How to Take the Factorial of Any Number
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- Опубліковано 7 чер 2024
- In this video, I walk through the derivation of an extension of the factorial function that works for any number: fractional, irrational, and even complex! This turns out to be a very important function, known as the gamma function, which has many surprising connections, one of which I explore in the last chapter of the video.
The animations in this video were made with Manim, an open-source Python library for making math animations, originally created by 3Blue1Brown. www.manim.community/
My Previous video: Extending the Harmonic Numbers to the Reals: • Extending the Harmonic...
Chapters:
0:00 Introduction
1:38 A few Disclaimers
3:58 The Recursive Formula
6:50 The Super Recursive Formula
8:45 A minor setback
10:28 Logarithms
15:21 Deriving the Solution
19:26 Our Constraints
20:25 History and Conventions
22:16 The Miracle
25:44 The End
I went for some more relaxing background music this time. Hopefully it doesn't put you to sleep!
Creative Commons music used in this video:
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Enchanted Journey by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 license. creativecommons.org/licenses/...
Source: incompetech.com/music/royalty-...
Artist: incompetech.com/
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Fluidscape by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 license. creativecommons.org/licenses/...
Source: incompetech.com/music/royalty-...
Artist: incompetech.com/
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Kevin MacLeod is a life-saver!
Other music used in this video:
- River by HarumachiMusic
- Night Music by Kevin MacLeod
- Moon Men by Jake Chudnow
And a couple of my own songs:
- / thanks-for-watching
- / the-fog
#SoME2 #Mathematics #Education
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.
Your videos are top-tier! Keep on doing what you're doing because whatever it is that you are doing is awesome!
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^^]
This video is very well done!
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.
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
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.”
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 ua-cam.com/video/_lb1AxwXLaM/v-deo.html) would lend itself to this approach, but I think you'd want something that goes from Reals to Reals.
@@ShankarSivarajan The Fibonacci numbers could be nice with Binet's Formula.
@@ShankarSivarajan can't u just use roots? Since they're the same as rational exponents
@@TheBlindfischLP oh what is that
@@TheBlindfischLP that sounds interesting. Lemme guess, does that formula contain all the Fibonacci numbers, and then the line approaches x times phi?
10:37 Okay, that caught me off guard lmao
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!
Thanks for shocking me as well
Maybe the animations take a long time
Bro has 3 videos wtf
10:36 made me laugh out loud. I love the Vsauce channel.
Right?!?? Me too. 😂
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
I second that. Also, being transparent when assumptions were made make this video even more valuable. I liked it a lot too.
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! :)
why i see so much of my clones?????????????????????????????????
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.
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
@@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.
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
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.
@@itellyouforfree7238 damn my man calm down
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!
Your submission was amazing!
0:14 "Plugging in different functions in a graphing calculator is a weird pastime"
*You know I'm something of a mathematician myself.*
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! 😍
Wow
So what did you choose to pursue specifically? And how's it going? And how r u doing
@@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! 🤩
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!
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
18:00 i dont even understand anything anymore im here for the animation ASMR
I lost him at 11:58 but still watched till the last. Dunno why, may be maths asmr🙃
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
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 UA-cam that I´ve seen about the derivation of the gamma function.
Felicidades amigo :)
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!
I really enjoy how you're so rigorous and show all subjective assertions
Stunning video. It will take me days, if not weeks, to recreate the math presented here, step by step. Thank you for posting!
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
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.
Really cool stuff and the connection with the previous video is just amazing.
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.
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!
What a great content! Dude, do not stop. Making math videos is absolutely your cup of tea
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.
this video is amazing man!
always nice to see math presented in such a neat way
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.
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
this is amazing! everything was so neatly and beautifully explained, especially with a subject where it's very easy to get lost in all the math. the music choices really great and added a lot to the video. thank you !
The taxicab running along the bottom when 1729 is mentioned at 20:37, chef's kiss! Overall, great video, keep 'em coming :)
Your videos are so relaxing and entertaining! Love rewatching them for some chill time! Keep up the good work!
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!
The Vsauce music-
The comment I was looking for. Fit in flawlessly and made me feel so at home... or did it? 🎶
@@colinsaska3467It definitely makes you feel at home, I know your address
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!
Man this video is amazing. The "prequel" was also quite interesting but the connection of the factorial with the harmonic series kinda blew my mind. I can't wait for the next video.
Great explanation! Definitely one of my favorite explanations in UA-cam. This video has some serious quality and I’m getting blue brown vibes form it too!
Great video! Fantastic animations. Thanks for all your effort. 👏👏👏
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!
What a truly awesome twist. Great explanation and pacing, too.
Finally a great video explaining not only the factorials many forms, but where the Euler-Mascheroni constant comes from.
Great video all round, definitely deserves a sub ❤
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.
Very cool, you made it seem almost obvious why factorials are extended the way they are!
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!
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.
A true challenger to 3Blue1Brown
I actually forgot I'd subscribed to you, but UA-cam 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.
This is the first video from your channel that was recommended to me, and I find it quite interesting and helpful. I like the recursive definition of factorial.
Never seen such a natural motivation for the gamma function. Love it!
There was nothing new here for me, but the concise line of reasoning and the editing is amazingly good. Thanks a lot
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
@@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)!
@@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
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.
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!
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.
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.
I lack words to express how great your video is. Both musically and mathematically... Thank you for this treat.
Such a great video. It's really cool that you first look at the differences of near-infinite terms of the function, to derive an infinite-series expression at finite values. Also, it just hit me that whenever a function is represented by an integral expression, that's basically an infinite series expansion in the real numbers.
Much awesome! I was genuinely looking for this exact kind of video at some point after watching yours on the harmonic numbers, and however disappointing didn't find any. So now that you made it into reality I'm super happy.
“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"
This is B.S.
It’s a joke
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!
Saw this after 3blue1brown's recommandation, very neat video! Great explainations, nice animations (that reminds 3blue1browns' style) and gives a cool new perspective of the gamma function!
I'll sure check out your other videos, this is quality content!
Can you make a video deriving Laplace’s Transform? Loved your explanation!
why is this in my recommended i literally have never watched anything about math before
You know you want it
If you don't come to maths, maths comes to you.
Wow!!!! First time I see this one full length !!! This is soo amazing , I'm so grateful for this piece of art. Thank you so much
this is one of the most high quality things I've ever seen. thank you. mind blown multiple times.
The vsauce music caught me off guard
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?
An astonishingly good video! I would put this on par with some of the best mathematics channels on UA-cam at this time. Nice work!
It is really sad to see that so much work was put in to such an amazing video, and so less have seen it!
This video has once again sparked up my love for graphs Thank You!
the visauce reference on 10:30 made me laugh
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.
Fantastic video. This was beautiful, and the introduction of the harmonic series at the end was shocking.
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!
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!
didnt expect to see you here (im logeton from frhd if you remember, i dont play that game anymore lol)
@@logestt didn't expect to see either of you here
@@1s3k3b5 lmao
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!
@@LinesThatConnect perhaps you could consider adding explanations in the closed captions
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
I'll let you know if I find out lol
I literally understand nothing but I can appreciate the amount of work put in. Nice job!
The animations are spot on. So satisfying to watch. Thanks.
That was a fantastic Vsauce "or is it" with the music
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...
Phenomenal video, had never seen this piece of math before and here it was presented so beautifully
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.
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
It's called fractional derivatives. I still do not know what uses a half derivative has other than mental masturbation.
some quantum fields or electricity related stuff use fractional derivatives
how dare you not call the oily macaroni constant by its true nsme
This is just an amazing video. Very well explained and a real joy to watch. Thanks very much!
Seriously a banger of a video. Currently in a stat mech course which has gamma functions and factorials all over the place so I'll definitely be looking at that math a bit differently from now on.
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.
"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.
@@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?
@@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.
Hey, Vsauce
Absolutely beautiful! I can't believe I just found your channel - as a video creator myself, I understand how much time this must have taken. Liked and subscribed 💛
This is genuinely such a good math channel
Truly an incredible channel. I am very uncomfortable while using limits and you made it muchhhhh easier to understand. Wall done. You are brutally underrated. You deserve my subscribe
this is beautiful, please don't stop making math videos.
You make it so intuitive. This is the reason why SoME exists. For creators to do exactly this. Thank you. Thank you. Thank you.
I love this. I will definitely continue watching if you continue to make these videos ☺️
bro your video is just so clean amazing !
This is incredibly delightful! Excellent presentation!
Absolutely brilliant and beautiful demonstration!
great job man! you are definately going to get a lot of subscribers
Such an awesome, well explained video! thank you for your hard work!!
This is the best video about Gamma function I've ever seen, thanks very much!
This a slick derivation of an important formula, and also good publicity for Manim!
this is really cool, keep making stuff!!
Please keep making more videos! They're so good!!!