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Concept
Hyperoperation
Summary
In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations in this context) 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.) 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's number and googolplexplex (e.g. is much larger than Skewes's 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.
The hyperoperation sequence is the sequence of binary operations , defined recursively as follows:
(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
The operations for n ≥ 3 can be written in Knuth's up-arrow notation.
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'.
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Related concepts (18)
Tetration
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though and the left-exponent xb are common. Under the definition as repeated exponentiation, means , where n copies of a are iterated via exponentiation, right-to-left, i.e. the application of exponentiation times. n is called the "height" of the function, while a is called the "base," analogous to exponentiation. It would be read as "the nth tetration of a".
Hyperoperation
In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations in this context) 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.
Conway chained arrow notation
Conway chained arrow notation, created by mathematician John Horton Conway, is a means of expressing certain extremely large numbers. It is simply a finite sequence of positive integers separated by rightward arrows, e.g. . As with most combinatorial notations, the definition is recursive. In this case the notation eventually resolves to being the leftmost number raised to some (usually enormous) integer power. A "Conway chain" is defined as follows: Any positive integer is a chain of length .