Tetration Calculator

Introduction to tetration and power towers

Multiplication repeats addition, exponentiation repeats multiplication, and tetration repeats exponentiation. Written with Knuth's double arrow, 2 ↑↑ 4 stacks four twos into a tower of exponents, 2²²². The first number is the base, the second is the height of the tower.

The rule that makes tetration violent is that a tower collapses from the top down: 2 ↑↑ 4 = 2^(2(22)) = 2¹⁶ = 65,536, not the boring 256 you would get evaluating bottom-up. Because one extra level replaces the whole exponent, 2 ↑↑ 5 = 2^65,536 already runs to 19,729 digits — the jump from a five-digit number to a nineteen-thousand-digit one is the operation's entire personality. Shrink the base below e^1/e ≈ 1.4447, though, and the same construction turns tame, creeping toward a finite ceiling however tall you build. This page is built to explore both regimes.

How to use this tetration calculator

The mode switch at the top of the form chooses which question you are asking:

• Tetrate evaluates a ↑↑ b for any positive base and any height from −1 up to 10¹⁵, including fractional heights such as 3.5, which almost no other calculator handles.
• Super-root fixes the height and finds the base: which number tetrated to height 4 gives 65,536? It answers 2.
• Super-log fixes the base and finds the height: how tall must a tower of 2s be to reach 65,536? It answers 4.

Presets load the classic landmarks. Results print a plain value while that stays readable, then fall back to scientific notation, log₁₀(value), and digit counts once printing becomes absurd — usually within a row or two. The optional step table lists each level of the tower, the infinite-tower checkbox adds a convergence verdict, approximate results carry a warning rather than rounding silently, and Copy LaTeX exports the calculation for a paper or homework write-up. Below the form, a cobweb plot animates the tower iteration and a limit curve collects every infinite-tower value; the fastest way to build intuition is to evaluate a tower and then drag the cobweb slider through the same base.

Notation and formulas for repeated exponentiation

Several notations circulate for the same idea. This page uses Knuth's up-arrows, where each extra arrow repeats the operation below it: a ↑ b is exponentiation, a ↑↑ b is tetration, a ↑↑↑ b is pentation. You will also meet Rudy Rucker's left-superscript ^ba (so ⁴2 = 2 ↑↑ 4) and the iterated-exponential form exp_a^b(1), which reads the tower as x → a^x applied b times to 1. The defining recursion, anchored at a ↑↑ 0 = 1, is:

Running it backwards gives a ↑↑ (−1) = 0 and then stops, since height −2 would need log_a(0) = −∞. Tetration is the fourth rung of the hyperoperation ladder:

Each rung makes the one below it look stationary. To keep climbing, the pentation calculator and hexation calculator continue the sequence, and the exotic math guidebook tours the wider landscape of large-number notation.

Worked examples, from 2 ↑↑ 4 to fractional heights

2 ↑↑ 4 by hand. Work top-down: 2² = 4, then 2⁴ = 16, then 2¹⁶ = 65,536. The intermediate values 4, 16, and 65,536 are themselves the towers 2 ↑↑ 2, 2 ↑↑ 3, and 2 ↑↑ 4 — which is exactly what the step table writes out for you.

1.35 ↑↑ 4, the quiet regime. With a base below e^1/e the tower barely moves: 1.35, 1.499, 1.568, 1.600, and a hundred levels only inch toward 1.6319…, the solution of x = 1.35^x. Same recursion, completely different story — the base, not the height, decides whether tetration explodes.

0.5 ↑↑ 8, a convergent small base. Bases below 1 approach their limit from alternating sides — 0.5, 0.707, 0.613, 0.654, … — with odd levels pushing down and even levels up until the sequence squeezes onto 0.641186. The cobweb plot below shows this as a spiral tightening around the fixed point.

2 ↑↑ 3.5, a fractional height. Under the linear approximation this page uses, the tower seeds at 2 ↑↑ 0.5 = 2^0.5 ≈ 1.414 and climbs three exponentiation steps to 2 ↑↑ 3.5 ≈ 81.17 — landing, as it must, between 2 ↑↑ 3 = 16 and 2 ↑↑ 4 = 65,536, though far nearer the bottom because nothing on a tower is linear.

