Compare hyperoperation levels
Select a base, second argument, and maximum rank. The visualizer calculates each supported operation using the same input pair.
What are hyperoperations?
Hyperoperations form a hierarchy of arithmetic operations. Each level generalizes the operation below it.
| Rank | Operation | Interpretation |
|---|---|---|
| 0 | Successor | Add one to the second argument |
| 1 | Addition | Repeated successor |
| 2 | Multiplication | Repeated addition |
| 3 | Exponentiation | Repeated multiplication |
| 4 | Tetration | Repeated exponentiation |
| 5 | Pentation | Repeated tetration |
| 6 | Hexation | Repeated pentation |
| 7 | Heptation | Repeated hexation |
| 8 | Octation | Repeated heptation |
The recursive definition
This visualizer uses the following common convention for hyperoperations:
For ranks n ≥ 4:
This recursive definition explains why each new rank can grow much faster than the previous one. Different mathematical sources sometimes use different indexing conventions, so always check the stated definition when comparing results.
How hyperoperation growth changes
Consider the progression from multiplication to exponentiation and tetration:
At the next level, pentation applies tetration recursively. The resulting numbers can become too large to write in ordinary decimal notation even for small arguments.
The higher ranks are useful in studying extremely fast-growing functions, large-number notation, and the limits of ordinary numerical computation.
Numerical limitations
- Base values are restricted to integers from 1 to 10.
- The second argument is restricted to integers from 0 to 6.
- Ranks from 0 through 8 can be selected.
- Numerical evaluation is limited to values up to approximately 10300.
- Recursive computation also has a call budget.
- Values outside the supported range are shown symbolically rather than approximated.
Related hyperoperation calculators
Frequently asked questions
What is the first hyperoperation?
In this convention, rank 0 is the successor operation, H₀(a,b) = b + 1. Rank 1 is addition, followed by multiplication and exponentiation.
What is tetration?
Tetration is rank 4 in this hierarchy. It represents repeated exponentiation and is commonly written using double up-arrow notation.
What comes after pentation?
Hexation is rank 6, followed by heptation, octation, and increasingly higher hyperoperations. Names for very high ranks are not universally standardized.
Why are higher hyperoperations difficult to calculate?
Each level uses the previous operation recursively. This causes extremely rapid growth, quickly exceeding the numerical range of ordinary computer data types.
Are all higher hyperoperations supported?
This visualizer supports ranks 0 through 8, subject to computation limits. It uses a specific recursive convention and does not claim to evaluate every possible generalized hyperoperation.