How to Take the Factorial of Any Number

[@LinesThatConnect]

Thanks for watching everyone! I'm overwhelmed by the response to this video - 100k views is more than I dared to hope for! I've got a couple quick clarifications: 5:26 - This cannot hold for _every_ x - only for values where the domain of the function allows the formula to make sense. It turns out that this excludes non-positive integers. Some people rightly pointed out that the recursive formula seems to imply that 0! = 0 * (-1)! = 0., but this assumes that (-1)! exists and is finite. In fact it was that exact formula that led to the conclusion that there must be an asymptote at -1. (6:33) 9:08 - We might guess that we can make the function behave better by taking its reciprocal, which would make it flatten out and rapidly approach 0. This is actually one of the first things I tried, but unfortunately it doesn't work. It would work the function approached any value _except_ for 0, but since the factorials are all about multiplication, and since 0 * anything = 0, we don't get any new information. 0:04 - So I wasn't actually in middle school. In my memory I was in the 8th grade, but I checked the Wayback Machine, and the version of the site I remember didn't exist until my first year of high school. 21:27 - The proof that I have the easiest time understanding is "Proof 2" on this ProofWiki page: proofwiki.org/wiki/Integral_Form_of_Gamma_Function_equivalent_to_Euler_Form Another note - This also works for complex numbers! You can just plug a complex number in for x, and it will converge. I made sure I never mentioned real numbers and instead said "any number" or "non-integer", so that I didn't accidentally exclude complex numbers.

[@Jacob.Peyser]

Your videos are top-tier! Keep on doing what you're doing because whatever it is that you are doing is awesome!

[@Memories_broken_]

As a learning student,I find this video really informative, thank you :) [also ,i suggest you to pin your comment as it might go unnoticed with several other comments^^]

[@fatitankeris6327]

This video is very well done!

[@michaelwaters1358]

at this point, you have a duty to the math world to keep producing videos. These two have been fantastic, I hope you can release another one sooner than 1 year from now.

[@mihirramaswamy1313]

0:45 Hey small mistake in the video. The factorial is defined as the product of all "natural" numbers up to that number, not "whole." Great video anyways. Congrats on getting featured on 3B1B

[@ToastyEggs]

You could make a series out of this where you explain how extensions of different discrete functions are derived! You could call it “Points that connect.”

[@ShankarSivarajan]

That'd be neat, but what other such function can you think of? The gamma function is the only one that comes to mind. If you pick, say, 2^x, you run into a problem. Let's say we understand exponentiation as repeated multiplication, and want to extend that from the natural numbers to the Reals. 2^1 = 2, 2^2 = 4 …. Declare by fiat 2^(x + 1) = 2^x × 2. ⇒ 2^(x - 1) = 2^x/2. ⇒ 2^0 = 1, 2^-1 = 1/2, etc. Great, but non-integers are what we're here for. So following the steps in this video, we get 2^x = e^(x ln 2). This is a nice result, but the problem is that this is circular as a definition of exponentiation. The best way to define it would be as its Taylor series expansion, but that's nowhere near as interesting. Maybe something like x^x (see kzbin.info/www/bejne/lZ3FYnSurL1_l68) would lend itself to this approach, but I think you'd want something that goes from Reals to Reals.

[@Henriiyy]

@@ShankarSivarajan The Fibonacci numbers could be nice with Binet's Formula.

[@mihailmilev9909]

@@ShankarSivarajan can't u just use roots? Since they're the same as rational exponents

[@mihailmilev9909]

@@Henriiyy oh what is that

[@mihailmilev9909]

@@Henriiyy that sounds interesting. Lemme guess, does that formula contain all the Fibonacci numbers, and then the line approaches x times phi?

[@theauthor8882]

I was shocked to see that you only have two videos. The production of this and the explanation were both fantastic. Keep it up, I'll be there to watch anything else you put out!

[@adamantii]

Thanks for shocking me as well

[@BambinaSaldana]

Maybe the animations take a long time

[@R4y-06]

Bro has 3 videos wtf

[@ethandennis368]

This was one of the most well put together math videos I have ever seen. Please do not stop making content because you truly have incredible potential as a math explainer

[@brunesi]

I second that. Also, being transparent when assumptions were made make this video even more valuable. I liked it a lot too.

[@tommasobonaccorsi8935]

I totally agree! Everything was so well explained and extremely clear, as a 12th grade student I understood almost everything. Keep up with the work! :)

[@Normal_user61]

why i see so much of my clones?????????????????????????????????

