How Big Are Factorials?

This explains how to estimate how big n! is, focusing on the number of decimal digits, and gives a simple practical rule: number of digits ≈ n·log10(n/e) + 2. Using 52! as an example, the rule yields about 69 digits while the exact value has 68, so the estimate is very close for ordinary uses; when dealing with enormous numbers, being off by a digit or two is usually acceptable. The piece then develops the mathematical justification behind that convenient formula.

The justification starts with the Gamma function Γ(n) = ∫0^∞ x^{n-1} e^{-x} dx, whose recurrence Γ(n+1) = n Γ(n) and value Γ(1)=1 reproduce factorials on the integers. Rewriting n! as Γ(n+1), changing variables x = n y, and applying Laplace’s method to the integral with f(y)=ln y − y (maximum at y=1) produces Stirling’s approximation n! ≈ √(2πn)(n/e)^n. Taking base‑10 logarithms of Stirling yields the digits formula log10(√(2πn)) + n·log10(n/e); the first term is small but contributes about two extra digits for n up to roughly 1,600. Practical mental-log tips (e.g., log10 2 ≈ 0.3, log10 3 ≈ 0.5) make the estimate easy to use without a calculator.