Hyper-E notation

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Hyper-E Notation (E# for short) is a notation for large numbers devised by Sbiis Saibian. It was first introduced on his Web book One to Infinity: A Finite Journey on November 19, 2011, and was generalized to Extended Hyper-E Notation (xE# for short).

E# is not primitive recursive, and specifically the function E(n) = En##n eventually dominates all primitive recursive functions. In fact, in the fast-growing hierarchy, n →E100##n dominates f_n for all n < ω and is itself dominated by f_ω.

E# and xE# form part of a larger notation, the Extensible-E System, that also encompasses Cascading-E Notation.

Nathan Ho and Wojowu proved termination for the rules of Hyper-E Notation.

Original definition

The original Hyper-E Notation consists of a sequence a_n of one or more positive integer arguments separated by hyperions (or hyper marks) #. We notate this as E[b]a₁#a₂#...#a_n. b is called the base — if it is omitted, as it often is, it defaults to 10. "E[b]d" is also equal to "b^d".

The three rules are as follows:

• Rule 1. If there are no hyperions:

  E[b]x = b^x.

• Rule 2. If the last entry is 1:

  E[b]a₁#a₂#a₃...#a_n#1 = E[b]a₁#a₂#a₃...#a_n

• Rule 3. Otherwise(iteration last):

  E[b]a₁#a₂#a₃...#a_n-2#a_n-1#a_n = E[b]a₁#a₂#a₃...#a_n-2#(E[b]a₁#a₂#a₃...#a_n-2#a_n-1#a_n-1)

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. Otherwise...
   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.

1. If there is only one argument x, the value of the expression is .
2. If the last entry is 1, it may be removed.
3. Otherwise...
   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.

The priority of the rules are stated in the original webpage below the definition, "There are 3 rules used. If your given a E# expression, you first check to see if rule 1 applies. If it does not you check for rule 2. If that doesn't apply you go to 3." It disables a_n being 1 in the 3rd rule and avoids an overlapping case classification.

Extended definition

Extended Hyper-E Notation 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, 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. 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)-1a_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 = #).

Examples

• E6 = E6#1 = 10⁶ = million
• E100 = E100#1 = 10¹⁰⁰ = googol

  This is an example of rule 1 with a one-entry expressions. Since the base defaults to 10, we can abbreviate E(10)100 as E100.

• E100#2 = E(E100#1) = E10¹⁰⁰ = 10¹⁰¹⁰⁰ = googolplex
• E100#3 = E(E100#2) = E10¹⁰¹⁰⁰ = 10^1010100 = googolduplex

  This is an example of rule 3 (rule 4 in the expansion) with a two-entry expressions. In the second expression, the parentheses can be omitted: E(E100#1) = EE100#1.

• E303#1 = E303 = eceton = centillion = 10³⁰³
• E303#2 = ecetonplex = EE303 = 10¹⁰³⁰³
• E303#3 = EEE303 = 10^1010303 = ecetonduplex
• E1#3 = EEE1 = 10¹⁰¹⁰ = trialogue
• E1#4 = EEEE1 = 10¹⁰¹⁰¹⁰ = tetralogue
• E1#10 = EEEEEEEEEE1 = 10^^10 = Decker
• E303#6 = EEEEEE303 = 10^{1010101010303} = ecetonquintiplex
• E1#100 = EEE...EEE1 (100 E's) = giggol

  Repeated application of rule 3: E1#100 = EE1#99 = EEE1#98 = ...

• E100#100 = EEE...EEE100 (100 E's) = grangol

  This is the same as E1#100, but with a different first entry.

• E100#101 = EEE...EEE100 (101 E's) = grangolplex

  E100#101 = EE100#100 = , hence the name.

• E100#100#2 = E100#(E100#100) = EEE...EEE100 (grangol E's) = grangoldex

  Now we enter three-entry expressions.

• E100#100#3 = E100#(E100#100#2) = E100#(E100#(E100#100)) = EEE...EEE100 (grangoldex E's) = grangoldudex

  Increasing the value of the third entry makes nesting deeper and deeper.

• E100#100#100#100 = E100#100#(E100#100#100#99) = E100#100#(E100#100#(E100#100#100#98)) = ... gigangol

  Four-entry expressions are similar — they create deeper and deeper nesting in the array level below them. This can also be written as E100##4; the beginning of the next level of the notation.

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

  Now we have arrived at Extended Hyper-E Notation. Two successive hyperion marks (deutero-hyperions) indicate repetition at the lower level.

• 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).

References

This article "Hyper-E notation" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Hyper-E notation. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

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