Compare growth at an input value
Choose an integer n to compare the functions at the same input. The selected value is saved in the page URL, so you can refresh the page or share your comparison without losing the result.
Results for n = 5
Function values and mathematical notation
Product of all positive integers up to n.
Repeated multiplication by a fixed base.
A power tower of 2 with height n.
The hyperoperation following tetration.
A higher hyperoperation following pentation.
Function comparison table
| Function | Notation | Value or representation |
|---|---|---|
| Factorial |
5!
|
120 |
| Exponential |
2^5
|
32 |
| Tetration |
2 ↑↑ 5
|
≈ 10^19,728.3018 |
| Pentation |
2 ↑↑↑ 5
|
Symbolic representation |
| Hexation |
2 ↑↑↑↑ 5
|
Symbolic representation |
Understanding the growth functions
1. Factorial growth
n! = 1 × 2 × 3 × ... × n
Factorial multiplies every positive integer up to n. It grows faster asymptotically than any fixed-base exponential function c^n, where c is a constant greater than 1.
2. Exponential growth
f(n) = 2^n
The input appears in the exponent. Exponential functions with a fixed base greater than 1 grow rapidly, but factorial eventually overtakes any fixed-base exponential.
3. Tetration
2 ↑↑ n = 2^(2^(...^2))
Tetration creates a power tower of n copies of the base 2. The tower is evaluated from the top down. For example, 2 ↑↑ 3 = 2^(2^2) = 16.
4. Pentation
2 ↑↑↑ n
Pentation is the next operation in the usual hyperoperation sequence. It extends tetration through another level of iteration and grows much faster.
5. Hexation and higher operations
2 ↑↑↑↑ n
Hexation follows pentation. Still higher hyperoperation ranks continue the sequence, producing growth rates far beyond ordinary exponentials and power towers.
Asymptotic growth order
For the functions used in this calculator, the following describes their eventual growth order as n becomes large:
-
Exponential:
2^n -
Factorial:
n! -
Tetration:
2 ↑↑ n -
Pentation:
2 ↑↑↑ n -
Hexation:
2 ↑↑↑↑ n
2^n << n! << 2 ↑↑ n << 2 ↑↑↑ n << 2 ↑↑↑↑ n
This is an informal asymptotic hierarchy. The symbols indicate eventual growth dominance, not a claim that every adjacent pair has this numerical order for every small integer input.
Why compare growth rates?
Growth-rate comparison is useful in number theory, combinatorics, algorithms, mathematical logic and googology. It helps explain why some functions become impractical to calculate even when their input values are relatively small.
Factorial values can be calculated for many ordinary inputs, while a modest tetration height can produce a number with an enormous number of digits. Pentation and higher hyperoperations go much further.
Frequently asked questions
1. What is a growth rate in mathematics?
A growth rate describes how quickly a function increases as its input becomes larger. Comparing growth rates helps distinguish functions that may look similar at small inputs but behave very differently for large inputs.
2. Does factorial grow faster than an exponential function?
The factorial n! eventually grows faster than c^n for every fixed constant c greater than 1. At small inputs, however, a particular exponential function can still have a larger numerical value.
3. What is the difference between exponentiation and tetration?
Exponentiation repeats multiplication, while tetration repeats exponentiation. For example, 2 ↑↑ 3 means 2^(2^2), which equals 16.
4. What comes after tetration?
Pentation is the next operation in the usual hyperoperation sequence, followed by hexation and still higher ranks.
5. Why are some results shown symbolically?
Many higher-operation values are too large to store or print as ordinary decimal numbers. Symbolic notation preserves the mathematical expression without presenting an unreliable approximation.
6. Does the growth hierarchy apply to every input?
The hierarchy describes eventual growth as the input increases. It does not mean that every function has a larger value at every small integer input.
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