Notation hyper-E étendue

La notation hyper-E étendue (Extended Hyper-E notation; E# en abrégé) est une notation pour les grands nombres imaginée par Sbiis Saibian. Il s'agit de la deuxième étape du système extensible-E.

Définition

The extended hyper-E notation is based on the notation hyper-E and allows multiple hyperions to appear between each entry. The number of hyperions following entry a_n is represented by h(n). For the sake of this definition, #ⁿ is a shorthand for n successive hyperion marks. For example, a full expression would be written E(b)a₁^h(1)a₂^h(2)...^h(n-1)a_n^h(n). Saibian uses @ to indicate the rest of the expression.

• Rule 1. If there are no hyperions:

  \(E(b)x = b^x\).

• Rule 2. If the last entry is 1:

  \(E(b) @ \#^{h(n-1)}{a_n}\#^{h(n)}1 = E(b) @ \#^{h(n-1)}{a_n}\).

• Rule 3. If \(h(n-1)>1\):

  \(E(b) @ \#^{h(n-2)}{a_{n-1}}\#^{h(n-1)}{a_n} = E(b) @ \#^{h(n-2)}{a_{n-1}}\#^{h(n-1)-1}{a_{n-1}}\#^{h(n-1)}{a_n-1}\).

• Rule 4. Otherwise:

  \(E(b) @ \#^{h(n-2)}{a_{n-1}}\#{a_n} = E(b) @ \#^{h(n-2)}(E(b) @ \#^{h(n-2)}{a_{n-1}}\#{a_n-1})\) (note \(\#^1\) = \(\#\)).

We can also rewrite it in plain English:

1. If there is only one argument x, the value of the expression is b^x.
2. If the last entry is 1, it may be removed.
3. Let h be the length of the last set of hyperion marks. If h > 1:
   1. Remove the last entry of the expression and call it r.
   2. Again remove the last entry of the expression; this time call it z.
   3. Repeat "z" r times with h - 1 hyperion marks in between each repetition. Append this to the end of the expression. (Restore a removed hyperion mark sequence to glue the two expressions together.)
4. If the last set of hyperion marks is of length one:
   1. Evaluate the original expression, but with the last entry decreased by 1. Call this value z.
   2. Remove the last two entries of the expression.
   3. Add z as an entry to the end of the expression. (Again, restore a removed hyperion mark sequence to glue the two expressions together.)

Approximation

We evaluate the recursion level of this notation with the hiérarchie de croissance rapide (HCR) with the Wainer hierarchy.

As written in the notation hyper-E, we have relationship

$a \uparrow^{c}b = E(a)\underbrace{1\#1\#...1\#}_{(1\#) \times (c-1)}b$

Comparing the recusrion level, adding a # corresponds to adding an uparrow, which is in HCR adding 1. adding 2 #s corresponds to adding 2 in HCR, and adding n #s corresponds to adding n in HCR.

Two hyperion marks (deutero-hyperions), ##n, is expanded as n times repetitions of #, and therefore equivalent to n or ω in HCR. Repetitions of n ##s corresponds to ω×n.

Three hyperion marks (trito-hyperions), ###n, is expanded as n times repetitions of ##, and therefore equivalen to ω×n or ω² in HCR. Repetitions of n ###s corresponds to ω²×n.

Therefore, roughly speaking, n+1 hyperion marks ##...## corresponds to ωⁿ in HCR.

Examples and googolisms

Some googolisms with this notation is shown for showing how the calculation proceeds.

• E100##100 = E100#100#100#...#100#100#100 with 100 repetitions of 100 = gugold

  By using the rough estimation shown above, it is approximated as f_ω(100).

• E100##100##100 = E100##100#100#...#100#100 with 100 repetitions of 100 = gugolthra

  We ignore the first ## until the second one has been expanded and all the 100s have been solved. By using the rough estimation shown above, it is approximated as f_ω×2(100).

• E100###100 = E100##100##...##100##100 with 100 repetitions of 100 = throogol

  Three hyperion marks (trito-hyperions) constitute a repetition of two hyperion marks. Remember, the double marks are solved from right to left. It is approximated as f_ω²(100).

• E100####100 = E100###100###...###100###100 with 100 repetitions of 100 = teroogol

  Quadruple hyperions decompose into triples. It is approximated as f_ω³(100).

• Godgahlah = E100#####...#####100 with 100 hyperion marks or E100#¹⁰⁰100

  Sets of 100 hyperion marks decompose into 99s, 99s decompose into 98s, etc. Also note that the superscript 100 means that there are 100 #'s, and should not be confused with E100#(¹⁰⁰100). It is approximated as f_ω⁹⁹(100) ≈ f_ω^ω(99).