HyperoperationIn 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.
TetrationIn 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".
PentationIn mathematics, pentation (or hyper-5) is the next hyperoperation after tetration and before hexation. It is defined as iterated (repeated) tetration (assuming right-associativity), just as tetration is iterated right-associative exponentiation. It is a binary operation defined with two numbers a and b, where a is tetrated to itself b-1 times. For instance, using hyperoperation notation for pentation and tetration, means 2 to itself 2 times, or .
Knuth's up-arrow notationIn mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations. Goodstein also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation. The sequence starts with a unary operation (the successor function with n = 0), and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n = 3), tetration (n = 4), pentation (n = 5), etc.
Large numbersLarge numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical mechanics. They are typically large positive integers, or more generally, large positive real numbers, but may also be other numbers in other contexts. Googology is the study of nomenclature and properties of large numbers.
