What Is The Factorial Of 1/2? SURPRISING (1/2)! = (√π)/2

[@MindYourDecisions]

We've come a long way! I posted this video exactly 5 years ago when I was just learning about animation. Now the video has 1 million views, and we're one of the best channels on KZbin. Thank you for believing in me, even when the channel was just starting!

[@RajaAnbazhagan]

I always enjoyed your videos. And i get the honest intentions of reviving this video. However, i was here for an explanation. Because i never knew that factorials are for rational numbers. But there isn't a lot of explanation. I would be so glad if you could re-do the video explaining the bits.

[@Joe_Payne]

Is there a function that lets you reverse it (i). Eg 6i = 3 , 24i = 4. I would love to know and it would be a great way to celebrate this video.

[@kirannambiar4983]

Your explanations are complicated

[@jatin1231]

I love your channel, and really appreciate your efforts. But one slight misunderstanding at 0:36 "product of all whole numbers upto n" will lead to 0. Please correct me if I missed something?

[@JordanMetroidManiac]

Jatin Agrawal I’m pretty sure zero is not a whole number.

[@yoyo-mv3wt]

π=3.14 ✖️ π=22/7 ✖️ π=((1/2)!)²×4 ✔️

[@highguardian13]

Yonas Gaming we need the big brain meme here

[@carterhoward2827]

Pi= ln(-1)/i

[@ffggddss]

@@felixjohanschistadjacobsen767 No, it *is* exact: [(½)!]²·4 = [(-½)!]² = π Fred

[@theeraofmywords9205]

Pi is pi

[@ranveergupta4]

HAHAHA........WELL PI CAN BE DEFINED IN MANY WAYS ,BUT THE CORRECT ONE IS - π=diameter/circumference .., BUT IF WE TALK ABOUT THE VALUE OF PI IT IS REALLY HILARIOUSLY PREDEFINED EVERY TIME .......

[@fdnt7_]

So, 0! is greater than 0.5!. Interesting...

[@clivegoodman16]

0!=1

[@emmyloufumera4381]

However, 0

[@kar.s3390]

😂😂😂

[@youtubesubtitles9741]

@@emmyloufumera4381 Thanks Newton.

[@gunhasirac]

Good news: it make more sense for negative input

[@fisher00769]

I was expecting some proof and method in this video, not just it telling me, "It's like this and we can prove it". I enjoyed some of your videos, but in all honesty this was a waste of time.

[@ffggddss]

+フィッシャー00769 Proof that ½! = ½√π; i.e., that (-½)! = √π; is in three steps: 1. The function f(x) = ∫₀ᴰᴰ tˣe⁻ᵀdt = x!, for all non-negative integers, x, and it obeys the factorial function's recursion relation, f(x) = xf(x-1), for all x, not just the integers. 2. It is continuous, save for all of the negative integers, where it has simple poles (behaves at each of them, like y = a/x in the neighborhood of x=0). So this f(x), defined that way, can be adopted as the "natural" extension of the ordinary factorial function into the entire real line. It plots as a nice, smooth curve that goes through each of the factorial function's integer values. A fact brought up here, a bit clumsily, perhaps, is that there is an established function, the gamma function, Γ(x), which is an exact match for f(x) = x!, except that, for historical reasons, is displaced by one unit: Γ(x) = (x-1)! And it is this gamma function that is used as the f(x) in this video. 3. Substitute t = u²; dt = 2udu : C = (-½)! = ∫₀ᴰᴰ (1/√t)e⁻ᵀdt = 2 ∫₀ᴰᴰ e^(-u²) du Then use this cute trick, involving double integration over the 1st quadrant, converted to polar coordinates: ¼C² = [ ∫₀ᴰᴰ e^(-x²) dx ] [ ∫₀ᴰᴰ e^(-y²) dy ] = ∫₀ᴰᴰ ∫₀ᴰᴰ e^-(x²+y²) dx dy = ∫[0,½π] ∫₀ᴰᴰ e^(-r²) r dr dθ = ½π ∫₀ᴰᴰ r e^(-r²) dr = ¼π ∫₀ᴰᴰ 2r e^(-r²) dr = ¼π ∫₀ᴰᴰ e⁻˅ dv = ¼π C = √π Proof of point #1 is fairly simple (for integral calculus, that is), using integration by parts: ∫u dv = uv - ∫v du where u = tˣ, du = xtˣ⁻¹dt ; dv = e⁻ᵀdt, v = e⁻ᵀ As for the proof of point #2, it will not fit in this margin ;-)

[@fisher00769]

ffggddss That's fair enough, thanks!!

[@TheAgamemnon911]

+ffggddss I see what you did there. :D

[@TranSylvainie]

+ffggddss "it will not fit in this margin"... Well trolled, Mr Fermat ;)

[@Idkkeponer1]

Totally agreed, he only explains what a factorial is, nothing else. I can simply look for it in the dictionary and get the same info there.

[@sleepingsaucer3801]

0! = 1 1! = 1 (1/2)! = 0.886... yeah... so what do you call the graph of this one?

[@ffggddss]

And don't forget that (-½)! = √π = 1.772454... And: lim[x → -1⁺] = ∞ lim[x → -1⁻] = -∞ As x increases from -1, x! drops from ∞ to 1 at x=0, dips to a minimum around (0.46163)! = .885603194..., then increases back up to 1 at 1! = 1, and goes upward to ∞ as x→∞. Even more fun is what it does below x = -1.

[@anonimzwx]

Ferdinand Rafanan x!=gamma(x+1), 0!=gamma(1)=1 , 1!=gamma(2)=1 , 1/2!=gamma(3/2)= sqrt(pi)/2. , function gamma(x+1) between 0 and 1/2 decrease and between 1/2 and infinite increase, also factorial start to be weird between -infinite and -1

[@ferencrarosi8101]

en.wikipedia.org/wiki/Gamma_function#/media/File:GammaAbsSmallPlot.svg ;) en.wikipedia.org/wiki/Gamma_function#/media/File:Gamma_plot.svg ;)

[@scikick]

​@@definitelynotofficial7350 Yeah.. Such a weird graph www.desmos.com/calculator/vqbkyikbg2

[@nellvincervantes3223]

Also 0!! Is not equal to 1.

[@bxyify]

I was waiting for a mathematical proof

[@bitterlemonboy]

you dont need a proof. what the hell is proof. i dont like it. just take your calculator and do it. its pi.

[@danielgates7559]

cupisukk Most mathematical paradoxes are caused by a lack of formal proof... Believing in intuition or calculator is not a good idea from this aspect.

