| Revision as of 13:56, 7 August 2009 editDouble sharp (talk | contribs)Autopatrolled, Extended confirmed users, Page movers, File movers, Pending changes reviewers102,065 edits ←Redirected page to Hyperoperation | Revision as of 07:50, 15 November 2009 edit undo4 (talk | contribs)3,692 editsNo edit summaryNext edit → | ||
| Line 1: | Line 1: | ||
| {{orphan}} | |||
| #REDIRECT ] | |||
| '''Hexation''' is the operation of repeated ], just as ] is the operation of repeated ] and is a ]. Like pentation and tetration, it has little real-world applications, and has few applications even in mathematics. It is non-commutative, and therefore has two inverse functions, which might be named the '''hexa-root''' and the '''hexa-logarithm (analogous to the two inverse functions for exponentiation: ] function and ]). Hexation is not ]. | |||
| The word "Hexation" was coined by ] from the roots ] (six) and ]. It is part of his general naming scheme for ]. | |||
| Hexation can be written in ] as <math>a \uarr \uarr \uarr \uarr b</math> or <math>a \uarr^{4} b</math>. | |||
| ==Extension== | |||
| It is not known how to extend hexation to ] or non-integer values. | |||
| ==Selected values== | |||
| As its base operation (pentation) has not been extended to non-integer heights, hexation <math>a \uarr^{4} b</math> is currently only defined for integer values of ''a'' and ''b'' where ''a'' > 0 and ''b'' ≥ 0. Like all other ] of order 3 (]) and higher, pentation has the following trivial cases (identities) which holds for all values of a and b within its domain: | |||
| *<math>1 \uarr^{4} b = 1</math> | |||
| *<math>a \uarr^{4} 1 = a</math> | |||
| Other than the trivial cases shown above, hexation generates extremely large numbers very quickly such that there is only one fnon-trivial case that produces numbers that can be written in conventional notation, as illustrated below: | |||
| *<math> 2 \uarr^{4} 2 = 4</math> | |||
| {{Hyperoperations}} | |||
Revision as of 07:50, 15 November 2009
 | This article is an orphan, as no other articles link to it. Please introduce links to this page from related articles; try the Find link tool for suggestions. |
Hexation is the operation of repeated pentation, just as pentation is the operation of repeated tetration and is a hyperoperation. Like pentation and tetration, it has little real-world applications, and has few applications even in mathematics. It is non-commutative, and therefore has two inverse functions, which might be named the hexa-root and the hexa-logarithm (analogous to the two inverse functions for exponentiation: nth root function and logarithm). Hexation is not elementary recursive.
The word "Hexation" was coined by Reuben Goodstein from the roots hexa- (six) and iteration. It is part of his general naming scheme for hyperoperations.
Hexation can be written in Knuth's up-arrow notation as or .
or 
.
Extension
It is not known how to extend hexation to complex or non-integer values.
Selected values
As its base operation (pentation) has not been extended to non-integer heights, hexation is currently only defined for integer values of a and b where a > 0 and b ≥ 0. Like all other hyperoperations of order 3 (exponentiation) and higher, pentation has the following trivial cases (identities) which holds for all values of a and b within its domain:
Other than the trivial cases shown above, hexation generates extremely large numbers very quickly such that there is only one fnon-trivial case that produces numbers that can be written in conventional notation, as illustrated below:
| Hyperoperations | |
|---|---|
| Primary | |
| Inverse for left argument | |
| Inverse for right argument | |
| Related articles | |