Calculate successor, addition, multiplication, exponentiation and tetration. Explore Knuth up-arrow notation and rapidly growing numbers.
Enter a non-negative base a, a non-negative value b, and choose a hyperoperation from H₀ through H₄.
Different mathematical sources sometimes use different hyperoperation indexing conventions, so this page states its convention explicitly.
Hyperoperations form a hierarchy of increasingly powerful arithmetic operations. Starting with successor, the hierarchy continues through addition, multiplication, exponentiation, tetration and higher operations.
The basic idea is that each level repeatedly applies the operation immediately below it.
| Level | Name | Notation | Meaning |
|---|---|---|---|
| H₀ | Successor | b + 1 | Add one to b. |
| H₁ | Addition | a + b | Repeated successor. |
| H₂ | Multiplication | a × b | Repeated addition. |
| H₃ | Exponentiation | a^b | Repeated multiplication. |
| H₄ | Tetration | a ↑↑ b | Repeated exponentiation. |
| H₅ | Pentation | a ↑↑↑ b | Repeated tetration. |
| H₆ | Hexation | a ↑↑↑↑ b | Repeated pentation. |
Tetration is the hyperoperation immediately above exponentiation. It repeatedly applies exponentiation to construct a power tower.
3 ↑↑ 3 = 3^(3^3)
= 3^27
= 7,625,597,484,987
The height of the tower is important.
This is why hyperoperations should not simply be converted to ordinary PHP floating-point numbers. Floating-point arithmetic loses exactness very quickly for enormous integers.
Donald Knuth's up-arrow notation provides a compact way to describe extremely large operations.
| Notation | Operation | Example |
|---|---|---|
| a ↑ b | Exponentiation | 2 ↑ 5 = 2⁵ = 32 |
| a ↑↑ b | Tetration | 2 ↑↑ 4 = 65,536 |
| a ↑↑↑ b | Pentation | Repeated tetration |
| a ↑↑↑↑ b | Hexation | Repeated pentation |
Adding another arrow does not merely increase the number. It moves to a substantially faster level of growth.
Addition → repeated successor
Multiplication → repeated addition
Exponentiation → repeated multiplication
Tetration → repeated exponentiation
Each new level repeatedly applies the operation below it. Consequently, increasing the operation level can cause an enormous increase in the result even when the inputs are small.
This is also why exact arbitrary-precision arithmetic and symbolic notation are important when building a hyperoperation calculator.
H₀(7,5) = 5 + 1 = 6
H₁(7,5) = 7 + 5 = 12
H₂(7,5) = 7 × 5 = 35
H₃(2,10) = 2¹⁰ = 1,024
H₄(2,4) = 2 ↑↑ 4 = 65,536
H₄(2,5) = 2^65,536
Hyperoperations connect naturally with arithmetic, number theory, combinatorics and rapidly growing sequences. Explore these related Summe.org calculators:
Evaluate finite sums using sigma notation.
Calculate arithmetic and geometric series.
Explore sums involving powers of integers.
Calculate sums involving rapidly growing factorials.
Explore sums involving binomial coefficients.
Explore Pascal’s triangle and its combinatorial patterns.
Study efficient addition sequences related to exponentiation.
Explore ways integers can be represented as sums.
Calculate triangular numbers and related sequences.
Calculate sums of consecutive squares.
Calculate sums of consecutive cubes.
Explore unusual and challenging summation problems.
Learn how common summation identities can be proved.
For summation and notation problems, see the Sigma Calculator , Sum of a Series Calculator , Sum of Powers Calculator , and Factorial Sum Calculator .
For combinatorics, explore the Binomial Sum Calculator , Pascal Triangle Calculator , and Number Partition Calculator .
For integer sequences and related arithmetic structures, see the Triangular Number Calculator , Sum of Squares Calculator , Sum of Cubes Calculator , Addition Chains Calculator , Digital Root Calculator , and Mod 9 Checker .
A hyperoperation is a hierarchy of arithmetic operations extending successor, addition, multiplication and exponentiation to tetration and higher operations.
On this page H0 is successor, H1 is addition, H2 is multiplication, H3 is exponentiation and H4 is tetration. Mathematical sources can use different indexing conventions, so the convention is stated explicitly.
Tetration is repeated exponentiation. For example, 3 ↑↑ 3 means 3^(3^3), which equals 7,625,597,484,987.
Knuth up-arrow notation provides a compact way to describe rapidly growing operations. One arrow represents exponentiation, two arrows tetration, three arrows pentation and four arrows hexation.
3 ↑↑ 3 = 3^(3^3) = 3^27 = 7,625,597,484,987.
Pentation comes after tetration in the commonly used hyperoperation hierarchy. It is represented by three up arrows: a ↑↑↑ b.
Pentation is the hyperoperation above tetration and can be understood informally as repeated tetration. Its values become enormous extremely quickly.
Each new level repeatedly applies the operation below it. Addition repeats successor, multiplication repeats addition, exponentiation repeats multiplication and tetration repeats exponentiation.
Small hyperoperations can be calculated exactly. For enormous values, displaying every decimal digit is impractical, so this calculator uses symbolic notation and size estimates instead of inaccurate floating-point arithmetic.