FANDOM


The world of big numbers can be confusing, so I thought about creating a nice little table that shows the progression of larger and larger number-realms in some of the simplest notations. I'll compare BEAF, my letter notation, and Denis Maksudov's extended arrows. I'm not using the Fast Growing Hierarchy here even though it is standard, because I fear it will only confuse people. Suffice to say that for the bigger numbers, the fast growing hierarchy is comparable to Maksudov's arrows.

So here we go:


Letter Notation My Array Notation BEAF Arrows Notes
E2 1|2 {10,2,1} 10↑2 One Hundred = 100
E3 1|3 {10,3,1} 10↑3 One Thousand = 1000
E4 1|4 {10,4,1} 10↑4 Ten Thousand = 10,000
E5 1|5 {10,5,1} 10↑5 One Hundred Thosuand = 100,000
E6 1|6 {10,6,1} 10↑6 One Million = 1,000,000
E7 1|7 {10,7,1} 10↑7 Ten Million = 10,000,000
E8 1|8 {10,8,1} 10↑8 One Hundred Million = 100,000,000
E9 1|9 {10,9,1} 10↑9 One Billion = 1,000,000,000
E10 = F2 1|10 = 2|2 {10,2,2} 10↑10 = 10↑↑2 Ten Billion = 1010
F3 2|3 {10,3,2} 10↑↑3 101010
F4 2|4 {10,4,2} 10↑↑4 10101010
F5 2|5 {10,5,2} 10↑↑5 1010101010
F6 2|6 {10,6,2} 10↑↑6 A power tower of 6 tens
F7 2|7 {10,7,2} 10↑↑7 A power tower of 7 tens
F8 2|8 {10,8,2} 10↑↑8 A power tower of 8 tens
F9 2|9 {10,9,2} 10↑↑9 A power tower of 9 tens
F10 = G2 2|10 = 3|2 {10,2,3} 10↑↑10 = 10↑↑↑2 A power tower of 10 tens, called the "Decker" by Jonathan Bowers
G3 3|3 {10,3,3} 10↑↑↑3 10↑↑10↑↑10 (evaluated from right to left)
G4 3|4 {10,4,3} 10↑↑↑4 10↑↑10↑↑10↑↑10
G5 3|5 {10,5,3} 10↑↑↑5 10↑↑10↑↑10↑↑10↑↑10
G6 3|6 {10,6,3} 10↑↑↑6 10↑↑10↑↑10↑↑10↑↑10↑↑10
G7 3|7 {10,7,3} 10↑↑↑7 10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑10
G8 3|8 {10,8,3} 10↑↑↑8 10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑10
G9 3|9 {10,9,3} 10↑↑↑9 10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑10
G10 = H2 3|10 = 4|2 {10,2,4} 10↑↑↑10 = 10↑↑↑↑2 10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑10↑↑10
H3 4|3 {10,3,4} 10↑↑↑↑3 10↑↑↑10↑↑↑10
H4 4|4 {10,4,4} 10↑↑↑↑4 10↑↑↑10↑↑↑10↑↑↑10
H5 4|5 {10,5,4} 10↑↑↑↑5 10↑↑↑10↑↑↑10↑↑↑10↑↑↑10
H6 4|6 {10,6,4} 10↑↑↑↑6 10↑↑↑10↑↑↑10↑↑↑10↑↑↑10↑↑↑10
H7 4|7 {10,7,4} 10↑↑↑↑7 10↑↑↑10↑↑↑10↑↑↑10↑↑↑10↑↑↑10↑↑↑10
H8 4|8 {10,8,4} 10↑↑↑↑8 10↑↑↑10↑↑↑10↑↑↑10↑↑↑10↑↑↑10↑↑↑10↑↑↑10
H9 4|9 {10,9,4} 10↑↑↑↑9 10↑↑↑10↑↑↑10↑↑↑10↑↑↑10↑↑↑10↑↑↑10↑↑↑10↑↑↑10
H10 = J4 4|10 = (1,0)|4 {10,10,4} 10↑↑↑↑10 =
10↑ω4

Jx = 10↑ωx = 10 [x arrows] 10

In Letter Notation, this is the only transition from one letter to the next where [this letter]10 ≠ [the next letter]2. Here H10=J4 rather than J2.

