Tetration Calculator
Build and read a power tower
Tetration is repeated exponentiation. A tower written as a ↑↑ b contains b copies of the base a, and it is evaluated from the top down. For example, 2 ↑↑ 4 means 2222, which is 216 = 65,536.
Use the presets for common landmarks, or enter a positive base and an integer height. The result shows the value when it is still printable, then falls back to scientific notation, logarithms, and digit counts once the tower becomes too large to write out.
What the calculator is doing
A finite tower is recursive. Level 1 is just the base. Level 2 is aa. Level 3 is a(aa). Every new level puts the whole previous tower into the exponent, so even tame-looking inputs can cross from ordinary arithmetic into large-number notation very quickly.
The calculator sends the main evaluation to a background worker so the page stays responsive. For manageable values it uses high-precision large-number arithmetic through OmegaNum. Once the raw value becomes impractical, it reports the scale instead: log₁₀(value), an approximate digit count, and a step table when requested.
How to read the output
- Value is the finite tower for the base and height you entered. Small results appear directly; large ones appear in scientific or up-arrow notation.
- Digit count is computed from ⌊log₁₀(value)⌋ + 1 whenever the value is positive and the logarithm is available.
- Partial towers show each level from 1 through the requested height. This is the best way to see where the scale changes from human-sized to astronomical.
- Infinite tower analysis studies the endless expression aaa.... That is a different question from a finite tower, but it is often the most interesting one.
The convergence window
For a positive real base, the infinite tower converges to a real number exactly in the interval e^(-e) ≤ a ≤ e^(1/e), roughly 0.065988 ≤ a ≤ 1.444668. Inside that window, the limit solves x = ax. At the upper endpoint, the limit is e; above it, the tower runs away instead of settling down.
Plain-text formula: lowerConvergenceBase = exp(-e); upperConvergenceBase = exp(1/e). Classification: 0 < a < exp(-e) oscillates / does not converge to a single real fixed point; exp(-e) <= a <= exp(1/e) converges; a > exp(1/e) diverges.
Bases between 0 and 1 deserve careful wording. They often alternate above and below the limit before settling, which is why the step table may look like it is bouncing. If the base drops below e-e, there is no single real fixed-point limit for the infinite tower.
Useful reference points
| Tower | Meaning | Scale |
|---|---|---|
| 2 ↑↑ 3 | 2(22) | 16 |
| 2 ↑↑ 4 | 216 | 65,536 |
| 2 ↑↑ 5 | 265,536 | about 19,729 digits |
| 3 ↑↑ 3 | 327 | 7,625,597,484,987 |
| 0.5 ↑↑ 8 | a small-base tower | near 0.642 |
| e1/e ↑↑ n | critical infinite-tower base | finite towers approach e slowly |
Worked example: 1.35 ↑↑ 4
Take 1.35 ↑↑ 4. The first two levels are still quiet: level 1 is 1.35, and level 2 is about 1.499. Level 3 rises to about 1.568, and level 4 lands near 1.600. Because 1.35 is below e1/e, the infinite tower has a real limit rather than diverging.
This kind of example is a good reminder that tetration is not always about incomprehensibly large numbers. The base controls the story. A base of 2 becomes enormous almost immediately; a base like 1.35 climbs toward a stable fixed point.
Practical limits
This page is built for exploration, not formal proof or arbitrary-precision research. Heights are capped at 64 to keep the browser responsive. Positive fractional bases are supported, but negative bases are rejected because real-valued tetration can become ambiguous as soon as fractional exponents enter the tower.
Frequently asked questions
Is tetration the same as writing a^b?
No. Ordinary exponentiation has one exponent. Tetration repeats exponentiation, so the exponent is itself a tower. That is why 2 ↑↑ 4 is 2^(2^(2^2)), not 2^4.
Why do you evaluate the tower from the top down?
Exponentiation is right-associative in power towers. If you evaluated 2^(2^(2^2)) from the bottom as ((2^2)^2)^2, you would get 256 instead of 65,536. Tetration uses the top-down convention.
Why does the output sometimes stop being a normal number?
The value may still be mathematically well-defined, but printing it is no longer useful. 2 ↑↑ 5 already has about 19,729 digits, and 2 ↑↑ 6 has a digit count that is itself roughly a 19,729-digit number. Logarithms communicate that scale cleanly.
What does the infinite tower result mean?
It asks what happens as the height grows without bound. When the limit exists, it must satisfy x = ax. The calculator reports that fixed point for bases in the convergence window and reports divergence outside it.
Can I trust the last decimal place?
For small towers, yes, within ordinary floating-point and library precision. For very large towers, the important result is the scale, not the final printed digit. Treat large scientific-notation outputs and logarithmic summaries as magnitude information.