How to Take the Factorial of Any Number

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.

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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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:

#SoME2 #Mathematics #Education

[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.


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.

[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.”

[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.

[theauthor]

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!

[theothetorch]

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

[ethandennis]

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

[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!

[Ratigan]

10:37 Okay, that caught me off guard lmao

[adityasriram]

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

[efedevirgen]

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

[andregonzales]

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

[arctic]

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!

[MNGN]

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! 😍

[aradziv]

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

[crispywhiskers]

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

[eriktempelman]

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!

[Enko]

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 YouTube that I´ve seen about the derivation of the gamma function.

Felicidades amigo :)

[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!

[rmw]

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