When does an infinite power tower converge?

Building the tower forever means iterating the sequence x₁ = a, x_n+1 = a^x_n. Euler settled its fate in the 18th century, and the answer is one of the prettiest results in classical analysis: the infinite tower converges exactly on the closed interval below, numerically 0.06599 ≤ a ≤ 1.44467, where the limit L solves the fixed-point equation on the right.

Plain-text formulas: lowerConvergenceBase = exp(-e); upperConvergenceBase = exp(1/e); limit = base^limit. A finite real infinite-tower limit requires the base to stay within those convergence bounds.

Why e^1/e? A limit is a crossing of the curve y = a^x with the diagonal y = x. Raising a lifts the curve; at the critical base it stops crossing and only kisses the line — tangency, which forces both a^x = x and a^x · ln a = 1 at once. Solving them together pins x = e and a = e^1/e. Above that base the curve clears the diagonal and the tower diverges; exactly at e^1/e it converges, with agonizing slowness, to e itself.

The √2 paradox. The equation x = (√2)^x is solved by both 2 and 4, yet the tower equals 2. Starting from 1, the iteration climbs to 2 and stops because 2 is attracting — the slope ln 2 ≈ 0.693 there is below 1, so nearby points get pulled in — while 4 is repelling, its slope 2 ln 2 ≈ 1.386 pushing every neighbor away. Press the √2 button on the cobweb plot to watch it. Below e^−e ≈ 0.066 the curve crosses too steeply and the sequence splits by parity into a stable two-cycle, so no single limit survives; the tetration base convergence analyzer explores that boundary in detail.

Super-logarithms and super-roots: tetration's two inverses

Roots recover the base of an exponentiation and logarithms recover the exponent, and tetration inherits the same pair of inverses. The super-root fixes the height and solves for the base. The order-2 super-root asks which a satisfies a^a = x; it has the closed form ssrt(x) = ln x / W(ln x) via the Lambert W function, so ssrt(4) = 2 and ssrt(27) = 3. Higher-order super-roots (which base gives a ↑↑ 4 = 65,536?) have no closed form and are found by numerical search. Because a^a bottoms out at e^−1/e ≈ 0.6922, targets below that have no real order-2 super-root, and the calculator says so.

The super-logarithm slog_a(x) fixes the base and asks how tall the tower must be: slog₂(65,536) = 4. Whole-number answers are exact; fractional ones reuse the linear approximation, since the two questions are mirror images. It is the smooth cousin of the iterated logarithm log* beloved by computer scientists — the count of how many logarithms it takes to drop a number to 1 — which is why the calculator reports huge towers in slog-like terms once ordinary notation gives out. Tetration is not an engineering tool: it lives in computability theory (the Ackermann function grows tetrationally), tower-type bounds from logic and Ramsey theory, and the number engines behind incremental games. Graham's number itself begins two rungs higher, at 3 ↑↑↑↑ 3.

Accuracy, approximations, and limitations

Integer-height towers use layered big-number arithmetic (the OmegaNum library) and are exact in structure; the printed digits of astronomical results carry magnitude, not certified decimals. Fractional heights and fractional super-logs use the piecewise-linear approximation — define a ↑↑ h = 1 + h for −1 ≤ h ≤ 0, then extend by the recursion. It is continuous and exact at integer heights but only once-differentiable, so it is a tetration, not the unique real-analytic one Kneser constructed for base e in 1950; those results carry a warning. Higher-order super-roots come from a numerical search at floating-point tolerance.

The infinite-tower analysis iterates x → a^x to a 10⁻¹⁴ tolerance over 80 steps; bases within about 10⁻³ of e^1/e converge so slowly the last digits may still be drifting, so the exact endpoints (e at the top, 1/e at the bottom) are handled specially. The step table stops at 100 levels and heights are capped at 10¹⁵ to keep the page responsive. Negative bases are rejected on purpose: with fractional exponents they leave the real numbers, and their complex continuation (the Shell–Thron region) is beyond what this page will compute.

Tetration: frequently asked questions