How big are factorials?

The other day, I found myself wondering how big 52! (52 factorial) is, and that led me to ponder how these could be estimated without a calculator or a computer.

It turns out there’s some fairly interesting math behind being able to estimate the size (number of digits) of a factorial reasonably accurately. This post will start by stating how to do the estimate, and if you’re curious you can read on for the math background.

Without further ado, the approximation is:

As an example, let’s use my original question, by estimating this for 52!

Well, 52 divided by e is... 20-ish? And \log_{10}(20) is about 1.3 [1] ; therefore our estimate comes out to:

The real answer is 68, so this is very close! In estimates like this - when you’re dealing with enormous numbers - being off by a couple of digits usually isn't a big deal.