[@EADgbeist]

(1/2)! is the same as inputting 3/2 into the gamma function which yields the integral explained wonderfully in this video: kzbin.info/www/bejne/aJmQlqGQbK1saZo

[@prathamyadav3105]

@@bitterlemonboy engineer

[@dangabrieldayon1276]

Is that you Professor Hardy?

[@willyw9080]

This explains nothing. You left the world with the gamma function and said...hey guys thats why it equals to sqrtpi/2. I was hoping for some proof or something more rigorous.

[@phenlex2148]

He has a marvelous proof, it's just that his margin is too narrow to write it there.

[@Sporkabyte]

willy w just evaluate the integral for 1/2. You would have to do integration by parts

[@minuprasad2590]

fermat's dialogue 😂😂😂😂😂

[@minuprasad2590]

Sporkabyte it's not that simple

[@Dinesh7219]

It is interesting to note here that 'Factorial "Zero" is one'..

[@gaviningrave1731]

There’s a video to proof this

[@Seltyk]

That works even without extending the function, because n! = (n+1)!/(n+1) and when n=0 that makes 1

[@tomjeffered1668]

Just watch this impressive Math channel kzbin.info/door/ZDkxpcvd-T1uR65Feuj5Yg

[@cuber456]

Because of the Gamma function. This video is meh at best because it takes you ~5 minutes to reach the same conclusion without really providing the link as to why the Gamma function is the solution. You just list some properties and claim the Gamma function has those same properties.

[@Donaldbeebi]

+cuber456 and theres no explanation about why 1/2! = square root of pi divided by 2 srsly

[@cuber456]

+Mister Dee True. While it isn't super difficult to understand, it is a little difficult to explain in this post. However, if you know some calc 2 and 3 then you might be able to follow along. You have to understand how and why the Gamma function is actually defined: en.wikipedia.org/wiki/Gamma_function#Main_definition As Wikipedia says, the Gamma function is defined by that integral because, via integration by parts, it satisfies the properties of f(1) = 1 and f(x+1) = x*f(x). If you replace f(x) with factorial notation like f(x) = (x-1)! then you can see that these properties are the same as the factorial properties. I won't show how it falls out through integration by parts here but if you know your calc 2 then this isn't hard to do. Plus, writing it in this post would suck. You need to understand the special case of the Gaussian integral: en.wikipedia.org/wiki/Gaussian_integral Evaluating the special case of the Gaussian integral requires some easy but non-obvious tricks. If you are able to understand that then it won't be too difficult to understand the proof of why Gamma[1/2] = Sqrt[Pi]: jekyll.math.byuh.edu/courses/m321/handouts/gammahalf.pdf The last thing to do is to put all everything we learned together. We want to find (1/2)! (I hate this abuse of notation because factorials are only for whole integers but whatever). We know that (x)! = Gamma[x+1]. So using properties of the Gamma function (1/2)! = Gamma[3/2] = (1/2)*Gamma[1/2] = Sqrt[Pi]/2 So yeah, if you were able to follow all of that then that is one way to arrive at the result.

[@FF-pv7ht]

when senpai goes unnoticed, wtf?

[@FF-pv7ht]

could you do a vid on this though? how he arrives at the function for once, and maybe some intuition how factorials make sense in these regards

[@ffggddss]

@@cuber456 If you're disturbed by the use of x! for non-integer x (and it *is* strictly speaking, not proper), there is a function defined for that: ∏(x) = Γ(x+1), ∀x ∈ ℂ ; and ∏(x) = x! for all integers x ≥ 0. Fred

[@garrytalaroc]

you did not even show us how to use the gamma function to get to square root of pie over 2.

[@duckymomo7935]

plug in 1/2 into x and integrate you know how to integrate (use integration by parts) math.stackexchange.com/questions/314527/help-evaluating-a-gamma-function

[@jiaminzhu406]

I don't think integration by parts will solve gamma(1/2). Essentially it cannot be expressed by elementary functions. math.stackexchange.com/questions/11859/why-cant-erf-be-expressed-in-terms-of-elementary-functions the link you give only proves that gamma(x) = (x-1)* gamma(x-1), it has nothing to do with evaluating gamma(1/2)

[@will100smith4]

write gamma fn in sin and cosine form then solve for 1/2 ez

[@justabunga1]

@@jiaminzhu406 this is a special function, which is a non-elementary function, called the error function denoted as erf(x). The integral from -infinity to infinity of e^(-x^2) dx is sqrt(pi). This is a Gaussian integral. The answer of sqrt(pi) comes from the integration using polar coordination even if the function cannot be expressed in terms of elementary functions.

[@unic·play]

@Victor Vega that's because the majority of pies are round

[@pulkitsingla7421]

Presh's videos have changed so much since this video was uploaded 6 years ago.. Great going man! Thanks for making maths so fun and interesting for us..

[@targetiitbcse1761]

I am relieved that I am not the only one watching Presh's old videos

[@ssaamil]

Woah I know right

[@richardproctor4396]

At 2:56, doesn't f(x+1)=(x+1)f(x) not xf(x)?

[@MindYourDecisions]

It is confusing, but f(x+1) = x*f(x) is written correctly. I had to verify the calculation when making the video. f(1) = 1 by definition f(2) = f(1 + 1) = 1*f(1) = 1 f(3) = f(2 + 1) = 2*f(2) = 2 f(4) = f(3 + 1) = 3*f(3) = 3x2 = 6 f(5) = f(4 + 1) = 4*f(4) = 4x3x2 = 24 Note this is the factorial offset by 1.

[@ffggddss]

Yes - if f were the factorial function, then the recursion formula would be f(x+1) = (x+1)f(x). But as Presh says, f is the gamma function, so f(x+1) = Γ(x+1) = x! = x(x-1)! = xΓ(x) = xf(x)

[@DaveyJonesLocka]

What you're saying and what you're meaning are not equivalent. When you wrote f(1)=1, you said you want 1! to be 1 for this function. So verbally, you are already stipulating that your desired function f(x) is the factorial function. When you wrote f(x+1) = xf(x), you're then anticipating f as the gamma function. Clearly, they're related, but what you say about f initially isn't agreeing with what you're writing about f.

[@ffggddss]

Davey Jones f(1)=1 does **not** demand that f be the factorial function; and specifically, it doesn't rule out f being the gamma function. Just from that single value, f can be either, because: Γ(1) = 0! = 1, and Γ(2) = 1! = 1 so that Γ(1) and 1! both equal 1.

