## Hyperoperation |

In **hyperoperation sequence**^{[nb 1]} is an infinite *hyperoperations* in this context)^{[1]}^{[11]}^{[13]} that starts with a *n* = 0). The sequence continues with the *n* = 1), *n* = 2), and *n* = 3).

After that, the sequence proceeds with further binary operations extending beyond exponentiation, using *n*th member of this sequence is named by *n* suffixed with *-ation* (such as *n* = 4), *n* = 5), hexation (*n* = 6), etc.)^{[5]} and can be written as using *n* − 2 arrows in

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

This can be used to easily show numbers much larger than those which

This recursion rule is common to many variants of hyperoperations (see below in definition).

The *hyperoperation sequence* is the

(Note that for *n* = 0, the

For *n* = 0, 1, 2, 3, this definition reproduces the basic arithmetic operations of

So what will be the next operation after exponentiation? We defined multiplication so that , and defined exponentiation so that so it seems logical to define the next operation, tetration, so that 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 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:

The hyperoperations can thus be seen as an answer to the question "what's next" in the

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*,

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

dansk: Hyperoperator

Deutsch: Hyper-Operator

español: Hiperoperación

Esperanto: Hiperoperatoro

français: Hyperopération

한국어: 하이퍼 연산

עברית: היפר-פעולות

日本語: ハイパー演算子

português: Hiperoperação

русский: Гипероператор

Simple English: Hyperoperation

slovenščina: Hiperoperacija

Türkçe: Hiperişlem

українська: Гіпероператор

中文: 超运算