In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations) that starts with the unary operation of successor (n = 0), then continues with the binary operations of addition (n = 1), multiplication (n = 2), and exponentiation (n = 3), after which 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:
Contents
- Definition
- Examples
- Special cases
- History
- Notations
- Variant starting from a
- Variant starting from 0
- Lower hyperoperations
- Commutative hyperoperations
- References
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:
This can be used to easily show numbers much larger than those which scientific notation can, such as 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).
Definition
The hyperoperation sequence
(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
So what will be the next operation after exponentiation? We defined multiplication so that
The H operations for n ≥ 3 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.
Examples
This is a list of the first seven (0th to 6th) hyperoperations. (Notice that in this article, we define 0⁰ as 1.)
Special cases
Hn(0, b) =
0, when n = 2, or n = 3, b ≥ 1, or n ≥ 4, b odd (≥ −1)1, when n = 3, b = 0, or n ≥ 4, b even (≥ 0)b, when n = 1b + 1, when n = 0Hn(a, 0) =
0, when n = 21, when n = 0, or n ≥ 3a, when n = 1Hn(a, −1) =
0, when n = 0, or n ≥ 4a − 1, when n = 1−a, when n = 21/a , when n = 3History
One of the earliest discussions of hyperoperations was that of Albert Bennett in 1914, who developed some of the theory of commutative hyperoperations (see below). About 12 years later, Wilhelm Ackermann defined the function
In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations, and also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation (because they correspond to the indices 4, 5, etc.). As a three-argument function, e.g.,
The original three-argument Ackermann function
Notations
This is a list of notations that have been used for hyperoperations.
Variant starting from a
In 1928, Wilhelm Ackermann defined a 3-argument function
Another initial condition that has been used is
Variant starting from 0
In 1984, C. W. Clenshaw and F. W. J. Olver began the discussion of using hyperoperations to prevent computer floating-point overflows. Since then, many other authors have renewed interest in the application of hyperoperations to floating-point representation. (Since Hn(a, b) are all defined for b = -1.) While discussing tetration, Clenshaw et al. assumed the initial condition
Lower hyperoperations
An alternative for these hyperoperations is obtained by evaluation from left to right. Since
define (with ° or subscript)
with
This was extended to ordinal numbers by Donner and Tarski,[Definition 1] by :
It follows from Definition 1(i), Corollary 2(ii), and Theorem 9, that, for a ≥ 2 and b ≥ 1, that
But this suffers a kind of collapse, failing to form the "power tower" traditionally expected of hyperoperators:[Theorem 3(iii)]
If α ≥ 2 and γ ≥ 2,[Corollary 33(i)]
Commutative hyperoperations
Commutative hyperoperations were considered by Albert Bennett as early as 1914, which is possibly the earliest remark about any hyperoperation sequence. Commutative hyperoperations are defined by the recursion rule
which is symmetric in a and b, meaning all hyperoperations are commutative. This sequence does not contain exponentiation, and so does not form a hyperoperation hierarchy.