[@ffggddss]

MindYourDecisions But at that point in the video, you are, as you say there, starting out to try to find the factorial of ½, and you've just graphed the factorial function for x = 1, 2, 3, and 4. So the f(x) you're introducing at that point, is supposed to be the factorial function, and obey the recursion f(x+1) = (x+1)f(x). If you wanted to relate this to the gamma function, you need to insert a lot more material before this point in the vid, to connect to where you are now. But it isn't necessary at this point; you can and should just be sticking with the factorial function. You could even replace your initial introduction of Γ(x) and its definition, with f(x) = ∫ [0..∞] Tˣ e⁻ᵀ dT and then point out that f has the nice properties you seek, and matches x! for all the non-negative integers; and that there is this established function, Γ(x), that is just f(x-1). So Richard was right, that you've got a logical disconnect in your exposition at this point.

[@swordofdoom1517]

The actual proof involves coverting the equation from x-y system to polar co-ordinates. Since Γ(n)=(n-1)!, we use Γ(3/2)=(1/2)!. This statement isnt exactly true but we use gamma functions as an extension of factorial. And since Γ(n+1)=nΓ(n) [its a property], we get Γ(3/2)=1/2Γ(1/2)=((π)^1/2)/2.

[@ffggddss]

One of the cutest (IMHO) uses of non-integer factorials, is expressing the capacity of an n-dimensional ball of radius r as: V(r,n) = (π r²)ᵏ / k! where k = n/2 Try it for n = 0, 1, 2, 3, 4, ... ! When n is odd, use the factorial recursion, k! = k·(k-1)! to find k!, knowing that (½)! = ½√π. You might be a bit surprised at the way square roots of π always cancel, leaving only integer powers of π.

[@MindYourDecisions]

Thanks, I was unaware of this.

[@twistedsim]

Do you get V(r,0) = 1? Does that mean that a 0-dimension ball always have a volume of 1 ? :p

[@dar3690]

+ffggddss so.,what 1/2 represents? can you help me solve this 280 x [6.9]1/2= ?

[@ffggddss]

+Simon Bouchard Yes, but it's kind of meaningless. It just means that a 0-dimensional object, always being just a point, just has a unit capacity.

[@ffggddss]

+Darko Zaric Is the ½ supposed to be an exponent? Cause otherwise, why not just put another "x" before it? And why is the 6.9 enclosed in brackets? Are those supposed to mean something besides just a grouping? Where did you get this? What is its context? Taken literally, it would just mean the product of those three numbers: 280 · 6.9 · ½ = 14·69 = 966

[@mangeurdecowan]

what about √(τ/8) just saying...

[@MrRoyalChicken]

At 3:15 it should be f(x+1) = (x+1)*f(x) try putting in 1 or 2 for x it doesn't work out the way it's shown

[@bwcbiz]

Yeah, I noticed that too...

[@dastardlyexpressions]

MrRoyalChicken That's because the gamma function is defined as f(x+1) = x * f(x). So x! = Gamma(x+1) not x! = Gamma(x). Actually Gauss defined his Pi function as f(x+1) = (x+1)f(x) such that Pi(x) = x! . However most mathematicians and all textbooks use Euler's gamma convention instead Gauss' Pi convention (which would actually simplify many equations but whatever)

[@cubicardi8011]

the marble racer What? 3!=3*2*1=6 4!=4*3*2*1=12*2=24

[@leif1075]

@@dastardlyexpressions But if you see the function like that you get the wrong answer ..so how do you reconcile this?

[@iScream2367]

Yes that's what I was thinking too

[@cillo71]

Nice video¡ A good introduction to the function. As a curiosity, the minimum of the gamma function at the positive axis is 0.88560319441, which is only a little less than the value 0.8862269...I understand that this minimum is found numerically.

[@andreshenao2358]

Instead of f(x + 1) = XF(x) it makes more sense f( x + 1) = (x + 1)F(x) according to the definition of factorial 20! = 20 times 19! Thanks.

[@mahdi2753]

Exactly!

[@kreativambience8720]

@andreshenao2358 , You got it right, this is also the method of how we can divide factorials.

[@BitcoinMotorist]

But, but, 0! = 1

[@tomszczypka7003]

Patrick Dukemajian I was thinking the same thing, it's convinent he left that out, cause that would suggest that there is a miminum between 1 and 0

[@BandanaDrummer95]

+Tom Szczypka the reason that it would suggest that there is a minimum between 0! and 1! when extended by the gamma function (so between 1 and 2) at 1.46163... (0.46163... if there were non-integer factorials) which yields the value 0.88560...

[@sephirothjc]

I know it's a convention but it helps me to think of it in a combinatorics way, if you have an empty set there's still the one way it can be arranged

[@martinschuh3327]

+Tom Szczypka There is a minimum between 1 and 2 of the gamma function. But that doesn't contradict any of the properties it should have, because no one said it has to be monotonically increasing.

[@Trias805]

+Tom Szczypka And there is. 0.000001! = 0,99999942278532415355498927168952, so more than 0.5!

[@ffggddss]

Even a bit neater: (-½)! = √π You can get that straightaway, by taking your result, along with the recursion, x! = x·(x-1)! - ½√π = (½)! = ½(-½)! ; thus, √π = (-½)! Also, the "extended" factorial function's definition is a bit more elegant than the gamma function's: x! = ∫₀⁰⁰ Tˣ e⁻ᵀ dT

[@davidodonohue362]

why should f(x + 1) = xf(x)? when x = 1, f(x+1) should equal 2, but you get 1*f(1) = 1*1 = 1. when x = 2, f(3) should equal 6, but you get 2*f(2) = 2 (using previous result) or 2*f(2) = 2*2 = 4 (using knowledge)

[@davidodonohue362]

+David O'Donohue unless x is the factorial you are trying to get, in which case it would make more sense to say f(x) = x*f(x-1)

[@GT6398]

+David O'Donohue The video is a mess really. The first and third "x" are equal, but the second "x" has a different value. The expression you created would be correct

[@mistymouse6840]

+David O'Donohue The video glosses over something, which is the gamma function isn't quite the same thing as the factorial function. In fact, gamma(x)=(x-1)! So gamma(x+1)=xgamma(x) but x!=x(x-1)! When the video talks about f(x), it's talking about the gamma function, not the factorial function (although it doesn't explicitly say so). I don't know why the gamma function was defined this way. In the definition of the gamma function as an integral, if you replace t^(x-1) with t^x,  you get the factorial function which seems more natural. P.S. Actually, the video does clarify this when it says gamma(x+1)=x!, but only after it has discussed f(x) for a while.

