_@)n] > [n([@([G,G,G,n,G^n]_@)n])n]

[n([G|h|G]_@)n] > [n([@([G,G,G,..(h-1)...G,n,G↑↑...(h-3)...↑↑h]_@)n])n] (for h ≥ 4)

[n([G||h||G]_@)n] > ≈[n([@([G|h+3|G]_@)n])n]

[n([G|||h|||G]_@)n] > ≈[n([@([G||2h+3||G]_@)n])n

And so on!

COMPERISON:

[n,n]~f_w^2(n)

[n,n,n]~f_w^2(f_w^2(n))

[n|n|n]~f_w^2+1(n)

[n||n||n]~f_w^2+2(n)

[n|||n|||n]~f_w^2+3(n)

[n(§n)n(§n)n]~f_w^2+w(n)

[n/n]~f_w^2+w2(n)

[n/n/n]~f_w^2+w2+1(n)

[n/n/n/n]~f_w^2+w2+2(n)

[n//n]~f_w^2+w3(n)

[n///n] > f_w^2+w4(n)

[n/n]_{n} > f_w^2+w^2(n)

[n(@)n] > f_w^4(n)

[n(@+1)n] > f_w^4+w(n)

[n(@+2)n] > f_w^4+w2(n)

[n(@+n)n] > f_w^4+w^2(n)

[n(@+@)n] > f_w^w+w^4(n)

[n(@+@+@)n] > f_w^{w^w+w^4}(n)

[n(4@)n] > f_w^{w^{w^w+w^4}}}(n)

[n(@^2)n] > f_ε₀(n)

[n(@^3)n] > f_εε₃(n)

[n(@^@)n] > f_φ₂,₀(n)

[n(@^@^@)n] > f_φφ₂,ω(n)

[n(@↑↑@)n] > f_Γφ₂,ω(n) (f_Γ₀(n) is also called f_φ₁,₀,₀(n))

[n(@↑↑↑@)n] > f_ΓΓΓ...Γφ₂,ω ~f_φ₁,₀,₂,ω(n)

[n(@↑↑↑↑@)n] >> f_φ₁,₁,₀,₂,ω

[n(@→@→5)n] >> f_φ₁,₁,₀,₂,₃,ω(n)

[n(@→@→@→2)n] >> f_ψ(Capital Omega^{Capital Omega^{n-1}})(n)

[a(g_@)b] >> f_ψ(CO^{CO^{CO^{CO^{...(ω)...^{CO}}}....}}})(n)   (CO - Capital Omega)

And so on!

OK,SO WHAT'S NEXT?

We went through a lot of new notation and terms here and with the last few parts we pretty much hit the barier to ultra-sezration and probably all sezration.I could extend it furthur,but we got pretty far beyond comprehensive level and it becomes quite tidious at this point.So,what I decided to do is to create a completely new function,that might(or might not) grow faster,than any other computable function. But,I will do it in another blog.

Boboris02 (talk) 17:40, October 11, 2016 (UTC)Boboris02Boboris02 (talk) 17:40, October 11, 2016 (UTC)