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Concept
Knuth's up-arrow notation
Summary
In 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.
Various notations have been used to represent hyperoperations. One such notation is .
Knuth's up-arrow notation is another.
For example:
the single arrow represents exponentiation (iterated multiplication)
the double arrow represents tetration (iterated exponentiation)
the triple arrow represents pentation (iterated tetration)
The general definition of the up-arrow notation is as follows (for ):
Here, stands for n arrows, so for example
The square brackets are another notation for hyperoperations.
The hyperoperations naturally extend the arithmetical operations of addition and multiplication as follows.
Addition by a natural number is defined as iterated incrementation:
Multiplication by a natural number is defined as iterated addition:
For example,
Exponentiation for a natural power is defined as iterated multiplication, which Knuth denoted by a single up-arrow:
For example,
Tetration is defined as iterated exponentiation, which Knuth denoted by a “double arrow”:
For example,
Expressions are evaluated from right to left, as the operators are defined to be right-associative.
According to this definition,
etc.
This already leads to some fairly large numbers, but the hyperoperator sequence does not stop here.
Pentation, defined as iterated tetration, is represented by the “triple arrow”:
Hexation, defined as iterated pentation, is represented by the “quadruple arrow”:
and so on.
Official source
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