[@spelunkerd]

+David O'Donohue Yes, I was going to make that remark but you beat me to it. f(x+1) should be =(x+1)f(x), not (x)f(x). For example, if x=3, then f(x+1)=6(3+1)=24.

[@duhboss1]

Dude I'm sorry but I don't think this is a very good video. It's a cool topic and everything, but it was kind of a chore to watch you explain stuff. You need to clean this up a little bit, and you could have a great video.

[@JohannaMueller57]

+duhboss1 true. also i dont want to see any "proof" or demonstration with windows' calculator. i dont want to see "x" as "times" in formulas either.

[@ZardoDhieldor]

This is a horrible explanation of the Gamma function. Motivating it using a calculator (on which it may have been implemented for a good reason) is a very bad idea. Also, you didn't really explain anything. This is not how it should be presented to your intended audience. As a student of mathematics, I give you a downvote.

[@akshakeel1]

Agree. Can you please explain what is not clarified.

[@gunhasirac]

Brian LA that’s not a logical argument. One doesn’t need to be able to do something to criticize something. If you’re trying to argue if one can’t do it should not have criticized, that’s is not a logical argument as well. Being able to do it makes the argument more plausible, but does not affect the validity of the argument, especially when the argument gives objective reasons.

[@ManishKumarIITG]

@Brian LA I don't need to make a movie to tell whether a movie is good or bad :)

[@torgo_]

Hmmm.. I was looking forward to seeing a direct calculation for (1/2)! But the clip just sort of abruptly ended. Can't you use the gamma function to calculate it? This video felt like a cliffhanger with so resolution.

[@SuperCamponotus]

You can find explanation in many books.

[@Pomme843]

I believe the properties of the new function at 2:57 should be: f(1) = 1 f(x+1) = (x+1) * f(x) or: f(1) = 1 f(x) = x * f(x-1)

[@juliettevlieghe8571]

+Pomme843 I agree ^^

[@Apostate1970]

Correct, and the second way of writing it is I think the most elegant. I'm really surprised they let such a glaring mistake as this go uncorrected before publication, and have never apparently bothered to make a correction.

[@aayushmishra2007]

Le calculator: (1/2)!= Domain error 😂 😂😂

[@Maybemiles_C]

😂😂😂

[@baliramkumarsinghpatel1816]

use HiPER Scientific Calculator

[@TimJSwan]

This video was absolutely superb. My favorite part is how you show why the gamma function is the uniquely desired real version of the factorial function.

[@billrussell3955]

Hello Presh, I have a puzzle for you. When is a straight line with points "a", "b". Points a and b terminate at the center of two other lines and are both parallel with and simultaneously perpendicular or orthogonal to both lines. We define as 1 unit line (a, b). The other two lines are of undefined units. Let me know what you think.

[@minhokim8263]

Great, it was very nice that you mentioned the 3 conditions of continuous factorial, and I appreciate it. Could you please explain more about smooth graph = convex ln{f(x)} or introduce some materials to read?

[@alessandromarchetti2766]

n!=(n)(n-1)(n-2)...(1) n=(1/2) n!=(1/2)(-1/2)(-3/2)(-5/2)...(-Infinity)(1) = (1/2)(-1/2)(-3/2)(-5/2)...(-Infinity) = (sqrt(pi))/2 The product of every number n=(1/2)-k with k being a natural number is then equal to the square root of pi divided by 2.

[@freemanthompson706]

I enjoy your videos. I'm probably wrong about my next observation, and if so, I hope the explanation is simple. At about 3:15 in this video, isn't f(x+1) = (x+1)*f(x)?

[@baptistepetiot7649]

No you are r

[@ziadalaoui2461]

no because x!=gamma(x+1) not x!=gamma(x)

[@ogle777]

Not sure if anyone has already posted this, but one way to actually solve that integral, after setting x=1/2, is by a couple substitutions from multivariable calculus. (Specifically, the integral of e^-t/sqrt(t) from 0 to infinity). Letting u = sqrt(t), you get u^2 = t, and 2udu = dt. Then the problem amounts to solving the integral of 1/2e^-(u^2) from 0 to infinity (the limits of integration are the same after substitution). This function doesn't have an elementary antiderivative so you can use multivariable calc to get the solution. Essentially, you need to "square" the integral; i.e. let Int = ∫e^-u^2 du, then Int^2 = 1/4∫∫e^-(u^2 + v^2) du dv, with both limits of integration going from 0 to infinity. Then there's the super useful switch to polar coordinates which makes the solution elementary: let r^2 = u^2 + v^2; then dudv = rdrdø, where ø is the angle from 0 to π/2 and r going from 0 to infinity (these new limits will span the same range as 0 to infinity for u and v; you can draw an u-v plane to better see this). Finally, substitute for u and v and integrate! Namely, the integral Int^2 = ∫∫re^-r^2 drdø from 0 to infinity and 0 to π/2. You should see that only π/2 remains, meaning that Int^2 = π/4; then square root to get Int, which was the original integral. Lemme know if you guys think I missed something! (Not sure why there are lines thru some of the expressions, but just ignore them)

[@toriknorth3324]

+kjnmn The lines are due to some formatting rules that youtube uses. The three types of formatting you can do that I know of are *bold*, _italics_, and -strikethrough-, which you would type out in a comment as ​*​bold​*​, ​_​italics​_​, and ​-​strikethrough​-​. To avoid having the youtube comment parser pick out the * _ - symbols you would need to put spaces around each of your - signs since having a - touching another character -will mess up however much text you have between- the two -. One way to get an invisible space is to use a unicode zero width space, like the one between these two quote marks: '​'. Other than that your integration looks good to me. (Edit like 30 times because OMG IT IS REALLY HARD TO GET ALL OF THE ZERO WIDTH SPACES WHERE THEY NEED TO GO)

[@katelikesrectangles]

I feel like you skipped a critical step or something, and then you're like "ta-dah! that's why!", leaving me no better off than when I started.

[@doctorduck3596]

One question, if the function must satisfy both conditions: - f(1)=1 - f(x+1)=xf(1) then, if x=1: f(1+1)=1f(1) -> f(2)=f(1) -> { if f(1)=1, then } -> f(2)=1, and f(2)=2

[@TechnoRaabe]

First video on this channel that I disliked. You didn't proof anything. You gave the function to us and said: "This is the unique solution." BUT WHY?

[@CharlesScalfani]

And a miracle occurs at 3:37.

[@psibarpsi]

It's called Proof By Miracle.

