Hexation Calculator
Enter a small integer base and a positive hexation level. Because hexation grows extraordinarily quickly, only small inputs can be evaluated exactly.
Hexation is evaluated recursively from the preceding hyperoperation.
What Is Hexation?
Hexation is a very high-level arithmetic operation in the hyperoperation hierarchy. It comes immediately after pentation.
In the common Knuth up-arrow correspondence, pentation uses three up arrows while hexation uses four:
Hexation: a ↑↑↑↑ b
Hexation grows far faster than addition, multiplication, exponentiation, tetration, and pentation.
Hyperoperation Hierarchy
Hyperoperations form a sequence in which each level is recursively constructed from the previous operation.
| Level | Operation | Notation | Example |
|---|---|---|---|
| 1 | Addition | a + b | 2 + 3 = 5 |
| 2 | Multiplication | a × b | 2 × 3 = 6 |
| 3 | Exponentiation | a ↑ b | 2³ = 8 |
| 4 | Tetration | a ↑↑ b | 2 ↑↑ 3 = 16 |
| 5 | Pentation | a ↑↑↑ b | 2 ↑↑↑ 3 = 65,536 |
| 6 | Hexation | a ↑↑↑↑ b | 2 ↑↑↑↑ 2 = 4 |
The terminology can vary between mathematical conventions, especially for indexing the first few hyperoperations. This page uses the convention in which tetration is the fourth hyperoperation, pentation the fifth, and hexation the sixth.
How Hexation Works
Hexation can be understood recursively using pentation.
Under the convention used by this calculator, positive-level hexation can be represented by:
H(a, n) = P(a, H(a, n − 1))
Here H represents hexation and P represents pentation.
Hexation Examples
Small examples demonstrate why the operation becomes impractical very quickly.
2 ↑↑↑↑ 2 = 2 ↑↑↑ 2 = 4
2 ↑↑↑↑ 3 = 2 ↑↑↑ 4
The last expression already involves a pentation calculation with an argument of 4, producing a number far beyond ordinary numerical representation.
Hexation and Knuth's Up-Arrow Notation
Donald Knuth's up-arrow notation provides a compact way to describe extremely fast-growing operations.
| Arrows | Operation |
|---|---|
| ↑ | Exponentiation |
| ↑↑ | Tetration |
| ↑↑↑ | Pentation |
| ↑↑↑↑ | Hexation |
Therefore, a compact representation of hexation is:
Why Hexation Grows So Quickly
Each hyperoperation builds upon the previous operation. Exponentiation repeats multiplication, tetration repeats exponentiation, pentation recursively uses tetration, and hexation recursively uses pentation.
Repeated multiplication.
Repeated exponentiation.
Recursive tetration.
Recursive pentation.
Hexation vs. Pentation
Pentation is represented by three up arrows:
Hexation adds another level to the hierarchy:
Consequently, even small hexation arguments can represent operations that are vastly larger than familiar large-number expressions.
Hexation vs. Tetration
Tetration is two up arrows:
Hexation uses four:
Two additional levels of recursion separate tetration from hexation, making their growth rates dramatically different.
Related Summe.org Calculators
Frequently Asked Questions
What is hexation?
Hexation is a high-level hyperoperation that comes immediately after pentation. In Knuth's up-arrow notation, it is commonly represented using four up arrows: a ↑↑↑↑ b.
What comes before hexation?
Pentation comes immediately before hexation. Pentation uses three up arrows in Knuth's notation, while hexation uses four.
What is 2 ↑↑↑↑ 2?
Under the recursive convention used on this page, 2 ↑↑↑↑ 2 = 4, because the second level reduces to 2 ↑↑↑ 2, which is 4.
Is hexation bigger than tetration?
Yes. Hexation is two hyperoperation levels above tetration. Its recursive structure uses pentation, whereas tetration uses exponentiation.
Can a computer calculate large hexation numbers?
Only very small cases can be represented explicitly. Hexation grows so rapidly that symbolic notation is generally more useful than storing the complete decimal expansion.
Is hexation the same as four-arrow notation?
Under the common Knuth up-arrow convention used here, hexation corresponds to four up arrows: ↑↑↑↑.