N Array Notation

N Array Notation (N.A.N) was created by Antfarmerguys. It is based off of Fast-growing hierarchy and Graham's number.

Definition

Level 1 Of N.A.N

The basics

N Array Notation is defined with N-a-{n} = N-a-1-{N-a-1-{...}} with n N's except for N-1-{n} which equals 10{N-1-{n-1}}10 in Arrow Notation except for N-1-{1} which equals 10^^^^^10. We can put this system inside of itself for example: N-N-N-...-{n}-{n}-{n} With n N's this can be represented with N-1,1-{n} then we an have N-2,1-{n} = N-1,1-{N-1,1-{...}} with n N's so we can define N-a,1-{n} = N-a-1,1-{N-a-1,1-{...}} with n N's but we an also put this system inside of itself for example: N-N-...,1-{n},1-{n} with n N's this equals

N-1,2-{n} so we can define N-1,b-{n} = N-N-...,b-1-{n},b-1-{n} so the second input (b) will change when the first input is put inside of itself n times so we can add a third input (c). Here is an example: N-1,1,1-{n} = N-1,N-1,...-{n}-{n} with n N's. So the third input (c) will change when the second input is put inside of itself n times. So we can keep adding new inputs for example: N-1,1,1,1,1-{n} = N-1,1,1,N-1,1,1,...-{n}-{n} with n N's.

Layers

Now we will need to count these inputs so we can use something called layers. Basically layers are the best way to count large amounts of inputs so to use layers we can define a new system: N-La-{n} a will choose the size of the layer here an example of a layer: N-1,1,...-{n} with n inputs that equals N-L1-{n} then here is a visual representation of N-L2-{n}:

As you can see layers are very powerful but we can continue into larger numbers with: N-L1,1-{n} which equals N-LN-L...-{n}-{n} with n N's this function is the same as N-1,1-{n} but it uses layers so we can easily add new inputs for example: N-L1,1,1,1,1-{n} we can even have layers of the inputs of layers this can be represented with: N-L(1)a-{n} which equals N-L1,1,...-{n} with the layer of a. Now we need to define N-L(2)a-{n} which equals N-L(1)1,1,...-{n} with the layer of a so now we can keep increasing the value of the number in the ()'s .

Layers Inside Of Layers

we can even have multiple inputs for example: N-L(1,1)1-{n} which as we've defined before equals N-L(N-L(...)1-{n})1-{n} with n N's we can even have layers in the ()'s for example: N-L(L1)a-{n} which equals L(1,1,...)1-{n} with the layer of a. *The L1 in the ()'s is NOT the layer a will always be the layer anything else is just what a is the layer of. Now we need to define N-L(L2)a-{n} to start we will define N-L(L1|1)a-{n} which equals N-L(L1)1,1,...-{n} with the the layer of a as you can see the | separate's the L1 and the 1 which as we've defined before in the N-L(b)a-{n} function once we reach N-L(L1|1,1,...)1-{n} with the layer of a this will equal to N-L(L2)a-{n} so now we can keep increasing the value of the number in the (L )'s we can even have something like N-L(L(1)b)1-{n} which equals N-L(L1,1,...)1-{n} with the layer of b. We can keep adding L's in the ()'s and then can create a new system: N-1La-{n} which equals N-L(L(...)1)1-{n} with the layer of a to define N-2La-{n} it would equal to N-L(L(...)1)1-{n} with the layer of a but with a 1 at the begining which looks like this: N-1L(L(...)1)1-{n} with the layer of a so we can keep increasing the value of b and we can even have multiple inputs before the L for example: N-1,1,...L1-{n} with the layer of b this equals N-LbL1-{n} as you can see the Lb comes before the L1 this just means that the Lb is the function behind the L1. Now we need to define N-1LL1-{n} which equals N-1L1-{n} but the function behind the L1 is L(L(...)1)1 which looks like this: N-L(L(...)1)1L1-{n} with the layer of a. Now we need to define N-1LLLa-{n} which equals N-L(L(...)1)1LL1-{n} with the layer of a so we can keep adding L's to this function until we get N-1L...L1-{n} with the layer of a which equals N-1L(1)La-{n}. Now need to define N-1L(1)(1)La-{n} which equals N-1L...L1L(1)L1-{n} with the layer of a now we can keep adding (1)'s until we reach N-1L(2)La-{n} which equals N-1L(1)...(1)L1-{n} with the layer of a Now need to define N-1L(2)(2)La-{n} which equals N-1L(1)...(1)L1L(2)L1-{n} with the layer of a now we can keep adding (2)'s until we reach N-1L(3)La-{n} so we can define N-1L(c)La-{n} = N-1L(c-1)...(c-1)L1-{n} with the layer of a. Now we can have multiple inputs in the ()'s for example: N-1L(1,1)La-{n}.

