Ask Uncle Colin: What’s $\sqrt{100!}$?

Dear Uncle Colin,

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

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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 Approximation1 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!

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. Hey! That’s your name! []

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I teach in my home in Abbotsbury Road, Weymouth.

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