Notation Array Notation

Notation Array Notation, although it sounds redundant, is a very quickly growing function. It catalogues all the levels of operation. It is also known as NaN (but do not get it confused with Not A Number)!!! The rules for the notation are as follows.

Rules
1. follows a similar rule to that of the Hyper-L notation parenthesis

2. Placed in a parenthesis

3. The first and last number (3,3) tells you which numbers are on each side of the operation

4. the brackets {} tell you the notation level (addition is 1, mult and exp is 2, etc.)

5. They also tell you which level of operation in the notation you are describing

An example: (4{3,2}2)

In this example, 4 and 2 are on both sides of the operation 3 tells you it is Up Arrow Notation 2 tells you there are 2 up arrows separating the 4 and 2

Levels
The levels are:

Level One (addition) = (x{1,1}x) Ex: (3{1,1}1) = 3+1 = 4

Level Two (mult and exp ) = (x{2,1}x) for mult, (x{2,2}x) for single exp Ex: (4{2,1}3) = 4*3 = 12; (4{2,2}3) = 4^3 = 64; (4{2,3}3) = 4^3^3 = 262144

Level Three: (arrow ) = (x{3,n}x) Ex: (10{3,5}6) = 10^^^^^6 (^ is an up arrow)

Level Four: (Conway ) = (x{4,n}x) Ex: (4{4,4}4) = 4->4->4->4 = Conway's Tetratet

Level 5: (Alpha) = (x{5,n}x) Ex: (3{5,6}4} = 3AAAAAA4

For those beyond notation:

Level 29: Not29 (3{29,3}3)

Level 30: Not30 (3{30,3}3)

Second Order NaN
PolyNaN Notation, or second order NaN, is a variety of the notation array notation that uses 2 parenthesis instead of one to express the level of the first bracket number in more detail. The form of PolyNaN usually follows like this: (n{X,n}n) (n{n,n}n) where X remains in the equation no matter what.

Three Bracket NaN is another second order NaN notation extension. It is used to compute numbers far larger than any of the other numbers by using a third bracket number- the compounder. This number can fill in for the other numbers many times over. An example follows: (n{x,y,z}n)

Three bracket rules: N represents the numbers on both sides of the operation. X represents the notation level Y represents the level of operation Z represents the level of compounding upon X

Compound rules: if the Z is equal to one, the function can be re-written without Z in normal NaN. If the Z is equal to 2, then the compound of the entire X can be boosted above the highest number achieved by typical NaN. This is because it compounds the function using the function itself. An example is (3{3,4,2}3). There are 3s on each side of a set of 3 up arrows. Then use the compound to make the arrows nested 3^^^^3 amount of times. This yields roughly, using the G function, G(3^^^^3).

Example: (3{4,4,2}3) 4 means chain arrow notation 3 means threes on either side of the up arrow notation 4 means there are 4 chained arrows in between the threes 2 means there is compounding of the entire function by a factor of 2

Directed Bracket, Double Comma
Directed three bracket NaN: A part of three bracket NaN in which a parenthesis is placed next to Z that can cause the function to become less generalised. Normally, placing a 2 in Z gives a specific number and it is difficult to change the notation without changing the number drastically. Instead, you can always use directed three bracket notation to define how many compounds to perform. The form looks like this: (n{X,Y,Z(q)}n).

An example: (10{3,5,2(100)}10) This usually, without Z, would mean 10^^^^^10. With Z, it would mean a compound of 10^^^^^10 an amount of times equivalent to 10^^^^^10. However, the 100 informs us that you should only compound it 100 times instead of 10^^^^^10 times.

Double Comma Notation: Double comma notation is another part of three bracket NaN that further increases the total expansion factor of the notation set as a whole. Having a double comma is to compound the compound again without having to add a number to Z. You can use directed three bracket notation in conjunction with double comma notation. You can also add as many commas as you wish to further recurse the function. The form is: (n{X,Y,,Z(q)})

An example: (10{3,3,,2}10) This usually means 10^^^10 without Z and 10^^^10 compounded 10^^^10 times with Z, but in a double comma it means 10^^^10 compounded 10^^^10 times which is compounded again this amount of times: (10^^^10 compounded 10^^^10 times). A triple comma would continue the compounding.