How Big Are Factorials? A Simple Estimate for 52!
Eli Bendersky explores how to estimate the number of digits in a factorial without a calculator. Using Stirling's approximation, he shows that 52! has about 68 digits, and derives the formula from the Gamma function and Laplace's method. The post includes a practical approximation and the mathematical background, making it accessible for curious readers.
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.
- svobodamartin
My favorite one is with the 52! seconds:
Start a timer that will count down the number of seconds from 52! to 0.
Then walk around the Earth’s equator with one step every billion years.
Then, after you make your way around the earth equator (by taking 1 step every billion of years), you take one drop of water out of the Pacific Ocean.
Then, you repeat the process of walking around the equator, and everytime you walk around, you keep draining one singular drop of water.
After the ocean is fully drained, you refill the ocean and put a piece of paper underneath you.
Now, you once again repeat this process of walking, draining, and placing papers.
After your stack of papers has reached the Sun, you repeat another 1000 times.
After all this, you have completed just about a third of the timer.
- smcin
This is restating Stirling's approximation, which has been known for three centuries (1730, de Moivre 1721).
- andrewla
This brings to mind the analysis in Bender & Orszag; they approach this through difference equations (a bit of a lost art in formal mathematics; very 19th-century feel) rather than integration.
Instead of introducing the gamma function, they instead start from the observation that log(F_n) - log(F_n-1) = log(n), so treating this difference as analogous to integration, it says that F_n ~= nlogn + n as the leading asymptotic behavior. This is clear just by substitution and algebra; no calculus necessary (though it helps to "know the answer beforehand").
From there you can treat the error term in this as F_n = n^n * e^n * E_n and plug that into the same relationship (F_n = n * F_n-1) to derive what that error term looks like asymptotically, and end up in the same place that the integration on the OP leads to.
- ninju
The author's casual mention of 52! at the opening of the article triggered an OLD webpage that I saw many years ago
https://czep.net/weblog/52cards.html
Anyone know how to determine the age of this page (it's got be at least 20yrs old)
- Sharlin
A quick and dirty approximation of the number of digits in n! is n lg n, which approximates n! from above, via the inequality
1 * 2 * … * n ≤ n * … * n.
(This approximation should be familiar to many from an algorithmics class.)
For a tighter bound, use n lg n - n/2, or a better approximation of ln 10 in place of 1/2 if you wish. This comes from Stirling's approximation which notes that
ln n! = n ln n - n + O(ln n).