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#1 2021-06-10 16:46:06

Jai Ganesh
Registered: 2005-06-28
Posts: 46,753

Knuth's up-arrow notation

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

. 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

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

stands for n arrows, so for example


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.


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