Hyperoperation - Definition

Definition

The hyperoperation sequence is the sequence of binary operations indexed by, defined recursively as follows:

$H_n(a, b) = \begin{cases} b + 1 & \text{if } n = 0 \\ a &\text{if } n = 1, b = 0 \\ 0 &\text{if } n = 2, b = 0 \\ 1 &\text{if } n \ge 3, b = 0 \\ H_{n-1}(a, H_n(a, b - 1)) & \text{otherwise} \end{cases}\,\!$

(Note that for n = 0, the binary operation essentially reduces to a unary operation 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

and for n ≥ 4 it extends these basic operations beyond exponentiation to what can be written in Knuth's up-arrow notation as

...

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 sequence: successor, addition, multiplication, exponentiation, and so on. Noting that

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).

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.

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