[@veggiet2009]

I wish you'd explained the gamma function itself

[@bitterlemonboy]

Gamma(n) = (n - 1)!

[@MindYourDecisions]

Pi Day Suprising Fact - The factorial of 1/2 equals (√π)/2. kzbin.info/www/bejne/h5mndaOJn9pmq7s

[@baekBlackbeen]

Complicated... I can't understand @-@

[@thribsilva]

+baek Blackbeen Neither do I. I kinda doubt this used true mathematic principles :/

[@motorhue]

+Thiago R. Silva but but.. He just showed that he does! Insert the gamma function formula in the other formula for factorial and u see it adds (multiplies) up

[@TheGamerofDeath666]

+motorhue It also shows that the curve should be declining as X approaches 0. However, 0! or Zero Factorial is equal to 1. Therefore it is a discontinuous function and only true on the plane (0, ∞), and not from [-∞, 0]. It is a good question to bring up, and I am curious as to how it really would work on the full 2D plane; possibly even weirder results on a 3D plane.

[@ffggddss]

+Cyrus Sly Fox Actually, x! can be defined using the gamma function, for all complex x, except for all the negative integers, each of which has a simple pole. ∞ x! = Γ(x+1) = ∫₀ tˣe⁻ᵀdt On the real line, it is continuous on the open interval (-1, ∞), and on every unit-length open interval between consecutive negative integers, at each of which it goes to ±∞. Interestingly, 1/x! is defined and continuous on the entire real line (and the complex plane, for that matter), because the gamma function has no zeros. But it oscillates out of control as you go down the -x axis.

[@conradgarcia4850]

Euler's equations are indeed breathtaking. Can I have a tutorial, sir?

[@ameerunbegum7525]

Everyone: This is pi. Me: 🥧 is pi.

[@danielstone7921]

For me, this is not true: My calculator outputs "Domain Error." 😂

[@1-.-.-.-..-.-.--.]

What class are you and if its school calculator its normal but university calculator huh…

[@amarjeetverma6965]

Use scientific calculator

[@OriginalSuschi]

The problem is, that we somehow defined the factorial function. So it can't really be proven, because it's a definition. Our properties were, that (also important: f(0)=1), f(1)=1, f(2)=2, f(3)=6 and so on. Just think about it. Long ago, the function a^x was only defined for positive integers (with or without x=0). After, we had to define what a^0 equals to and what it means to evaluate a^(1/n) (yeah, exactly: It's the nth root of a). Every other positive fraction can be evaluated with power rules, which were actually also somehow defined to make absolute sense. E.g.: a^(3/2)=sqrt(a^3) and much more. After this, it was important to define the function a^x for negative numbers as well. When looking at the graph, it makes sense, that the function converges to 0 when x=-infinity (When you don't believe, use the first and second derivative, that's already enough to prove it) Or do it like that: a^3->a^2->a^1->a^0->a^-1... 8. 4. 2. 1. 1/2. ... We see the pattern, that we just always divide by two when setting a=2 After that we can see: 2^-1=1/2, which is 1/(2^1). After evaluating this with many other integers for a, we get our rule, which is a^-x=1/(a^x). It's basically the same as that with the factorial function, just way harder. Searching for patterns, evaluate numbers, experimenting and so on. So why did everyone want to see a "prove"?

[@kevinlim8477]

1:14 Photomath said: undefined

[@trananswers]

At 3:14, possibly error, f(x+1) = (x+1)f(x) by definition

[@AllanPoeLover]

我還以為只有我覺得怪怪的

[@hypisk-simplysplendid442]

Brother 0 factorial is 1. 1 factorial is 1 but. 0.5 factorial is less than 1 this means it is not strictly increasing or strictly decreasing. How it can be convex?

[@Blaqjaqshellaq]

It isn't that function that's convex but the natural logarithm of the function.

[@BrianGreene-i4o]

Good video. I started looking up the factorial function after being spellbound by the natural number e function.

[@alexfresh8951]

But 0!=1 The graph doent make sense

[@thomaskember4628]

I am a retired programmer. I still read books on computer languages in which the factorial function is used to introduce recursion. For factorial n the base case is always 0 and its factorial is given as 1. Since factorial 1 is 1, the factorial of every number between 0 and 1 must also be 1. But my calculator works out square root of pi divided by 2 to be 0.88622.. which is less than 1. How can this be? Also if the square root of 0 is not 1, what is it? It can’t be 0, because that would make the factorial of any integer in the recursive function equal to 0.

[@SlimThrull]

I couldn't help but notice something that looked similar to Euler's Equation in there. Did Euler have something to do with finding the Gamma Function?

[@MindYourDecisions]

SlimThrull Euler did play a role in this, according to the history part in Wikipedia's entry: en.wikipedia.org/wiki/Gamma_function#History

[@SlimThrull]

MindYourDecisions Ah, I should have looked there. Thank you. :)

[@gameguy8101]

But the curve concept does not work, because if 1/2 is approx equal to 0.88, then since that is not one the curve is messed up. 0! Is one and 1! Is one, so using a curve we can deduce that (0+x)! If 0

[@klikkolee]

gameguy8101 why do you say that values of gamma(x+1) for 0

[@BlueCosmology]

+gameguy8101 x^2 = 1 at x=-1 and x=1. It is not always equal to 1 between -1 and 1.

[@alistermatheson4967]

You use pi as some sort of legitimisation of your calculation. However, once stated then pi is ignored! There is no explanation of how this infinite special number fits into the factorial of non-whole numbers! I suggest that you have entered into a circular argument that justifies your position. I look forward to your peer reviewed publication

[@martind2520]

Alister Matheson No he didn't. π was never used as a justification, it was used as a hook to arouse interest in the subject. Then the subject was explained with its own justification followed by us being told how to find the √π/2 result for ourselves.

[@Arkalius80]

+Alister Matheson He wasn't trying to justify or prove anything. He was just iterating the interesting fact that GAMMA(1.5) = root(pi) / 2.

[@Jackcabbit]

+Arkalius80 True, he was just saying how there's an interesting case there. Alas, he doesn't go into exactly why it is that way, and it's unfortunate because it's such a big hook to this video.

[@cbernier3]

+Jackalope That's because you need to understand calculus II to solve that integral. Most of his audience does not understand Calculus.

[@SuperCamponotus]

@@Jackcabbit Video had purpose to give you idea. It really did .You will not understand more detailed explanation, anyway. Otherwise the explanation is part of any maths book and you will already know it if you have any interest in mathematics.

