Hyperoperation

This article is about the arithmetic concept. For the group theory hyperoperation concept, see Hyperstructure.

In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations in this context) 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.) 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:

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

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:

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

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. $50[50]50$ 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).

This recursion rule is common to many variants of hyperoperations.

Definition

Definition, most common

The hyperoperation sequence $H_{n}(a,b)\colon (\mathbb {N} _{0})^{3}\rightarrow \mathbb {N} _{0}$ is the sequence of binary operations $H_{n}\colon (\mathbb {N} _{0})^{2}\rightarrow \mathbb {N} _{0}$, defined recursively as follows:

$H_{n}(a,b)=a[n]b={\begin{cases}b+1&{\text{if }}n=0\\a&{\text{if }}n=1{\text{ and }}b=0\\0&{\text{if }}n=2{\text{ and }}b=0\\1&{\text{if }}n\geq 3{\text{ and }}b=0\\H_{n-1}(a,H_{n}(a,b-1))&{\text{otherwise}}\end{cases}}$

(Note that for n = 0, the binary operation essentially reduces to a unary operation (successor function) by ignoring the first argument.)

For n = 0, 1, 2, 3, this definition reproduces the basic arithmetic operations of successor (which is a unary operation), addition, multiplication, and exponentiation, respectively, as

${\begin{aligned}H_{0}(a,b)&=b+1,\\H_{1}(a,b)&=a+b,\\H_{2}(a,b)&=a\times b,\\H_{3}(a,b)&=a\uparrow {b}=a^{b}.\end{aligned}}$

The $H_{n}$ operations for n ≥ 3 can be written in Knuth's up-arrow notation.

So what will be the next operation after exponentiation? We defined multiplication so that $H_{2}(a,3)=a[2]3=a\times 3=a+a+a,$ and defined exponentiation so that $H_{3}(a,3)=a[3]3=a\uparrow 3=a^{3}=a\times a\times a,$ so it seems logical to define the next operation, tetration, so that $H_{4}(a,3)=a[4]3=a\uparrow \uparrow 3=\operatorname {tetration} (a,3)=a^{a^{a}},$ with a tower of three 'a'. Analogously, the pentation of (a, 3) will be tetration(a, tetration(a, a)), with three "a" in it.

${\begin{aligned}H_{4}(a,b)&=a\uparrow \uparrow {b},\\H_{5}(a,b)&=a\uparrow \uparrow \uparrow {b},\\\ldots &\\H_{n}(a,b)&=a\uparrow ^{n-2}b{\text{ for }}n\geq 3,\\\ldots &\\\end{aligned}}$

Knuth's notation could be extended to negative indices ≥ −2 in such a way as to agree with the entire hyperoperation sequence, except for the lag in the indexing:

$H_{n}(a,b)=a\uparrow ^{n-2}b{\text{ for }}n\geq 0.$

The hyperoperations can thus be seen as an answer to the question "what's next" in the sequence: successor, addition, multiplication, exponentiation, and so on. Noting that

${\begin{aligned}a+b&=(a+(b-1))+1\\a\cdot b&=a+(a\cdot (b-1))\\a^{b}&=a\cdot \left(a^{(b-1)}\right)\\a[4]b&=a^{a[4](b-1)}\end{aligned}}$

the relationship between basic arithmetic operations is illustrated, allowing the higher operations to be defined naturally as above. The parameters of the hyperoperation hierarchy are sometimes referred to by their analogous exponentiation term; so a is the base, b is the exponent (or hyperexponent), and n is the rank (or grade), and moreover, $H_{n}(a,b)$ is read as "the bth n-ation of a", e.g. $H_{4}(7,9)$ is read as "the 9th tetration of 7", and $H_{123}(456,789)$ is read as "the 789th 123-ation of 456".

In common terms, the hyperoperations are ways of compounding numbers that increase in growth based on the iteration of the previous hyperoperation. The concepts of successor, addition, multiplication and exponentiation are all hyperoperations; the successor operation (producing x + 1 from x) is the most primitive, the addition operator specifies the number of times 1 is to be added to itself to produce a final value, multiplication specifies the number of times a number is to be added to itself, and exponentiation refers to the number of times a number is to be multiplied by itself.

Definition, using iteration

Define iteration of a function f of two variables as

$f^{x}(a,b)={\begin{cases}f(a,b)&{\text{if }}x=1\\f(a,f^{x-1}(a,b))&{\text{if }}x>1\end{cases}}$

The hyperoperation sequence can be defined in terms of iteration, as follows. For all integers $x,n,a,b\geq 0,$ define