Calculate a Multiple Up-Arrow Expression
Choose a base, number of arrows, and height. The calculator evaluates the expression using the standard Knuth up-arrow recurrence.
Multiple Up-Arrow Reference Table
The number of arrows determines the hyperoperation level.
| Arrows | Operation | Hyperoperation rank | Notation |
|---|---|---|---|
| 1 | Exponentiation | 3 | a ↑ b |
| 2 | Tetration | 4 | a ↑↑ b |
| 3 | Pentation | 5 | a ↑↑↑ b |
| 4 | Hexation | 6 | a ↑↑↑↑ b |
| 5 | Heptation | 7 | a ↑↑↑↑↑ b |
| 6 | Octation | 8 | a ↑↑↑↑↑↑ b |
| 7 | Nonation | 9 | a ↑↑↑↑↑↑↑ b |
| 8 | Rank 10 hyperoperation | 10 | a ↑↑↑↑↑↑↑↑ b |
How Multiple Up-Arrow Calculations Work
Knuth up-arrow notation generalizes exponentiation. One arrow represents exponentiation, two arrows represent tetration, and additional arrows represent increasingly higher operations.
Two arrows: tetration
Tetration is repeated exponentiation. For example, with positive integer height, 2 ↑↑ 3 represents 2 raised to the power of (2 raised to the power of 2), which equals 16.
Three arrows: pentation
Pentation iterates tetration according to the up-arrow recurrence.
Four arrows: hexation
Hexation is the next hyperoperation level above pentation.
General recurrence
Here, n represents the number of arrows. The recurrence applies to n ≥ 2 and positive integer heights, with the one-arrow operation reducing to exponentiation.
Frequently Asked Questions
1. What is a multiple up-arrow expression?
It is a mathematical expression containing two or more Knuth up-arrows between values. More arrows represent higher levels of repeated operations.
2. What does two-arrow notation mean?
Two arrows represent tetration, a repeated exponentiation operation. For example, 2 ↑↑ 3 = 2^(2^2) = 16.
3. What is the difference between tetration and pentation?
Tetration uses two arrows and iterates exponentiation. Pentation uses three arrows and iterates tetration.
4. Why are the input limits restricted?
Multiple up-arrow operations grow extremely quickly. Even small inputs can exceed the numerical range of ordinary computer floating-point calculations. Limits help protect the calculator from excessive computation and overflow.
5. Does the calculator support decimal heights?
No. This calculator uses integer heights because its recurrence is implemented for discrete iterations. Fractional tetration and fractional higher up-arrow operations require additional mathematical definitions.
6. Can the result be an approximation?
Yes. PHP uses floating-point arithmetic for numerical results. Values that fit within the supported range may be displayed as decimals or scientific notation, and are not necessarily exact integer representations.