[@programablenuance]

A true challenger to 3Blue1Brown

[@LordBrainz]

Only this guy actually gives formulas for when we already have the intuition built, y'know, for people that can think better with written stuff

[@FundamSrijan]

True , but lol , he uses the manim library of 3b1b 😂😂😂

[@Carl-Gauss]

2:30 This so true! Lectures in university are usually about proving as many theorems, lemmas and formulas as possible during certain period despite the fact that it completely misses the point of sharing a proof with students. The fact itself that you’d shown a certain proof to a student doesn’t matter, what matters is student understanding why formula or theorem is the way it is and gaining additional intuition about the topic.

[@kylaxial]

yeah... I first experienced this with the quadratic formula they gave to me. but at least they told me about the similarities with the vertex finding equation... which they also just gave to me

[@casualoutlaw540]

​@@kylaxial Most schools usually force you to factorize and complete the square before the quadratic formula, so it's not as magical as the gamma function which is given to you and then you maybe see a proof that it works using integration by parts, and that's about it.

[@katakouzina]

a lot of theorems lemmas blabla, do not have a "logical" explanation. it is what it is, because the proof (lines of implies) is true. or if there is some kind of eplanation it can only be understood from the clever ones

[@itellyouforfree7238]

That's because understanding is the student's job. Given the amount of topics that have to be covered in a fixed amount of time, there is no other way. The teacher gives an explanation (proves a theorem, lemma, etc.) then the students can go home and think about it for as long as they wish. If they don't do that it's because they are lazy. It's unrealistic to believe that university lectures can be so complete to satisfy every student and have each of them completely understand everything on the spot. This is not how it's meant to be. If a Calculus 1 course were to be organized such that every student completely understand everything in class, if would take ~1000 hours in total (and some students won't even get it after 5000 hours...) instead of ~100. Stop bullshitting university: it's the most efficient way to learn a significan amount of knowledge, much more efficient than youtube or crappy paid courses.

[@ShanBojack]

@@itellyouforfree7238 damn my man calm down

[@soapycanthandle]

I call uppercase sigma bigma

[@puljatko]

You are sick man…

[@PriyanshSuthar-sffmp]

Too clever

[@SpeedyMcMichael]

nuh uh

[@Snidbert]

bigma balls

[@FlamingSpiral20]

Bigma

[@morphocular]

This was very well done! I actually used the gamma function in my own SoME2 submission and wished I could have included a derivation of it, at least as a side resource. But now I can just point to this video!

[@tommero6584]

Your submission was amazing!

[@M1551NGN0]

0:14 "Plugging in different functions in a graphing calculator is a weird pastime" *You know I'm something of a mathematician myself.*

[@enbyarchmage]

Dude, your channel is out of this world! I already considered this video one of the best math-related ones I've seen in a long while, several mitutes befor its end. However, when I saw the definition of gamma appear so naturally from the derivative of x!, I literally started screaming "It's gamma! GAMMA!" before the limit even appeared. This video reminded me of how much I - who dropped of a STEM major in favor of a Humanites one - still love math, and why. Thank you so, so, SO much! 😍

[@mihailmilev9909]

Wow

[@mihailmilev9909]

So what did you choose to pursue specifically? And how's it going? And how r u doing

[@enbyarchmage]

@@mihailmilev9909 That was SO kind/cute of you to ask! 😁 I'm a History undergrad now. In spite of all of the stress (LOTS of dense, often boring stuff to read), I feel like I'm where I was always meant to be. Life was never so meaningful! 🤩

[@arctic7526]

Out of all the submissions for SoME2, I can say that this one is definitely my favorite. It was easy to follow along and had amazing explanations. Very cool proof too!

[@Rot8erConeX]

4:18 I *love* the bounce you give the ends of the function when you condense it. It's a little tactile decision that shows you that a *person* made the video in order to show others something cool, rather than a textbook company making a video because they want all teachers teaching the same thing.

[@efedevirgen5835]

18:00 i dont even understand anything anymore im here for the animation ASMR

[@prince-hb8qk]

I lost him at 11:58 but still watched till the last. Dunno why, may be maths asmr🙃

[@aradziv89]

Man, this is the type of video I like most. Simple enough to appeal to inexperienced viewers, yet doesn't linger on the simple and teaches me something new... far enough than what I already know but touching on the familiar... great explanation, and great visuals! Knows when something is irrelevant, but throws it in for the curius. Bravo man

[@mattgsm]

I really enjoy how you're so rigorous and show all subjective assertions

[@eriktempelman2097]

Great stuff! I'm in design engineering and there we often use the "forget-me-nots" for beam deflection in bending. Few around me know the beautiful maths behind it. And if you know that, you appreciate those formulas so much more!

