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Generalization of addition, multiplication, exponentiation, tetration, etc.
In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations in this context) that starts with
Hyperoperation
Arithmetic operation
. Tetration is the next hyperoperation after exponentiation, but before pentation. Along with the other hyperoperations, tetration is used for the
Tetration
Method of notation of very large integers
introduced the specific sequence of operations that are now called hyperoperations. Goodstein also suggested the Greek names tetration, pentation, etc
Knuth's_up-arrow_notation
Elementary operation on a natural number
Successor operations are also known as zeration in the context of a zeroth hyperoperation. In this context, the extension of zeration is addition, which is defined
Successor_function
Algebraic structure equipped with at least one multivalued operation
called a hyperoperation. The largest classes of the hyperstructures are the ones called H v {\displaystyle Hv} – structures. A hyperoperation ( ⋆ ) {\displaystyle
Hyperstructure
Quickly growing function
extends these basic operations in a way that can be compared to the hyperoperations: φ ( m , n , 3 ) = m [ 4 ] ( n + 1 ) φ ( m , n , p ) ⪆ m [ p + 1 ]
Ackermann_function
Topics referred to by the same term
Hindi Hypercube, the n-dimensional analogue of a square and a cube Hyperoperation, an arithmetic operation beyond exponentiation Hyperplane, a subspace
Hyper
Large number coined by Ronald Graham
{\displaystyle f^{4}(n)=f(f(f(f(n))))} . Expressed in terms of the family of hyperoperations H 0 , H 1 , H 2 , ⋯ {\displaystyle {\text{H}}_{0},{\text{H}}_{1},{\text{H}}_{2}
Graham's_number
Typographical mark (^)
common in mathematics. The upward-pointing arrow is now used to signify hyperoperations in Knuth's up-arrow notation, with a single arrow representing exponentiation
Caret
Ordinal-indexed family of rapidly increasing functions
hierarchy coincide with those of the Grzegorczyk hierarchy: (using hyperoperation) f0(n) = n + 1 = 2[1]n − 1 f1(n) = f0n(n) = n + n = 2n = 2[2]n f2(n)
Fast-growing_hierarchy
Performing order of mathematical operations
completely stable. Common operator notation (for a more formal description) Hyperoperation Logical connective#Order of precedence Operator associativity Operator
Order_of_operations
Addition, multiplication, division, ...
relation that corresponds to a binary operation is a univalent relation. Hyperoperation Infix notation Operator (mathematics) Order of operations "Algebraic
Operation_(mathematics)
Arithmetic operation
Iterating tetration leads to another operation, and so on, a concept named hyperoperation. This sequence of operations is expressed by the Ackermann function
Exponentiation
Two raised to an integer power
first 21 of them are: Also see Fermat number, Tetration and Hyperoperation § Lower hyperoperations. All of these numbers over 4 end with the digit 6. Starting
Power_of_two
function Associated Legendre functions Meijer G-function Fox H-function Hyperoperations Iterated logarithm Super-logarithms Tetration Lambert W function: Inverse
List of mathematical functions
List_of_mathematical_functions
English mathematician (1912–1985)
introduced a variant of the Ackermann function that is now known as the hyperoperation sequence, together with the naming convention now used for these operations
Reuben_Goodstein
Natural number
\uparrow 2)} , or as the pentation 2 [ 5 ] 3 {\displaystyle 2[5]3} (hyperoperation notation). 65536 is a superperfect number – a number such that σ(σ(n)) = 2n
65,536
Means of expressing certain extremely large numbers
(equivalent to the last-mentioned property) The square brackets denote hyperoperation. a → b → 3 → 2 {\displaystyle a\to b\to 3\to 2} = a → b → 3 → ( 1 +
Conway_chained_arrow_notation
Programming language
exponentially, yet is computable by a LOOP2 program. Addition is the hyperoperation H 1 {\displaystyle H_{1}} . H 1 ( x , y ) {\displaystyle H_{1}(x,y)}
LOOP_(programming_language)
Growth of quantities at rate proportional to the current amount
and hyperbolic growth lie more classes of growth behavior, like the hyperoperations beginning at tetration, and A ( n , n ) {\displaystyle A(n,n)} , the
Exponential_growth
Arithmetic notation system
a fixed number of recursions, notably Knuth's up-arrow notation and hyperoperation notation. Mathematical notation Mark Cutler, Physical Infinity, 2004
Cutler's_bar_notation
Functions in computability theory
{E}}^{1}\subsetneq {\mathcal {E}}^{2}\subsetneq \cdots } because the hyperoperation H n {\displaystyle H_{n}} is in E n {\displaystyle {\mathcal {E}}^{n}}
Grzegorczyk_hierarchy
Operations on ordinals that extend classical arithmetic
exponentiation, including ordinal versions of tetration and other hyperoperations. See also Veblen function. Every ordinal number α can be uniquely written
Ordinal_arithmetic
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