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This notation is based on the Steinhaus-Moser Notation. For now let's call this BA and not BAL. (BASE, ARRAY, LEVEL; no LEVEL for now :P )

Definition

Let \# denote rest of array. Let a and b be natural numbers.

Rules:

  • a\{0\} = a (a is called the BASE)
  • a\{\#, 0\} = a\{\#\} (I chose 0 for some reason)
  • a\{\underbrace{0, 0, \cdots, 0}_{\text{n}}, b + 1, \#\} = a\{\underbrace{a, a, \cdots, a}_{\text{n}}, b, \#\}
  • a\{b, \#\} = a^a\{b-1, \#\}

Obviously, the ARRAY is inside the curly braces.

Others

I found that:

  • \text{Mega} = 2\{0, 2\} = 256\{256\}

a\{0, b\} represents a inscribed in a b+3-gon in Steinhaus-Moser Notation. Therefore,

  • \text{Moser} = 2\{0, 2\{0, 2\}-3\} = 2\{0, \text{Mega}-3\}

Numbers

Bal Series

Cannibal

\text{Cannibal} = 0

It ate itself.

Unibal

\text{Unibal} = 1\{1\} = 1

Yes, I had to define this. :)

Bibal

\text{Bibal} = 2\{2, 2\}

This is equal to 256 inside a pentagon.

Tribal

\text{Tribal} = 3\{3, 3, 3\}

Yay, it's in the dictionary! LOL

Quadribal

\text{Quadribal} = 4\{4, 4, 4, 4\}

Skip a few ...

Decabal

\text{Decabal} = 10\{10, 10, 10, 10, 10, 10, 10, 10, 10, 10\}

King Bal

\text{King Bal} = 2218\{\underbrace{2218, 2218, \cdots, 2218}_{2218}\}

I like this one. XD

That's it for now. In the next part, I will redefine these numbers to make them look simpler. I'll post Part 2 sometime later. :)

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