Estimating the Size of Huge Factorials
Factorials grow astonishingly fast. 52! is a number with 68 digits, and even rough estimates of its magnitude are surprisingly useful. You can estimate the number of digits in a factorial without a calculator using a straightforward approximation:
For 52!, the estimate works out to about 67 or 68 digits, which matches the exact value. When dealing with numbers this large, being off by a digit or two is usually irrelevant. The approximation is built on two pieces of mathematics worth understanding: the Gamma function and Stirling’s approximation.
The Gamma Function: Extending Factorials to Reals
The Gamma function is defined for real x as an integral that has no general analytic solution. However, it has a crucial recurrence property. By applying integration by parts, one can show:
This demonstrates that Γ(x+1) = x Γ(x). Combined with the special case Γ(1) = 1, an induction argument gives:
This means the Gamma function is an interpolation of the factorial over all positive reals. A plot of the Gamma function over a small range shows its rapid growth on a log scale:
Stirling’s Approximation
Stirling’s approximation is a well-known formula that estimates factorials with good accuracy even for small values:
Deriving it from the Gamma function starts with the integral definition. After a change of variables, the integrand takes a form suitable for Laplace’s method, which approximates integrals of the shape:
where is a large parameter and 
is twice-differentiable. Laplace’s method gives:
with being the global maximum of 
.
Applying this to the Gamma function involves setting f(t) = ln(t) - t. This function is twice differentiable with a global maximum at . Substituting the needed derivatives into Laplace’s approximation produces Stirling’s formula exactly.
Counting Digits from Stirling’s Formula
To estimate the number of digits in n!, take the base-10 logarithm of Stirling’s approximation:
The correction term in the logarithm is not multiplied by n itself, so its relative importance shrinks as n grows. Still, it contributes a few digits and should be kept for better accuracy.
