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

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Type 0.5! in your calculator to see what the factorial of one-half is. The result will be 0.886..., and the exact answer is the square root of pi divided by 2--amazing! How is this possible, when the factorial of a number n is defined as n! = n(n-1)(n-2)...1 and this definition only makes sense for whole numbers?

The calculator result is not an error, and in this video I explain how the factorial can be extended beyond the whole numbers for all real numbers by the gamma function. Once we extend the factorial function beyond whole numbers, you can see why the factorial of one-half is equal to the square root of pi divided by 2.

Bohr-Mollerup theorem

Applications

Numerical computation

Alternative ways to extend the factorial function

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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 YouTube. Thank you for believing in me, even when the channel was just starting!

MindYourDecisions
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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.

fisher
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π=3.14 ✖️
π=22/7 ✖️
π=((1/2)!)²×4 ✔️

yoyo-mvwt
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So, 0! is greater than 0.5!. Interesting...

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

sleepingsaucer
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I was waiting for a mathematical proof

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

Dinesh
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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.

willyw
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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.

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

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

richardproctor
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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.

ZardoDhieldor
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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

MrRoyalChicken
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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.

andreshenao
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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.

duhboss
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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..

pulkitsingla
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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.

swordofdoom
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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.

ffggddss
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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

ffggddss
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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)

Pomme