[@justabunga1]

Rather than the traditional formula n!=n(n-1)(n-2)...*3*2*1 for n to be positive integers, the factorial notation can be changed into an integral formula, n!=integral from 0 to infinity of t^n*e^-t dt. If you plug in 0 into the integral formula, the answer will be 1. If it's a non-integer like the example in the video here, the answer is sqrt(pi)/2. Negative integers are always undefined.

[@phucminhnguyenle250]

yet you explain nothing about the value of gamma(1/2). -_-

[@phucminhnguyenle250]

*****​ it is not a normal integral, and hard to solve for someone who doesn't know techniques like double integral. That's why he should put more explanation for it.

[@quocanhnguyenle4952]

Meh, try Googling the "Gaussian integral", and learn some calc 2 or 3.

[@Ckombactman]

Factorial's domain are Natural Numbers. 1/2 is a rational number so its factorial is just not defined. The gamma function is just like factorial but extended to real numbers. So here's the explanation.

[@funnyhappypopcorn]

Can you at least explain it a little? The video literally shows the formula for the gamma function with no proof or context, ergo the video has no educational value.

[@محمد23-و3ع]

F(x+1)=(x+1)f(x) 3:20

[@Robi2009]

So does that means that Bruce Banner was playing with some factorials when he got radiated, so function became gamma as he did?

[@MANOJYADAV-gn6kg]

That was the gamma rays .......🙄🙄🙄

[@ricoseb]

@@MANOJYADAV-gn6kg ok daddy

[@Eirmas]

I love your math videos man, keep it up

[@AshgabatKetchumov]

There is no proof because... *_The proof is left as an exercise to the reader_*

[@alexandrubragari1537]

Lol 😂😂

[@SaeedAcronia]

2:56 You made a mistake! f(1+1)=1*f(1)=1 which is wrong. The correct form of the equation should be f(x+1)=(x+1)*f(x)

[@awseomgyhero]

my calculator just says error =/

[@benzenering2183]

The Epic Gamer mine too

[@soelinhtet6064]

Out of date. what a pity!!!¡¡¡

[@justabunga1]

Type in your calculator as (1/2)! or 0.5!. It should come as an approximation answer (exact answer is sqrt(pi)/2).

[@yvesdelombaerde5909]

At 3:30, should it be f(x+1)=(x+1)*f(x) ?

[@complex314i]

I thought we were going to see you do the integral.

[@vedangpathak]

At 3:01 The second property should be f(x+1) = (x+1) f(x) not x f(x)

[@alberteinsteinthejew]

I'm still waiting for the punchline, but it didn't happen...

[@3dplanet100]

Hey, interesting! Check this out!: 1! = 1 0! = 1 0.5! = 0.88... 0.5 is between 0 and 1. If both 0! and 1! equals to 1, the answer to 0.5! should be in between as well which would be 1. But is not in between (1). It's 0.88... Then this is a *PARADOX*

[@isaacstruhl5379]

Ok, but don't we define 0! to be 1? Or is what I learned about Taylor series wrong...

[@panobato5984]

Yes 0! Is one

[@Woodside235]

+Isaac Struhl 0! is still equal to one. The extended factorial dips below 1 between x=0 and x=1. www.wolframalpha.com/input/?i=fact%28x%29+from+x+%3D+0+to+2

[@MrHatoi]

I think the question in the title is misleading. The video doesn't address why (1/2)! equals what it does, but why we're able to take factorials of non-counting numbers. I understand that it's used for an example, but it makes the video seem like it's about something completely different.

[@christianvikkels4801]

Nice video! But it's very strange... 1! = 1 0.5! = 0.8862 0! = 1

[@Ma2Ju]

Just look at the graph of the gamma function for all comlex numbers (wikipedia). The video states something simplyfied there. (You can define it for negativ values as well for example.)

[@tomarchelone]

Ma2Ju did you look at it? Cause it is clearly seen, that gamma(1/2) > 1

[@kurtschluss6718]

@@tomarchelone "1/2!" = gamma(1/2 + 1) = gamma (1.5)

[@frentz7]

Also "smooth" is not the same as "logarithmically convex", you have an error there. ("log convex" means what it sounds like : that log f(x) is a convex function; it doesn't mean f(x) is smooth, or even differentiable at all)

[@Lastrevio]

Ok, I,m in 7th grade and the thing at 3:46 scared the shit out of me. You shoul have put a jumpscare warning out here.

[@longjohnsilverthorne7766]

+Lastrevio I'm in AP Calculus, and I was still pretty confused so don't worry.

[@TrinoElrich]

+Lastrevio gg trying to learn what it means or how to calculate it any time soon. Lul, this is calc material. And, in all honesty it is pretty basic calc.

[@TrinoElrich]

+PeltedPancake and "those scary stuff" are just integrals haha

[@TrinoElrich]

"Well, we students down here..." Dude, relax lul. No need to try and seem all proper on here haha. Also, don't cut yourself short, man. You're not inferior in any way. You just aren't at this level of math yet, so don't worry lul.

[@TrinoElrich]

+PeltedPancake mastered the algebra of exponents and roots -lul woah just take it easy, man. and you teached yourself sigma?! lul chill man

[@professorpoke]

At 0:33 you said factorial of a number n is equals to the product of all whole numbers upto n, which is in fact wrong since whole numbers also include zer0. And if you include zer0 in the product you will end up having zer0 everytime. So instead of saying whole numbers, you must have said natural numbers to exclude zer0. Correct me if I am wrong.

[@DanDart]

you have the most accurate times new Roman pen in the world

[@DanDart]

I mean sans... I'm so glad you don't do this anymore

[@joshix833]

@@DanDart why did you correct yourself one year later?

[@DanDart]

@@joshix833 well kinda both, I am a sparse viewer so I thought I was wrong

[@joshix833]

@@DanDart k

[@IslandForestPlains]

This is a very interesting introduction to the Gamma function. Thanks for posting! The music is annoying and distracting for me, though, and I should like it better if there were no music during the explanations.

[@JonathanBartlesSWBGaming]

Bad definition of n factorial. 0 is a whole number...

[@BandanaDrummer95]

+Jonathan Bartles (SWB Gaming) How many ways can you arrange a set of zero objects? Answer: 1 (the null set)

[@JonathanBartlesSWBGaming]

BandanaDrummer95 what's your point? That has nothing to do with my statement. He defined factorial as the product of all the whole numbers up to the given integer so by his definition, 4! = 0 * 1 * 2 * 3 * 4 = 0. Troll harder next time

[@BandanaDrummer95]

I wasn't trying to troll, I just used the definition that he had given, that a factorial is the number of ways that a set of that many objects could be arranged in a line, which happens to nicely correspond to the product of the natural numbers less than and equal to that number (his use of the term whole number is just another in his long line of mistakes in this video).

