Convert to Knuth Up-Arrow Notation
Enter a rank, operation name, or supported expression. The converter identifies the operation and generates its corresponding notation.
Hyperoperation to Up-Arrow Reference
The converter uses the convention that rank 3 corresponds to one arrow, rank 4 to two arrows, and each subsequent rank adds one arrow.
| Rank | Operation | Knuth up-arrow notation | Arrow count |
|---|---|---|---|
| 0 | Successor | b + 1 |
Not applicable |
| 1 | Addition | a + b |
Not applicable |
| 2 | Multiplication | a × b |
Not applicable |
| 3 | Exponentiation | a ↑ b |
1 |
| 4 | Tetration | a ↑↑ b |
2 |
| 5 | Pentation | a ↑↑↑ b |
3 |
| 6 | Hexation | a ↑↑↑↑ b |
4 |
| 7 | Heptation | a ↑↑↑↑↑ b |
5 |
| 8 | Octation | a ↑↑↑↑↑↑ b |
6 |
| 9 | Nonation | a ↑↑↑↑↑↑↑ b |
7 |
| 10 | Rank 10 hyperoperation | a ↑↑↑↑↑↑↑↑ b |
8 |
How Knuth Up-Arrow Notation Works
Knuth up-arrow notation is a mathematical notation for expressing extremely rapidly growing operations. It extends ordinary exponentiation by using multiple arrows between two values.
One arrow: exponentiation
One arrow represents ordinary exponentiation.
Two arrows: tetration
Two arrows represent repeated exponentiation, commonly understood through power towers for positive integer heights.
Three arrows: pentation
Three arrows represent iteration of the two-arrow operation.
Four arrows: hexation
Four arrows represent the next level of iteration above pentation.
General relationship
This relationship applies to ranks 3 and above under the rank convention used by this calculator. Successor, addition, and multiplication are shown using their ordinary arithmetic notation.
Supported Input Formats
| Input | Meaning | Output |
|---|---|---|
H3 |
Rank 3 | a ↑ b |
H4 |
Rank 4 | a ↑↑ b |
tetration |
Tetration | a ↑↑ b |
pentation |
Pentation | a ↑↑↑ b |
a^^b |
Two-arrow expression | a ↑↑ b |
a^^^^b |
Four-arrow expression | a ↑↑↑↑ b |
a^b |
Exponentiation | a ↑ b |
a+b |
Addition | a + b |
a*b |
Multiplication | a × b |
This is a symbolic converter. It supports simple variable or numeric operands and does not parse arbitrary algebraic expressions or evaluate their numerical values.
Frequently Asked Questions
1. What is Knuth up-arrow notation?
It is a notation introduced by Donald Knuth to describe operations that grow much faster than ordinary exponentiation. Additional arrows represent higher levels of operation.
2. How many arrows does tetration use?
Tetration uses two arrows. It is written as a ↑↑ b and corresponds to rank 4 in the hyperoperation hierarchy used here.
3. How is pentation written?
Pentation uses three arrows: a ↑↑↑ b. It corresponds to rank 5.
4. Can addition and multiplication be written with up-arrows?
Standard Knuth up-arrow notation begins with exponentiation. Addition and multiplication are therefore normally written using their ordinary symbols rather than arrows.
5. Does the converter calculate enormous numbers?
No. It converts supported symbolic expressions and identifies their arrow counts. It does not evaluate tetration, pentation, or higher operations numerically.
6. What is the maximum rank supported?
This version supports ranks 0 through 10. For ranks 3 through 10, that corresponds to one through eight Knuth up-arrows.