[@theothetorch8016]

10:36 made me laugh out loud. I love the Vsauce channel.

[@DavidLindes]

Right?!?? Me too. 😂

[@reamartin6458]

“I can show that Mascheroni is actually an imaginary number masquerading as an irrational, I have a proof of this theorem, but there is not enough space in this margin"

[@azzteke]

This is B.S.

[@whydoineedname4932]

I'll get you some space

[@reamartin6458]

💗

[@rmw6151]

Stunning video. It will take me days, if not weeks, to recreate the math presented here, step by step. Thank you for posting!

[@aditya95sriram]

The taxicab running along the bottom when 1729 is mentioned at 20:37, chef's kiss! Overall, great video, keep 'em coming :)

[@cancercurry7278]

I'm just a year 8 student, but this video is just amazing, I've probably watched it 20 times by now and I still enjoy it because it turns the topic of something as simple to understand such as factorials in a more complex topic, but making the explanations simple enough to be understood by those who are inexperienced by touching on a few of the finer details so that it's understandable. Thanks for the great content. I hope to see more videos produced by you in my recommended.

[@Ratigan2]

10:37 Okay, that caught me off guard lmao

[@elliasortega4757]

Hey vsauce here!

[@HenryY8]

Or is it?

[@woo-wer]

Or did it?

[@baptiste5216]

Really cool stuff and the connection with the previous video is just amazing.

[@5ucur]

Found you in one of my treks down the maths rabbit hole. You immediately deserved a subscription! :D You're one of those people who make maths fun again :D

[@Enko97]

I study math at college and well I gotta say that I LOVED the two videos on your channel, so I subscribed right away. Keep it up pal, you´re doing an amazing job. I really liked your content. This video without exaggeration is the best video out there on KZbin that I´ve seen about the derivation of the gamma function. Felicidades amigo :)

[@richardconlin707]

The fact that this is free to watch is ridiculous. Insanely high quality.

[@mebamme]

I actually forgot I'd subscribed to you, but KZbin went and recommended me this video 30 seconds after you uploaded it. (: You're on the way to being one of my favorite math channels! Original topics, and great presentation.

[@JobBouwman]

There was nothing new here for me, but the concise line of reasoning and the editing is amazingly good. Thanks a lot

[@iwunderful3117]

Maybe you can explain to me why (-1)! Inevitably has you dividing by zero when plugged into the given formula. Because it seems to me that he just replaced -1 with 0 and divided by that

[@shophaune2298]

​@@iwunderful3117 I'm late and not who you replied to but: x! = (x-1)! * x Let x = 0 0! = (-1)! * 0 1 = (-1)! * 0 1/0 = (-1)!

[@dipankarhowladar9949]

​@@iwunderful3117yeah let gamma x+1=(x)gamma(x) from here (x)!=x(x-1)! now putting 0 in x (0)!=0(-1)! i.e 1/0=(-1)! that tends to me infinity

[@Krunschy]

Can't overstate how much I appreciate this video. When I first got to know the gamma function I was in the same boat as you were, desperately wanting to know how one would ever think that up. I got a bit into it, but eventually it just became too much work for me. But I never stopped wondering. Being able to finally achieve an understanding thanks to such a great presentation... it is almost cathartic.

[@crispywhiskers9365]

this video was simply amazing! the humor, the math and the understanding, everything was it's absolute forefront! looking forward to more of what this channel has to offer :D

[@eclipse6859]

That was a fantastic Vsauce "or is it" with the music

[@cmilkau]

Never seen such a natural motivation for the gamma function. Love it!

[@KristOFF-T]

These videos are AMAZING! Captions, animations, explainations, sound quality, etc. all 10/10. I can imagine how many time and hard work you're putting in these. Can't wait for the next one.

[@bon12121]

You make it so intuitive. This is the reason why SoME exists. For creators to do exactly this. Thank you. Thank you. Thank you.

[@2003ToyotaVitz]

why is this in my recommended i literally have never watched anything about math before

[@awesomethegreatamazing2651]

You know you want it

[@haabyalexis]

If you don't come to maths, maths comes to you.

[@klembokable]

Maybe it's time?

[@luisa.machado6595]

This is very easy to understand given a decent background in pre-college math A suggestion: when going from one step to another, please keep the previous step in sight, and give us about 2 or 3 seconds to take it in

[@a52productions]

I did the exact same thing in middle school (or maybe high school, I don't remember). I think Desmos was a big part of making me interested in math, as well as training my visual intuition.

[@soupy5890]

As a self-teaching highschool student, I really appreciate these presentations of wicked and mysterious maths that both presents ideas and some of the actual working-through-it

[@diplomaticfish]

Very cool, you made it seem almost obvious why factorials are extended the way they are!

[@aayushbajaj2260]

this is one of the most high quality things I've ever seen. thank you. mind blown multiple times.

[@jgperes]

this video is amazing man! always nice to see math presented in such a neat way

[@JCake]

Hey just to make you aware, I find videos like these super fascinating, but I always struggle to follow the plot. But your video was so easy to follow and rewarding to watch, I just had to mention how great I found it. 20/10

[@jakebruner2719]

Loved this video!!! Also, as a fellow Manim-learner, you’ve really gone above and beyond with this. I can tell you’ve spent hours upon hours mastering it; no easy feat!

[@themandel2017]

What a legend to explain the gamma function understandably to many people. It feels something like kindergarten now for I didn't think how to derive it.

[@ery5757]

Another great video! I am just so used to using the Gamme function instead of the factorial and I never wondered why that was allowed. But it was great to see the derivation!

[@fuzionfury7119]

I literally understand nothing but I can appreciate the amount of work put in. Nice job!

[@AnimaStation-F0RG3T]

10:38 Hey, Lines That Connect here!

[@EricLeePiano]

8:34 moments like this where you explain little things that most teachers don't explain, makes a huge difference to me, congrats u earned a sub

[@danilofigueiredo4532]

Thank you for making and sharing such an amazing video with your brilliant explanation! I just now have become aware of this python library created by 3Blue1Brown that you used for the animations. I will learn more about that. I see your inspirations, and also liked that @Vsauce vibe at 10:30... Your content is indescribably necessary, sir.

[@HazhMcMoor]

When i first see the title i thought this will be just another gamma function video so i skip it. But when this wins the entire some2 i have to look at this video again and turns out it's much better than I ever expected. You really deserve the win.

[@wlan246]

10:22 "...or is it?!" Brilliant VSauce reference. "Michael here!" Laughed out loud.

[@raph2550]

I have been pretty invested from the beginning of the video, but when you introduced the logarithms, I had to stop the video and to it by myself. You are doing a great job!

[@nothingtoseehere8063]

The vsauce music caught me off guard

[@minato232]

This is the best video about Gamma function I've ever seen, thanks very much!

[@andregonzales8630]

lmao the “sit back and enjoy the animations” had me 😂

[@kaifenjoyer]

What a great content! Dude, do not stop. Making math videos is absolutely your cup of tea

[@pianoconlatte]

I lack words to express how great your video is. Both musically and mathematically... Thank you for this treat.

[@terdragontra8900]

22:28 To anyone who feels the logarithmic derivative to be “arbitrary”, note that it is the same as f’(x)/f(x). In other words, instead of giving an infinitesimal absolute rate of change, it’s the infinitesimal RELATIVE rate of change, the rate of growth of f as a fraction of the current value.

[@brown56765]

A highschool friend and I thought it would be fun to figure out if you could find the "half derivative" of a function (take the half derivative twice and you get the derivative), and our Calculus teacher agreed to give us some extra credit if we compiled our findings into a small paper. We quickly fell into the fractional calculus rabbit hole, and the Gamma function quickly became our best friend Good times XD

[@StevenSiew2]

It's called fractional derivatives. I still do not know what uses a half derivative has other than mental masturbation.

[@Shyguy5104]

some quantum fields or electricity related stuff use fractional derivatives

[@lamshywy8920]

"The factorials are defined as product and that's that...or is it🤨🧐" I love that😅

[@spacelightning6587]

Ooh, that was an excellent video! I haven't seen this version before; I only knew about the gamma function. As for 0! = 1, there is another fun way that sort of relates back to the "number of ways to rearrange a set" definition we are often first presented with. The symmetric group on N objects is defined as the number of bijective self-maps for a set of size N under function composition. Since that is basically the fancy-pants algebra way to define permutations, it is not surprising that there are N! such functions. Well, let's think about our good friend the empty set, which is the only set of size 0. If we look at all the key bits in defining a function (left-total, univalent), we vacuously satisfy them all if we consider a function from the empty set to itself (this is often called the empty function). It is the identity function on the empty set and is the only bijective self-map (easy exercise) for the empty set, so the symmetric group on 0 objects had exactly 1 element. Hence 0! = 1.

