Ask Uncle Colin: What's 100!?

Dear Uncle Colin,

How would I work out $\sqrt{\ln (100!)}$ in my head?

- Some Tricks I’d Really Like In Number Games

Hi, STIRLING, and thanks for your message!

I don’t know how you’d do it, but I know how the Mathematical Ninja would!

Stirling’s Approximation ¹ says that $\ln (n!)\approx n\ln (n)-n+\frac{1}{2}\ln (2\pi n)$.

The Mathematical Ninja would expect you to know that $\ln (100)\approx 4.6$ and that $200\pi \approx 628$, the square root of which is a shade over 25. (25.07 or so, but that’s more accurate than we want here.)

So we have $100\ln (100)-100+\frac{1}{2}\ln (200\pi )\approx 460-100+\ln (25)$, which is somewhere in the region of 363.

The square root of that is a little over 19, probably in the region of 19.1.

(According to this calculator - if the Mathematical Ninja asks, you saw nothing, understood? - the correct answer is around 19.07. I like to think they’d be impressed!)

Hope that helps!

- Uncle Colin

* Edited 2018-02-24 to fix LaTeX in the title. Thanks, @christianp!

Footnotes:

1. Hey! That’s your name!