[@JonathanBartlesSWBGaming]

Ohhhh, then yes, you are correct. And thank you for correctly differentiating between whole numbers and natural numbers! You wouldn't believe how many people say 0 is a natural number, it's quite irksome

[@BandanaDrummer95]

+Jonathan Bartles (SWB Gaming) I will admit that I have to double check every time to make sure I know which set includes 0 and which doesn't.

[@royalbeatlemaniac7108]

Proving the factorial function’s integral form takes a formal definition of exponentials and a bunch or theorems about improper integrals, uniform convergence, differentiating under the integrals sign. Therefore it would probably take him at least a hour to go through the 6-7 hard proofs in order to prove the gamma function.

[@michailbialkovicz878]

You're just wrong. (1/2)! isn't sqrt(pi)/2, gamma(3/2) is sqrt(pi)/2, but (1/2)! is simply undefined

[@temrock7033]

Yeah. Factorial is only defined for non-negative integers.

[@Woodside235]

+Michail Bialkovicz Factorial can be extended without issue to be gamma(x+1). I once did it myself to write a paper about half derivatives (I bet you'd also say that's undefined). No serious mathematician would say it's undefined.

[@michailbialkovicz878]

Woodside No serious mathematician watches furry porn though

[@Woodside235]

Michail Bialkovicz Mathematicians can be quite depraved, have you asked any?

[@Woodside235]

Michail Bialkovicz What.

[@paolosalerno6633]

Sooo....where is the golden ratio?

[@markcrites7060]

Video fails on several levels. most importantly, it falls short of answering the original question.

[@sanjibsinha4120]

I tried it on my calculator and subtract pi from it but It was not 0

[@fedewar96]

The factorial function is defined in the natural numbers, the Gamma function in just an approximation. You CAN'T compute the factorial of a real number.

[@someonesomeone4099]

Federico I think it can ... It’s factorial but extended to all numbers.

[@yosefmacgruber1920]

@@someonesomeone4099 Can it be extended to the complex numbers?

[@someonesomeone4099]

Yosef MacGruber Yeah I think I remember that the domain of the gamma function includes the complex numbers

[@yosefmacgruber1920]

@@someonesomeone4099 Gamma function ≠ to factorial function? Is the domain of the factorial function then the same? So what then about the quaternions? Every complex number can be written in the form of (a,b,0,0) right?

[@someonesomeone4099]

Yosef MacGruber Well I don’t know much about quaternion (yet)😂, but the reason that gamma function is said to be the “extended factorial” is because it is defined in a way that kept an essential property of factorial for a less restricted domain. It is true that factorial is at first only defined for natural numbers, but that’s before the gamma function was found. In other words, the factorial function is like a part of the gamma function. It is a way to compute the gamma function with natural number inputs.

[@johntate6537]

Many infinite series naturally produce terms with 'factorials' of non-integers, which is the practical justification for the existence of the gamma function. I've always found the x-shift of the gamma function weird, since all you have to do is undo it again whenever you want to make a calculation, but I suspect it has something to do with the fact that there are a string of singularities in the values that the gamma function takes, and the way they have it set up arranges it so that the gamma function is well defined for all positive values, has a singularity at zero and then graphs in a complete different, but kind-of repeating form through it's values for negative argument. I remember going over the proof for the root-pi by two answer at uni. It relies on a really neat trick to take an otherwise unsolvable integral (algebraically unsolvable - of course your can do a numerical integration) by shifting the function into two dimensions and restating it in polar co-ordinates, which is where the pi comes from. An extra term appears that suddenly makes the integral solvable by inspection. It's the kind of thing to warm a maths nerd's heart.

[@vaibhavwaghmare153]

I am here for proof

[@fatihaouardi7530]

I adore your videos!!! Please continue we want more maths

[@christophegragnic8681]

Protip: use the right character: × instead of x.

[@Xnoob545]

× is times x is a letter Lol

[@yosefmacgruber1920]

@@Xnoob545 But isn't • times? So where does dot-product and cross-product enter into this? Those presume something more complicated than simple reals, such as matrices or vectors or something?

[@Xnoob545]

@@yosefmacgruber1920 WHAT DOES THAT EVEN MEAN

[@deepampurkayastha3040]

Hey, does this mean we'll have a U-kind of structure between 0 and 1 in graph chart? Because 0!,1!=1 but (1/2)!=0.886226925

[@henrylee4374]

What is t equal to?

[@ham1533]

a variable for you to integrate

[@frentz7]

Wow that was really different. Math people call this "hand waving." Besides the nervous stuttering, you first (1) jumped to the "here's the magic function we need" Gamma function, and not only, but (2) wrote (1/2) ! = Gamma (3/2) = sqrt(pi) / 2, with no explanation at all for where the "sqrt(pi) / 2" came from.

[@jojojorisjhjosef]

4:07 ded

[@plislegalineu3005]

Jäß

[@BandanaDrummer95]

Please understand that when you thought you told your computer to do the factorial of 1/2, you actually told it to approximate the gamma function at 1/2 +1. Also, when you tell your calculator to perform other rather complicated math functions, it just approximates as best as it can with built in functions.

[@TheReligiousCrap]

+BandanaDrummer95 Just like sin , cos and other trigonometric stuff

[@ZoeTheCat]

The brain has only a small problem getting used to Real factorials. However, the fact that the sum of the positive integers is equal to - (1/12).....takes a bit longer. In fact, I'm still not quite there and I've worried about it for decades ;-)

[@realityversusfiction9960]

