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' number and googolplexplex (e.g. ${\displaystyle 50[50]50}$ is much larger than Skewes’ 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 (see below in definition).

Definition

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

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

(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

{\displaystyle {\begin{aligned}H_{0}(a,b)&=b+1\,\!,\\H_{1}(a,b)&=a+b\,\!,\\H_{2}(a,b)&=a\cdot b\,\!,\\H_{3}(a,b)&=a^{b}\,\!,\end{aligned}}}

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

The H operations for n ≥ 3 can be written in Knuth's up-arrow notation as

{\displaystyle {\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:

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

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

{\displaystyle {\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;[14] so a is the base, b is the exponent (or hyperexponent),[12] and n is the rank (or grade).[6], and ${\displaystyle H_{n}(a,b)}$ is read as "the bth n-ation of a", e.g. ${\displaystyle H_{4}(7,9)}$ is read as "the 9th tetration of 7", and ${\displaystyle H_{123}(456,789)}$ 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.

Other Languages
español: Hiperoperación
Esperanto: Hiperoperatoro
français: Hyperopération
한국어: 하이퍼 연산
português: Hiperoperação
Simple English: Hyperoperation
slovenščina: Hiperoperacija
Türkçe: Hiperişlem
українська: Гіпероператор