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.

Definition

The extended hyper-E notation is based on the hyper-E notation 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^hm₁a₂^hm₂...^hm_n-1a_n^hm_n. Saibian uses @ to indicate the rest of the expression such as Bowers uses # to indicate the rest of the array.

The difference between original and extended notation is that extended Hyper-E notation allows more than one consecutive #'s.

• 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 \(\#^1\) = \(\#\)).

It seems similar to linear array notation rules. 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.)

Examples

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

  This expression decomposes into Ea##b expressions by applying rule 4 repeatedly.

• 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.

• 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.

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

  Quadruple hyperions decompose into triples.

• 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).