Sorry you have wasted decades of your lifetime worrying , about something that in realty is essentially and relatively simple in nature. Originally. Pi = 22/7 as an improper fraction, or 3 whole units and 1/7th of 1 whole unit. Or 3 whole diameter units, with 1/7th of one whole diameter unit remaining. Proof. THE LENGTH OF ONE DEGREE OF A CIRCLE Reference: “Estimating The Wealth” Encyclopedia Britannica. A Babylonian cuneiform tablet written some 3,000 years ago treats problems about dams, wells, water clocks, and excavations. It also has an exercise in circular enclosures with an implied value of π pi = 3. The contractor for King Solomon's swimming pool, who made a pond 10 cubits across and 30 cubits around (1 Kings 7:23) used the same value. Given a Square measuring 120 centimeter’s times 120 centimeter’s 1. Use one right angle of the 120 x 120 cm square as a Diameter Line 2. Multiply the 120 cm Diameter Line by 3 3. The Circles Circumference will measure 360 Centimeter’s in length 4. A Circle has 360 Degrees to its circumferential length; therefore each degree is 1 centimeter in length. 5. Therefore all Circles have a circumferential that is exactly three times its Diameter Line length; The Ancient Sumerian masters of geometry and mathematics defined this empirical reality, more than 2000 years before the plagiarizing Greeks. However one question for me does remain begging, did they also manage to achieve this? Twelve Steps From The Cube, To The Sphere Calculating the surface area and volume of a 6 centimeter diameter sphere, obtained from a 6 centimeter cube. 1. Measure the (a) cubes height to obtain its Diameter Line, which in this case is 6 centimeter’. 2. Multiply 6 cm x 6 cm to obtain the square area of one face of the cube; and also add them together to obtain the length of perimeter to the square face = Length 24 cm, Square area 36 sq cm. 3. Multiply the square area, by the length of diameter line to obtain the cubic capacity = 216 cubic cm. 4. Divide the cubic capacity by 4, to obtain one quarter of the cubic capacity of the cube = 54 cubic cm. 5. Multiply the one quarter cubic capacity by 3. to obtain the cubic capacity of the Cylinder = 162 cubic cm. 6. Multiply the area of one face of the cube by 6, to obtain the cubes surface area = 216 square cm. 7. Divide the cubes surface area by 4, to obtain one quarter of the cubes surface area = 54 square cm. 8. Multiply the one quarter surface area of the cube by 3, to obtain the three quarter surface area of the Cylinder = 162 square cm. CYLINDER TO SPHERE 9. Divide the Cylinders cubic capacity by 4, to obtain one quarter of the cubic capacity of the Cylinder = 40 & a half cubic cm. 10. Multiply the one quarter cubic capacity by 3, to obtain the three quarter cubic capacity of the Sphere = 121 & a half cubic cm, to the volume of the Sphere. 11. Divide the Cylinders surface are by 4, to obtain one quarter of the surface area of the Cylinder = 40 & a half square cm. 12. Multiply the one quarter surface area by 3 to obtain the three quarter surface area of the Sphere = 121 & a half square cm, to the surface area of the Sphere Confirmation by Weight Given that the 6 Centimeter Diameter Line Sphere was obtained from a Wooden Cube weighing 160 grams, prior to it being turned on a wood lathe into the shape of a sphere The Cylinder of the Cube would weigh 120 grams The waste wood shavings would weigh 40 grams Given that the Cylinder weighed 120 grams The waste wood shavings would weigh 30 grams. Note: And ironically you can also obtain this same result by volume, using Archimedes Principle. As To The Proofs OF The Pythagoras Theorem In any right triangle the sum of the square on the hypotenuse, is equal to the sum of the squares on the other two sides. Incorrect Given a right triangle whereby both the base line, and the vertical line of the triangle each measuring 12 units. The sum of the two squares will be 288 squares The sum of the square on the hypotenuse measuring 17 units will be 289 squares. One square greater than the sum of the squares, on the other two sides.

[@Ma2Ju]

This is NOT an regular summation! Therefore it isn't possible to imagine how it ends up at -1/12. You can manipulate the way you sum up so that there will be a result (like using zeta function regul. or Ramanujan's summation, which you may know), but the infinite Sum still diverges. This means you just force your way to a result, which does have useful properties, but is not equal to the infinite sum you have defined.

[@ZoeTheCat]

Reality Versus Fiction Did you really just use a ream of virtual paper to prove the value of pi? lol.

[@ZoeTheCat]

Ma2Ju sorry I didn't hit the humor button. I am well aware that it is a divergent series and not a fair comparison. The harmonic series occurs at the Landau pole of the RZF. I know a bit about this thing. Nevertheless, the -1/12 result is the basis of some String theories and is used to explain some aspects of the Casimir Effect and Regularization of QED during renormalization. It is still a somewhat stunning result.

[@Ma2Ju]

Sorry on my side for underestimating you. The problem is that this result is often explained totally wrong (on KZbin). Just wanted to make sure you (or other readers) didn't misunderstand.

[@xyz.ijk.]

Six years later this is still one of your best videos even though it is short.

[@youmah25]

fractionnal calculus

[@faizshaikh8681]

Factorial in practical terms is number of ways of arranging things. If there is 1 thing we can arrange in 1 way (1! =1). If there are 0 things the numbers of ways of arrangement is also in 1 way (0! =1) . (1/2)! Means arranging 1/2 thing. Now here the same 1/2 thing doesn't exist so thus the calculator gives error. We can consider this 1/2 thing as arranging 1 whole thing which again concludes that (1/2) ! =1 Note that we cannot have fractions as the answer as factorial is no of ways of arranging things and will always be a whole number. Practically, the factorial for 0 through 1 and all the decimals inbetween will be 1.

[@katzen3314]

What about 0!?

[@dlevi67]

What about it? 0! = 1.

[@katzen3314]

dlevi67 That doesn't fit the curve for the fractional factorials.

[@dlevi67]

Doesn't fit what curve? There is nothing that says that the Gamma function should be monotonically increasing below 1 [or rather, below 2, since Gamma(n) = (n-1)!], and as it happens, it doesn't. It still goes through the points 0,1 and 1,1, just like the factorial. [Or rather, it goes through the points 1, 1 and 2, 1 which correspond to the factorials of 0 and 1 respectively]

[@katzen3314]

dlevi67 oh yeah, my mistake.

[@finlaymcewan]

A way of thinking about it is to remember that factorial gives you the number of ways you can arrange an amount of objects. It's quite philosophical, but 0!=1 because there's only one way to arrange 0 objects

[@shirkepraveen8744]

Hi Presh I like your videos very as I am self studying mathematics, you have beautiful way of simplifying the things P.S. but your subtitles has black background which hides the details most of the times, of you change it to transparent, it would be better! Thanks for your contribution to society

[@bricepilard5267]

Just discovering this video, following your earlier post... I was vey curious to understand how (1/2)! could be equal to SQR(pi)/2, i was expecting some trigonometry. And I have to admit that I did't understnd the explanation ! I need to revise some lessons :D

[@alanr4447a]

And Gamma function times x is called the Beta function. x! = Beta(x) which is what the calculator gives you.

[@DavidFMayerPhD]

Analytic Continuation is the key and it is UNIQUE.

[@troybingham6426]

Unless I'm missing something then.. by the definition of a factorial, it actually isn't (√π)/2. We're just kind of going to say that it is so it will fit with that gamma function.