J5 (1,0)|5 {10,10,5} 10↑ω5 10↑↑↑↑↑10
J6 (1,0)|6 {10,10,6} 10↑ω6 10↑↑↑↑↑↑10
J7 (1,0)|7 {10,10,7} 10↑ω7 10↑↑↑↑↑↑↑10
J8 (1,0)|8 {10,10,8} 10↑ω8 10↑↑↑↑↑↑↑↑10
J9 (1,0)|9 {10,10,9} 10↑ω9 10↑↑↑↑↑↑↑↑↑10
J10 = K2 (1,0)|10 = (1,1)|2 {10,10,10} 10↑ω10 = 10↑ω+12 10↑↑↑↑↑↑↑↑↑↑10, called "tridecal" by Bowers because it is {10,10,10} in BEAF
K3 (1,1)|3 {10,3,1,2} 10↑ω+13

K works like the Graham's function:

K3 = 10 [K2 arrows] 10 = 10 [10↑↑↑↑↑↑↑↑↑↑10]

K4 (1,1)|4 {10,4,1,2} 10↑ω+14 K4 = 10 [K3 arrows] 10
K5 (1,1)|5 {10,5,1,2} 10↑ω+15 A Graham-style number with 5 layers of arrows
K6 (1,1)|6 {10,6,1,2} 10↑ω+16 A Graham-style number with 6 layers of arrows
K7 (1,1)|7 {10,7,1,2} 10↑ω+17 etc.
K8 (1,1)|8 {10,8,1,2} 10↑ω+18
K9 (1,1)|9 {10,9,1,2} 10↑ω+19
K10 = L2 (1,1)|10 = (1,2)|10 {10,10,1,2} 10↑ω+110 Graham's Number has 64 layers of arrows, so it will be about K64 or {10,64,1,2}
L3 (1,2)|3 {10,3,2,2} 10↑ω+23 L works as an iterated Graham function. So L3 is has L2 (=K10) layers of arrows in it!
L4 (1,2)|4 {10,4,2,2} 10↑ω+24 L4 has L3 layers of arrows in it
L5 (1,2)|5 {10,5,2,2} 10↑ω+25 L5 has L4 layers of arrows in it
L6 (1,2)|6 {10,6,2,2} 10↑ω+26 etc
L7 (1,2)|7 {10,7,2,2} 10↑ω+27
L8 (1,2)|8 {10,8,2,2} 10↑ω+28
L9 (1,2)|9 {10,9,2,2} 10↑ω+29
L10 = M2 (1,2)|10 = (1,3)|2 {10,10,2,2}

10↑ω+210

Iterating the L function gives us (1,3)|x.

So (1,3)|2 = L10.

(1,3)|3 = L(L10)

(1,3)|4 = LL(L10)

and so on

M3 (1,3)|10 {10,10,3,2} 10↑ω+310

Mx is defined as (1,x)|10.

So M3 = (1,3)|10.  = LLLLLLLLL10

M4 (1,4)|10 {10,10,4,2} 10↑ω+410

The function (1,4)|x is produced by iterating (1,3) x times. So:

(1,4)|2 = (1,3)|10

(1,4)|3 = (1,3)|(1,3)|10

(1,4)|4 = (1,3)|(1,3)|(1,3)|10

and so on.

And M4 = (1,4)|10 = (1,3)|(1,3)|(1,3)|(1,3)|(1,3)|(1,3)|(1,3)|(1,3)|(1,3)|10

M5 (1,5)|10 {10,10,5,2} 10↑ω+510 (1,4)|(1,4)|(1,4)|(1,4)|(1,4)|(1,4)|(1,4)|(1,4)|10
M6 (1,6)|10 {10,10,6,2} 10↑ω+610 (1,5)|(1,5)|(1,5)|(1,5)|(1,5)|(1,5)|(1,5)|(1,5)|10
M7 (1,7)|10 {10,10,7,2} 10↑ω+710 (1,6)|(1,6)|(1,6)|(1,6)|(1,6)|(1,6)|(1,6)|(1,6)|10
M8 (1,8)|10 {10,10,8,2} 10↑ω+810 (1,7)|(1,7)|(1,7)|(1,7)|(1,7)|(1,7)|(1,7)|(1,7)|10
M9 (1,9)|10 {10,10,9,2} 10↑ω+910 (1,8)|(1,8)|(1,8)|(1,8)|(1,8)|(1,8)|(1,8)|(1,8)|10
M10 = N2

