Introduction
The ordinary arithmetical operations of addition, multiplication and exponentiation are naturally extended into a sequence of hyperoperations as follows.
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,
To extend the sequence of operations beyond exponentiation, Knuth defined a “double arrow” operator to denote iterated exponentiation (tetration):
For example,
Here and below evaluation is to take place from right to left, as Knuth's arrow operators (just like exponentiation) are defined to be right-associative.
According to this definition,
- etc.
This already leads to some fairly large numbers, but Knuth extended the notation. He went on to define a “triple arrow” operator for iterated application of the “double arrow” operator (also known as pentation):
followed by a 'quadruple arrow' operator (also known as hexation):
and so on. The general rule is that an -arrow operator expands into a right-associative series of -arrow operators. Symbolically,
Examples:
The notation is commonly used to denote with n arrows.
Read more about this topic: Knuth's Up-arrow Notation
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