Hyperoperation


In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations that starts with a unary operation. The sequence continues with the binary operations of addition, multiplication, and exponentiation.
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, pentation, hexation 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:
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.
This recursion rule is common to many variants of hyperoperations.

Definition

The hyperoperation sequence is the sequence of binary operations, defined recursively as follows:
For n = 0, 1, 2, 3, this definition reproduces the basic arithmetic operations of successor, addition, multiplication, and exponentiation, respectively, as
The H operations for n ≥ 3 can be written in Knuth's up-arrow notation as
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 will be tetration, with three "a" in it.
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, and n is the rank, and moreover, is read as "the bth n-ation of a", e.g. is read as "the 9th tetration of 7", and 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 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

Below is a list of the first seven hyperoperations.
nOperation,
Hn		DefinitionNamesDomain
0orhyper0, increment, successor, zerationArbitrary
1orhyper1, additionArbitrary
2orhyper2, multiplicationArbitrary
3orhyper3, exponentiationb real, with some multivalued extensions to complex numbers
4orhyper4, tetrationa ≥ 0 or an integer, b an integer ≥ −1
5hyper5, pentationa, b integers ≥ −1
6hyper6, hexationa, b integers ≥ −1

Special cases

Hn =
Hn =
Hn =
Hn =
Hn =
Hn =
Hn =

History

One of the earliest discussions of hyperoperations was that of Albert Bennett in 1914, who developed some of the theory of commutative hyperoperations. About 12 years later, Wilhelm Ackermann defined the function which somewhat resembles the hyperoperation sequence.
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. As a three-argument function, e.g.,, the hyperoperation sequence as a whole is seen to be a version of the original Ackermann function — recursive but not primitive recursive — as modified by Goodstein to incorporate the primitive successor function together with the other three basic operations of arithmetic, and to make a more seamless extension of these beyond exponentiation.
The original three-argument Ackermann function uses the same recursion rule as does Goodstein's version of it, but differs from it in two ways. First, defines a sequence of operations starting from addition rather than the successor function, then multiplication, exponentiation, etc. Secondly, the initial conditions for result in, thus differing from the hyperoperations beyond exponentiation. The significance of the b + 1 in the previous expression is that =, where b counts the number of operators, rather than counting the number of operands as does the b in, and so on for the higher-level operations.

Notations

This is a list of notations that have been used for hyperoperations.
NameNotation equivalent toComment
Knuth's up-arrow notationUsed by Knuth, and found in several reference books.
Hilbert's notationUsed by David Hilbert.
Goodstein's notationUsed by Reuben Goodstein.
Original Ackermann functionUsed by Wilhelm Ackermann
Ackermann–Péter functionThis corresponds to hyperoperations for base 2
Nambiar's notationUsed by Nambiar
Superscript notationUsed by Robert Munafo.
Subscript notation Used for lower hyperoperations by Robert Munafo.
Operator notation Used for lower hyperoperations by John Donner and Alfred Tarski.
Square bracket notationUsed in many online forums; convenient for ASCII.
Conway chained arrow notationUsed by John Horton Conway

Variant starting from ''a''

In 1928, Wilhelm Ackermann defined a 3-argument function which gradually evolved into a 2-argument function known as the Ackermann function. The original Ackermann function was less similar to modern hyperoperations, because his initial conditions start with for all n > 2. Also he assigned addition to n = 0, multiplication to n = 1 and exponentiation to n = 2, so the initial conditions produce very different operations for tetration and beyond.
nOperationComment
0
1
2
3An offset form of tetration. The iteration of this operation is different than the iteration of tetration.
4Not to be confused with pentation.

Another initial condition that has been used is , due to Rózsa Péter, which does not form a hyperoperation hierarchy.

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. While discussing tetration, Clenshaw et al. assumed the initial condition, which makes yet another hyperoperation hierarchy. Just like in the previous variant, the fourth operation is very similar to tetration, but offset by one.
nOperationComment
0
1
2
3
4An offset form of tetration. The iteration of this operation is much different than the iteration of tetration.
5Not to be confused with pentation.

Lower hyperoperations

An alternative for these hyperoperations is obtained by evaluation from left to right. Since
define
with
This was extended to ordinal numbers by Donner and Tarski, by :
It follows from Definition 1, Corollary 2, 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:
If α ≥ 2 and γ ≥ 2,
nOperationComment
0increment, successor, zeration
1
2
3
4Not to be confused with tetration.
5Not to be confused with pentation.
Similar to tetration.

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.
nOperationComment
0Smooth maximum
1
2This is due to the properties of the logarithm.
3
4Not to be confused with tetration.

Numeration systems based on the hyperoperation sequence

used the sequence of hyperoperators to create systems of numeration for the nonnegative integers. The so-called complete hereditary representation of integer n, at level k and base b, can be expressed as follows using only the first k hyperoperators and using as digits only 0, 1,..., b − 1, together with the base b itself:
Unnecessary parentheses can be avoided by giving higher-level operators higher precedence in the order of evaluation; thus,
level-1 representations have the form b X, with X also of this form;
level-2 representations have the form b X Y, with X,Y also of this form;
level-3 representations have the form b X Y Z, with X,Y,Z also of this form;
level-4 representations have the form b X Y Z W, with X,Y,Z,W also of this form;
and so on.
In this type of base-b hereditary representation, the base itself appears in the expressions, as well as "digits" from the set. This compares to ordinary base-2 representation when the latter is written out in terms of the base b; e.g., in ordinary base-2 notation, 6 = 2 = 2 2 1 2 1 1 2 0 0, whereas the level-3 base-2 hereditary representation is 6 = 2 1 . The hereditary representations can be abbreviated by omitting any instances of 0, 1, 1, 1, etc.; for example, the above level-3 base-2 representation of 6 abbreviates to 2 2 2.
Examples:
The unique base-2 representations of the number 266, at levels 1, 2, 3, 4, and 5 are as follows: