Tetration - Notation

Notation

There are many different notation styles that can be used to express tetration. Some of these styles can be used for higher iterations as well (hyper-5, hyper-6, and so on).

Name	Form	Description
Standard notation		Used by Maurer and Goodstein ; Rudy Rucker's book Infinity and the Mind popularized the notation.
Knuth's up-arrow notation		Allows extension by putting more arrows, or, even more powerfully, an indexed arrow.
Conway chained arrow notation		Allows extension by increasing the number 2 (equivalent with the extensions above), but also, even more powerfully, by extending the chain
Ackermann function		Allows the special case to be written in terms of the Ackermann function.
Iterated exponential notation		Allows simple extension to iterated exponentials from initial values other than 1.
Hooshmand notation
Hyper operator notation		Allows extension by increasing the number 4; this gives the family of hyper operations
ASCII notation	a^^n	Since the up-arrow is used identically to the caret (^), the tetration operator may be written as (^^).
Bowers' array notation	{a,b,2}

One notation above uses iterated exponential notation; in general this is defined as follows:

with n "a"s.

There are not as many notations for iterated exponentials, but here are a few:

Name	Form	Description
Standard notation		Euler coined the notation, and iteration notation has been around about as long.
Knuth's up-arrow notation		Allows for super-powers and super-exponential function by increasing the number of arrows; used in the article on large numbers.
Ioannis Galidakis' notation		Allows for large expressions in the base.
ASCII (auxiliary)	a^^n@x	Based on the view that an iterated exponential is auxiliary tetration.
ASCII (standard)	exp_a^n(x)	Based on standard notation.
J Notation	x^^:(n-1)x	Repeats the exponentiation. See J (programming language)