10,821 Pages

## Rules

An array contains and zero or more entries, which can be either numbers or other arrays

If an array contains exactly one entry, that is the value of the array

If an array is immediately preceeded by a superscript, that is the class of the array. Otherwise, the class is either that of the containing array, if applicable, or else 1. It will always be a nonnegative integer.

$[a,b,...]=[a\uparrow^{n}b,...]$, where n is the class of the array and ... represents the rest of the array

An empty array has a value of 1

### Example:

$[2,[^{2}[2,3],7]]$

$[2,[2^{2^{2}},7]]$

$[2,[16,7]]$

$[2,[16^{7}]]$

$[2,268435456]$

$2^{268435456}\approx1.44\times10^{80807124}$

## Properties

An array with a nonzero class whose first entry is 1 will have a value of 1

If the last entry is one, it may be dropped

## The ARR function

$ARR(a,x,d)=\underset{x}{[\underbrace{ARR(a,x,d-1),...,ARR(a,x,d-1)}]};ARR(a,x,0)=a$

If a call to this function is immediately preceeded by a superscript, that is the class of the arrays within this function. Othrwise, if the call is an entry in a larger array, the arrays within this function have the same class as that array. If neither is true, the arrays within this function have a class of 1.