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

. 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.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.]]>