Hyperoperation

In mathematics, the hyperoperation sequence^{[nb 1]} is an infinite sequence of arithmetic operations (called hyperoperations in this context)^[1][11][13] that starts with a unary operation (the successor function with n = 0). The sequence continues with the binary operations of addition (n = 1), multiplication (n = 2), and exponentiation (n = 3).

After that, the sequence proceeds with further binary operations extending beyond exponentiation, using right-associativity. For the operations beyond exponentiation, the nth member of this sequence is named by Reuben Goodstein after the Greek prefix of n suffixed with -ation (such as tetration (n = 4), pentation (n = 5), hexation (n = 6), etc.) ^[5] and can be written as using n − 2 arrows in Knuth's up-arrow notation. Each hyperoperation may be understood recursively in terms of the previous one by:

${\displaystyle a[n]b=\underbrace {a[n-1](a[n-1](a[n-1](\cdots [n-1](a[n-1](a[n-1]a))\cdots )))} _{\displaystyle b{\mbox{ copies of }}a},\quad n\geq 2}$

It may also be defined according to the recursion rule part of the definition, as in Knuth's up-arrow version of the Ackermann function:

${\displaystyle a[n]b=a[n-1]\left(a[n]\left(b-1\right)\right),\quad n\geq 1}$

This can be used to easily show numbers much larger than those which scientific notation can, such as Skewes's number and googolplexplex (e.g. ${\displaystyle 50[50]50}$ is much larger than Skewes's number and googolplexplex), but there are some numbers which even they cannot easily show, such as Graham's number and TREE(3).^[14]

This recursion rule is common to many variants of hyperoperations.

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