(1,10)|10

= (2,0)|10

{10,10,10,2} 10↑ω×210 N2 = (2,0)|10 which then expands to:

(1,9)|(1,9)|(1,9)|(1,9)|(1,9)|(1,9)|(1,9)|(1,9)|(1,9)|10

N3 (3,0)|10 {10,10,10,3} 10↑ω×310 N3 = (3,0)|10 which then expands to:

(2,9)|(2,9)|(2,9)|(2,9)|(2,9)|(2,9)|(2,9)|(2,9)|(2,9)|10

The last (2,9) expands to a sequence of (2,8)'s, the last of which expands to a sequence of (2,7)'s and so on.

N4 (4,0)|10 {10,10,10,4} 10↑ω×410 etc
N5 (5,0)|10 {10,10,10,5} 10↑ω×510
N6 (6,0)|10 {10,10,10,6} 10↑ω×610
N7 (7,0)|10 {10,10,10,7} 10↑ω×710
N8 (8,0)|10 {10,10,10,8} 10↑ω×810
N9 (9,0)|10 {10,10,10,9} 10↑ω×910
N10 = P2

(10,0)|10

= (1,0,0)|10

{10,10,10,10} = 4 & 10 10↑ω210 Jonathan Bowers also calls this number the "General" (as in the rank in the army)
P3 (1,0,0,0)|10 {10,10,10,10,10} = 5 & 10 10↑ω310
P4 (1,0,0,0,0)|10 6 &10 10↑ω410
P5 (1,0,0,0,0,0)|10 7 & 10 10↑ω510
P6 (1,0,0,0,0,0,0)|10 8 & 10 10↑ω610
P7 (1,0,0,0,0,0,0,0)|10 9 & 10 10↑ω710
P8 (1,0,0,0,0,0,0,0,0)|10

10 & 10

10↑ω810 Bowers calls this number an "Iteral"
P9 (1,0,0,0,0,0,0,0,0,0)|10

11 & 10

10↑ω910
P10 = Q2 (1,0,0,0,0,0,0,0,0,0,0)|10 12 & 10 10↑ωω10
Q3 (this is beyond the limit of my array notation) 1010 & 10 10↑ω↑↑310 Starting here, the BEAF expressions aren't precisely equal to those given by Letter Notation.
Q4 101010 & 10 10↑ω↑↑410
Q5 10↑↑4 & 10 10↑ω↑↑510
Q6 10↑↑5 & 10 10↑ω↑↑610
Q7 10↑↑6 & 10 10↑ω↑↑710
Q8 10↑↑7 & 10 10↑ω↑↑810
Q9 10↑↑8 & 10 10↑ω↑↑910
Q10 = R2 10↑↑9 & 10 10↑ε₀10
R3 (this is beyond the limit of the well-defined portion of BEAF) 10↑φ(ε₀,0)10
R4 10↑φ(φ(ε₀,0),0)10
R5 10↑φ(φ(φ(ε₀,0),0),0)10
R6 10↑φ(φ(φ(φ(ε₀,0),0),0),0)10 = 10↑Γ₀6
R7 10↑Γ₀7
R8 10↑Γ₀8
R9 10↑Γ₀9
R10 = S2 10↑Γ₀10 The current limit of my letter notation
(S3) 10↑φ(1,0,0,0)10 From here the letters only serve as a rough ruler, rather than define specific numbers
(S4) 10↑φ(1,0,0,0,0)10
(S5) 10↑φ(1,0,0,0,0,0)10
(S6)

10↑φ(1,0,0,0,0,0,0)10 =

10↑SVO6

(S7) 10↑SVO7
(S8) 10↑SVO8
(S9) 10↑SVO9
(S10) = (T2) 10↑SVO10
(Tn) 10↑LVOn
(Vn) 10↑ψ(ψ₁(0))10
(Wn) 10↑ψ(Ωω)10
(Xn) 10↑ψ(ΩΩ)10
(Yn) 10↑ψ(ψ𝐈(0))10

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