Big Factorials - Numberphile

Large factorials and the use of Stirling's Approximation. Featuring Professor Ken McLaughlin.
More links & stuff in full description below ↓↓↓

We filmed this during his time at the Mathematical Sciences Research Institute.

NUMBERPHILE

Videos by Brady Haran

[_thank_you_]

on an unrelated note, Prof. McLaughlin's voice is one of the best in the game hands down. i want him to tell me everything will be okay while explaining combinatorics to my kids

[jeojavi]

3:18
"But you don't get it exactly right"
"That's exactly right"
Loved this

[Dysiode]

I love how clear Professor is, making sure to explain terms that may be unfamiliar and even just stating what he's going to do! "I'm going to rewrite that whole ratio, " makes it clear your brain doesn't have to be paying precise attention for new information. It would be an absolute joy to take a class with him

[jan_kulawa]

Back when I started following this channel, I struggled to follow Matt Parker's calculations, but was nonetheless interested in the results enough to pursue the subject further. Today, from the title alone, I correctly guessed the video was about Stirling's asymptotic approximation for factorials. Cheers from Brazil! And thanks for being a substantial (a mathematician would say, non-trivial) part of my life.

[younesabid]

- ...it's a bit of a shortcut but you don't get it exactly right!
- That's exactly right.

[patback]

Prof. McLaughlin: It's off by almost 2, that doesn't seem so close, right?

Me, an engineer: Seems close enough!

[PeterGaunt]

I love your knack of asking exactly the right questions of the people you meet.

[TheWaterMiners]

For those we are still confused about the part where their ratio gets closer to one while their difference increases, think about it like this: 1 and 2 have a ratio of 0.5 and a difference of one. 900 and 1000 have a ratio of 0.9, which is closer to 1, but a difference of 100, which is much larger.

[jonrutherford]

For me, as a non-mathematician, this was nothing short of thrilling. I had no idea there was a way apart from barebones multiplication to find the value of a factorial -- at least a very close approximation. To hear of its value to mathematicians was a bonus! Very well done, and many thanks.

[TateFM]

Professor McLaughlin does such a fantastic job breaking this down and making it really interesting for even just a layman (but appreciator) of math. His passion reminds me so much of a professor who taught me Calculus I in college too. The passion is infectious which is what makes this channel so special.

[PhilipSmolen]

I remember using Sterling's approximation in school. It was nice to be able to use Big O notation with a factorial. And we used to prove that a sorting algorithm couldn't do better than O(n · log(n)).

[jaerik]

Professor McLaughlin is a great educator. More of him please!

[aperson]

For physics the number of particles in a typical system is on the order 10^22 so this approximation is incredibly accurate. Also taking the log makes it much nicer to work with.

[markday]

I really like the way the professor explains things. I'd like to see more videos with him.

[gsurfer]

As a computational chemist, I can confirm Stirling's approximation is very handy. However, when manipulating the mathematical formulae for thermodynamics we stick to the factorial form as it often simplifies to N+1 etc.

[emmanuelagudo]

Something magical about worded problem solving questions that involve Factorials. The logic is so fluid, it is so beautiful. Imagine riding a super fast elevator for quite odd structures, or yet kind of navigating a complex query but then it is arranged in a certain fashion that can be ultimately deduce from Fibonacci sequence. Amazing! thank you!

[Kaepsele]

One instance where this pops up in physics, even at the Bachelors level, is in the calculation of entropy of a system. Entropy is a measure of how many ways a system can be in terms of the positions and momenta of its particles and still be the same volume, temperature and pressure. An important thing is that if you exchange the position and momentum of two indistinguishable particles, you do not actually get a new state. So to prevent that you count the same state multiple times, you have to divide by the number of ways you can permute the particles of the system, which is N!. For a macroscopic system, N is of the order of avocados constant, so around 10^24. You can imagine why the approximation is kinda useful here, especially since you have to take the logarithm to get to the entropy.

[cariboubearmalachy]

"You don't get it exactly right"
"That's exactly right."

[samstarlight]

3:20
"You dont get it exactly right"
"That's exactly right"
Got a laugh out of me

[jacemandt]

In other words: the limit of their ratio is 1, but the limit of their difference isn't 0. The limit of the difference isn't any number, because the difference keeps growing as N gets very big.

Taking a difference vs. taking a ratio both measure "sameness", but in different ways.