[@suicraft8395]

What a show, i have seen a lot of math videos related with this topic, but yours is kinda special becausr it made rhe connection between a lot of thing i have seen. This video is not just a divulgation video, is a piece of art.

[@TheZorbeck]

Very good stuff. But I still cannot grasp the fact that the difference between two diverging series (Hn and ln(N) )can converge, into gamma in this case (the Euler Mascheroni constant). This is just blowing my mind, it is counter-intuitive...

[@unneccry2222]

well another half a year has passed when are we getting another awsome video?

[@estebanvasquez-giraldo5770]

This is a great video, thank you sooo much! I have also thought a lot about the definition of the gamma function and I didn't know this infinite product representation, just the integral form you showed by Euler, it would be great if you could make a video explaining the connection between those 2. I learnt a lot from this video, again, thanks!

[@IamYee]

Wow... I was totally impressed by how you derive this beautiful factorial formula. It was one of the most satisfying math videos in YT! I'm looking forward to your future works!

[@SzanyiAtti]

Please do make more videos if your time allows, I have really enjoyed them so far, especially because they had been about questions I often wondered about, but never took the time to dive deeper into them.

[@drjjpdc]

I enjoy math videos even if I have to work to grasp them. I did have a year and a half of college calculus, but that was 50 years ago and my brain is a bit slower now so I am thankful for the rewind button on my laptop. Keep it up.

[@stevenschilizzi4104]

Great video! Fantastic animations. Thanks for all your effort. 👏👏👏

[@bernatjordacarbonell6483]

The VSAUCE reference was such a great, little detail.... Great video by the way, it seems understandable for highschoolers and I (graduated mathematician) enjoyed it A LOT. I will steal some of your didactic methods

[@rube9169]

Love your derivations. This was a bit hard to follow. Maybe include relevant definitions you found earlier on screen when using them to further derive the solution... if that makes sense lol. Just as mind-blowing as the last. Can't wait to see more! I remember almost deriving the general solution for some formula while trying to solve a difficult problem in an ECE class. My method was close, but I hit a point where I couldn't go on. It was still super satisfying to understand the formula a bit deeper by trying to get more general solutions. You take that to such a higher level though and I love it!

[@logestt]

didnt expect to see you here (im logeton from frhd if you remember, i dont play that game anymore lol)

[@1s3k3b5]

@@logestt didn't expect to see either of you here

[@logestt]

@@1s3k3b5 lmao

[@LinesThatConnect]

I did what I though would get the fanciest animations, which isn't quite the best priority in hindsight. I'll keep this in mind for future videos!

[@pricklesthecactus6183]

@@LinesThatConnect perhaps you could consider adding explanations in the closed captions

[@yesno7889]

Truly appreciate the cultured reference to Vsauce at 10:36

[@JLM-iq7gz]

Thank you for this very interesting video. The characterization of the gamma function is called Bohr-Mollerup's theorem. A far-reaching generalization of this theorem was recently published in the OA book "A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions". What about making a video about this generalization?

[@jakebruner2719]

And, although some might complain about the pacing, I loved it. It was just right where I could gut check most of the calculations and understand what was going on without making it too laborious or making it too quick to follow!

[@richardpike8748]

Exactly same here

[@richardpike8748]

Though this agreement between ourselves probably does not hold for everyone, as others' intuition or depth of knowledge in mathematics is not all the same

[@octopus1003]

That was absolutely amazing! I didn't understand everything, since I'm a highschool student, but it is extremely interesting (probably I will understand more if I watch it a few more times)! I wanted to point that out that not just te explanation was incredible but the animations looks so nice and your voice is so good to listen to that this video feels as a mathematical piece of art form a museum! I'm looking forward to see more video from you!

[@Codefan321]

I watched this video and understood EVERYTHING. You have explained this perfectly, I have liked this video and subscribed. You have done an amazing job and have satisfied my curiosity for how this works. Thank you!

