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Concept
Tetration
Summary
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".
It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration.
Tetration is also defined recursively as
allowing for attempts to extend tetration to non-natural numbers such as real and complex numbers.
The two inverses of tetration are called super-root and super-logarithm, analogous to the nth root and the logarithmic functions. None of the three functions are elementary.
Tetration is used for the notation of very large numbers.
The first four hyperoperations are shown here, with tetration being considered the fourth in the series. The unary operation succession, defined as , is considered to be the zeroth operation.
Addition n copies of 1 added to a combined by succession.
Multiplication n copies of a combined by addition.
Exponentiation n copies of a combined by multiplication.
Tetration n copies of a combined by exponentiation, right-to-left.
Note that nested exponents are conventionally interpreted from the top down: 3^{5^7} means 3^{\left(5^7 \right)} and not \left(3^5 \right)^7.
Succession, (an+1 = an + 1), is the most basic operation; while addition (a + n) is a primary operation, for addition of natural numbers it can be thought of as a chained succession of n successors of a; multiplication (a × n) is also a primary operation, though for natural numbers it can analogously be thought of as a chained addition involving n numbers of a. Exponentiation can be thought of as a chained multiplication involving n numbers of a and tetration () as a chained power involving n numbers a.
Official source
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