Various notations have been used to represent hyperoperations. One such notation is . Another notation is , an infix notation which is convenient for ASCII. The notation is known as 'square bracket notation'.
Knuth's up-arrow notation is an alternative notation. It is obtained by replacing in the square bracket notation by arrows.
Exponentiation for a natural power is defined as iterated multiplication, which Knuth denoted by a single up-arrow:
Tetration is defined as iterated exponentiation, which Knuth denoted by a “double arrow”:
Expressions are evaluated from right to left, as the operators are defined to be right-associative.
According to this definition,
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. The general rule is that an -arrow operator expands into a right-associative series of ()-arrow operators. Symbolically,
In expressions such as , the notation for exponentiation is usually to write the exponent as a superscript to the base number . But many environments — such as programming languages and plain-text e-mail — do not support superscript typesetting. People have adopted the linear notation for such environments; the up-arrow suggests 'raising to the power of'. If the character set does not contain an up arrow, the caret (^) is used instead.
The superscript notation doesn't lend itself well to generalization, which explains why Knuth chose to work from the inline notation instead.
is a shorter alternative notation for n uparrows. Thus .
Knuth's arrows have become quite popular.}
Writing out up-arrow notation in terms of powersEdit
Attempting to write using the familiar superscript notation gives a power tower.
If b is a variable (or is too large), the power tower might be written using dots and a note indicating the height of the tower.
Continuing with this notation, could be written with a stack of such power towers, each describing the size of the one above it.
Again, if b is a variable or is too large, the stack might be written using dots and a note indicating its height.
Furthermore, might be written using several columns of such stacks of power towers, each column describing the number of power towers in the stack to its left:
And more generally:
This might be carried out indefinitely to represent as iterated exponentiation of iterated exponentiation for any a, n and b (although it clearly becomes rather cumbersome).
The Rudy Rucker notation for tetration allows us to make these diagrams slightly simpler while still employing a geometric representation (we could call these tetration towers).
Finally, as an example, the fourth Ackermann number could be represented as:
Some numbers are so large that multiple arrows of Knuth's up-arrow notation become too cumbersome; then an n-arrow operator is useful (and also for descriptions with a variable number of arrows), or equivalently, hyper operators.
Some numbers are so large that even that notation is not sufficient. The Conway chained arrow notation can then be used: a chain of three elements is equivalent with the other notations, but a chain of four or more is even more powerful.
= , Since = = , Thus the result comes out with
= or (Petillion)
Even faster-growing functions can be categorized using an ordinal analysis called the fast-growing hierarchy. The fast-growing hierarchy uses successive function iteration and diagonalization to systematically create faster-growing functions from some base function . For the standard fast-growing hierarchy using , already exhibits exponential growth, is comparable to tetrational growth and is upper-bounded by a function involving the first four hyperoperators;. Then, is comparable to the Ackermann function, is already beyond the reach of indexed arrows but can be used to approximate Graham's number, and is comparable to arbitrarily-long Conway chained arrow notation.
These functions are all computable. Even faster computable functions, such as the Goodstein sequence and the TREE sequence require the usage of large ordinals, may occur in certain combinatorical and proof-theoretic contexts. There exists functions which grow uncomputably fast, such as the Busy Beaver, whose very nature will be completely out of reach from any up-arrow, or even any ordinal-based analysis.
Without reference to hyperoperation the up-arrow operators can be formally defined by
One can alternatively choose multiplication as the base case and iterate from there. Then exponentiation becomes repeated multiplication. The formal definition would be
for all integers with .
Note, however, that Knuth did not define the "nil-arrow" (). One could extend the notation to negative indices (n ≥ -2) in such a way as to agree with the entire hyperoperation sequence, except for the lag in the indexing:
The up-arrow operation is a right-associative operation, that is, is understood to be , instead of . If ambiguity is not an issue parentheses are sometimes dropped.
Computing can be restated in terms of an infinite table. We place the numbers in the top row, and fill the left column with values 2. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.
We place the numbers in the top row, and fill the left column with values 3. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.
We place the numbers in the top row, and fill the left column with values 4. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.
We place the numbers in the top row, and fill the left column with values 10. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.
For 2 ≤ b ≤ 9 the numerical order of the numbers is the lexicographical order with n as the most significant number, so for the numbers of these 8 columns the numerical order is simply line-by-line. The same applies for the numbers in the 97 columns with 3 ≤ b ≤ 99, and if we start from n = 1 even for 3 ≤ b ≤ 9,999,999,999.