[@oricraft6747]

The Vsauce music-

[@colinsaska3467]

The comment I was looking for. Fit in flawlessly and made me feel so at home... or did it? 🎶

[@SaagarNayak]

​@@colinsaska3467It definitely makes you feel at home, I know your address

[@Snowy_bearsss]

@@colinsaska3467 😃

[@Kowzorz]

"I hereby decree" is such a useful teaching tool/phrase.

[@element1192]

I noticed you drew the Hadamard gamma function at 3:10! What's the use of that particular function besides extending the factorials to the negative integers? I've been dying to know

[@LinesThatConnect]

I'll let you know if I find out lol

[@DrunkenUFOPilot]

This a slick derivation of an important formula, and also good publicity for Manim!

[@bagelnine9]

(0:04) Same.

[@govindagarwal3310]

this is beautiful, please don't stop making math videos.

[@FalcoGer]

if n! = (n-1)! * n, then obviously 0! is 0. 0! = (-1)! * 0. Since any number multiplied by 0 results in 0, 0! must be 0. given this 1! = (0)! * 1, must be 0, and so any number factorial must be 0. On the other hand 1! is defined as the product of all integer numbers from 1 to 1, which is obviously 1. The only reasonable conclusion to make here is that 0! is not defined, since that will cause a contradiction.

[@LinesThatConnect]

"0! = (-1)! * 0. Since any number multiplied by 0 results in 0, 0! must be 0." That would be true, except (-1)! is undefined, which I neglected to mention until 6:30. In fact, since there is a vertical asymptote at -1, we can think of (-1)! * 0 as infinity * 0, which is indeterminate, so the contradiction is avoided.

[@mahendrabende5458]

@@LinesThatConnect maybe i am missing something here , But upon putting this inderminate value of (-1)! We got back into the equation of 0!=(-1)!*0 We must conclude that 0! Does not exist as well?

[@FalcoGer]

@@LinesThatConnect at that point we have not yet defined what factorial means for any other number other than positive integers. We're trying to determine just that. Since you determine it from that recursive definition it is unreasonable to assume (-1)! is undefined, since that would lead to every factorial of a natural number to be undefined using that very same recursive definition. That is in contrast to the definition we started with. This contradiction leads to the only logical conclusion that the recursive definition doesn't work in all cases and we can't use it as a basis for a general definition. Even if we define (-1)! to be NaN, that leaves us with 0! = (-1)! * 0 = NaN, 1! = 0! * 1 = NaN, 2! = 1! * 2 = NaN, etc, using n! = (n-1)! * n, for n element of R Only if we add, by decree, that 0! = 1 and exempt 0! from the recursive function, things can make sense.

[@D7A1]

Amazing! Every bit of the video and of course the math. I feel you'll inspire a lot of people and your channel will be very popular. Keep going!

[@lucaswiese6]

0:10 what is that website (url)?

[@LinesThatConnect]

www.desmos.com/

[@bigbri64]

You had me at “the more you know...” Thank you for this journey!

[@JojoJere]

10:41 *Vsauce music*

[@TheBooker66]

Very good video, and very good channel overall. I watched this video for the first time over a year ago, and just came back for a second watch, after watching your video about the harmonic numbers. Will definitely go on to watch your other videos, and await new ones.

[@simply_paul]

What about Stirlings formula

[@LiftingPhilosophy]

YEAH WHAT ABOU5 STRILINGS FORMULA

[@yerivalpolanco1448]

Tier S video. Since this channel is pretty much unknown. I was really expecting to see a good yet not a great video. It was simply amazing.

[@theidioticbgilson1466]

how dare you not call the oily macaroni constant by its true nsme

[@roeyshapiro4878]

What a truly awesome twist. Great explanation and pacing, too.

[@vaakdemandante8772]

Hey, Vsauce

[@JasonKwann]

This is the first mathematical video I could fully enjoy watching it

[@derendohoda3891]

love the bit about gamma, really great video

[@mikehills7247]

WOW this is the most exciting video ive seen. I have been working with factorial of non integers for decades and am planning to submit an entry to some3 on a queuing theory i developed dealing with callers who abandon the queue before service. Ill be showing experimentally why the gamma function is a representation of real life

[@Redstonmaster]

My god what is this surpise crossover of videos!!! I felt chills on my back during the gamma appearance 😳

[@ilyasotnikov8229]

For the first time ever, I kind of understood what is going on on the screen, amazing video!

[@snabol]

I just seen the and came to like it and CAN'T BELIEVE THIS DOESN'T HAVE MILLIONS OF VIEWS

[@pelayomedina2174]

I literally saw the image and recognized the gamma function