Stirling's approximation says that for large n, n! ≈ √(2πn) · (n/e)ⁿ. The appearance of both π and e in a formula about counting permutations is striking. For n = 10 the error is under 1%. For n = 100 it is under 0.1%. The formula improves without bound as n grows.
Abraham de Moivre found in 1730 that n! ≈ C·√n·(n/e)ⁿ for some constant C. James Stirling identified C = √(2π) the same year. The √(2π) arises from the Gaussian integral: when deriving Stirling via the Gamma function, the integral ∫e^(-t²)dt = √π appears, carrying π into the formula.
The logarithmic form is used throughout physics: in statistical mechanics, Boltzmann's entropy formula S = k·ln(W) requires ln(N!) for huge N (moles of particles). Stirling gives ln(N!) ≈ N·ln(N) - N, making it tractable. The full asymptotic series adds corrections: n! = √(2πn)(n/e)ⁿ · exp(1/(12n) - 1/(360n³) + ⋯)
Plotted on a log scale, the exact factorial (blue) and the Stirling approximation (red dashed) are indistinguishable past n=5. The factorial grows super-exponentially, far faster than any exponential, and Stirling captures this growth precisely.