How big are factorials?

An exploration of how to estimate the number of digits in a factorial like 52! without a calculator, starting with a simple approximation formula and then diving into the underlying math. The derivation covers the Gamma function as a continuous interpolation of the factorial, its integration-by-parts recurrence relation, and how Stirling's approximation is derived from the Gamma function using Laplace's method. The post concludes by showing how to compute the number of digits of a factorial from Stirling's formula.

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