The ¬ Function

Then we can define N-1L¬La-{n} which equals N-1L(1L(...)L1)L1-{n} with the layer of a this  basically means the N-1L(1)L1-{n} function apllied to N-1L(1,1,...)La-{n} with the layer of a this is the limit of level 1 of N.A.N.

Level 2 Of N.A.N

The L/b/a Function

In this level of N.A.N we get much more complex functions. The first function is N-L/1/a-{n} this equals N-La-{n} and now that we have defined this we will create a set we will call this set U and it will contain the functions that we define with N-L/b/a-{n} for example N-L/1/a-{n} this equals N-La-{n} because U = {∅} then we add this function to U. Then we can define N-L/2/a-{n} equals N-the first function that doesn't contain U-{n} which equals N-L(1)a-{n} so we add this to U. U now equals {La,L(1)a} so N-L/3/a-{n} equals N-1La-{n} because 1La is the first function that doesn't contain {La,L(1)a} then N-L/4/a-{n} equals N-1L(1)La-{n} for the same reasons then N-L/5/a-{n} equals N-1L¬La-{n} now we can define N-1L¬¬La-{n} which based on the rules of N.A.N would equal N-1L¬L1-{n} but the function behind the L¬L1 is 1L(1L(...)L1)L1  which looks like this: N-1L(1L(...)L1)L1L¬L1-{n}. Based on the rules of N.A.N we can keep increasing the value of b in the N-L/b/a-{n} function.

The La#b Functions

Now we need to define N-L1#1-{n} which equals N-L/L/.../1/1-{n} with the layer of a this basically means the N-L/1/1-{n} apllied to N-L/1,1,.../1-{n} with the layer of a Now we need to define N-L2#1-{n} which equals N-L/L/.../1/1-{n} function apllied to N-L/L/.../1/1-{n} with the layer of a. Here is a visual representation of N-L2#1-{n}:

The ⸻ in this image just means that the function below (or right) of the ⸻ is apllied to the function above (or left) of the ⸻. As you can see this resembles L2 from level 1 of N.A.N. So N-La#1-{n} will resemble La from level 1 of N.A.N. From now on when we use ⸻'s in a format that resembles La we will call it resemblance to La.

Now we need to define N-La#2-{n} which equals N-L/L/.../1/1#1-{n} with resemblance to La. Now we can have something like N-La#L1-{n} which equals N-La#1,1,...-{n} with n 1's. So now we need to define N-La##1-{n} which equals N-L1#L1#...-{n} with resemblance to La. Now we can define N-La##b-{n} equals N-L/L/.../1/1#b-1-{n} with resemblance to La. Now we can define N-La###b-{n} equals N-L1##L1##...###b-1-{n} with resemblance to La so we can keep adding #'s until we get to N-La#^1|1-{n} which equals N-L1#...#1-{n} with resemblance to La now we can define N-La#^b|1-{n} equals N-L1#...#^b-1|1-{n} with resemblance to La. Now we need to define N-La#^1^1|1-{n} which equals N-L1#^L1#^...|1|1-{n} with resemblance to La Now we can define N-La#^1^d|1-{n} equals N-L1#^L1#^...^d-1|1|1-{n}. Now we can define N-La#^1^1^1|1-{n} equals N-L1#^1^L1#^1^...|1|1-{n} with resemblance to La. So we can keep adding ^1's until we reach N-La#^^1|1-{n} which equals N-L1#^1^1^...|1-{n} with resemblance to La. Now we need to define N-La#^^1^1|1-{n} which equals N-L1#^^L1#^^...|1|1-{n} with resemblance to La now we can keep adding ^1's until we reach N-La#^^1^^1|1-{n} which equals N-L1#^^1^1^1^...|1-{n} with resemblance to La. Now based on the rules of N.A.N we can keep adding ^^1's until we reach N-La#^^^1|1-{n} which equals N-L1#^^1^^1^^...|1-{n} with resemblance to La.

The La#¬c|b Function

Now like in level 1 of N.A.N the last function we will define is the ¬ function in this level of N.A.N it is the N-La#¬1|1-{n} now before we define this we first need to define multiple ⸻ functions the first function is this new section is the 2⸻. We've been using functions with resemblance to La but we can simplify this into just a 2⸻ function with La at the bottom here is a visual representation of N-L2#1-{n} with the 2⸻ function:

As you can see 2⸻ is very powerful now we need to define 3⸻ which is basically 2⸻ but all the ⸻'s are 2⸻'s. So now we can keep adding ⸻'s to this function until we N-La#¬1|1-{n} here is a visual representation of N-La#¬1|1-{n}:

The reason that it doesn't tell you how many ⸻'s are in each section is because the text that tells you how many ⸻'s are in each section is in the 3rd dimension this is the limit of level 2 of N.A.N.

*The number of ⸻'s in each section is La.