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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 http://en.wikipedia.org/wiki/Bohr%E2%80%93Mollerup_theorem Applications https://www.math.washington.edu/~morrow/336_10/papers/joel.pdf Numerical computation http://www.rskey.org/CMS/index.php/the-library/11 Alternative ways to extend the factorial function http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunctionMJ.html If you like my videos, you can support me at Patreon: https://www.patreon.com/mindyourdecisions Connect on social media. I update each site when I have a new video or blog post, so you can follow me on whichever method is most convenient for you. My Blog: https://mindyourdecisions.com/blog/ Twitter: https://twitter.com/preshtalwalkar Facebook: https://www.facebook.com/pages/Mind-Your-Decisions/168446714965 Google+: https://plus.google.com/108336608566588374147/posts Pinterest: https://www.pinterest.com/preshtalwalkar/ Tumblr: https://preshtalwalkar.tumblr.com/ Instagram: https://instagram.com/preshtalwalkar/ Patreon: https://www.patreon.com/mindyourdecisions Newsletter (sent only for big news, like a new book release): http://eepurl.com/KvS0r If you buy from the links below I may receive a commission for sales. This has no effect on the price for you. My Books "The Joy of Game Theory" shows how you can use math to out-think your competition. (rated 3.9/5 stars on 32 reviews) http://amzn.to/1uQvA20 "The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias" is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 4.6/5 stars on 3 reviews) http://amzn.to/1o3FaAg "Math Puzzles Volume 1" features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.4/5 stars on 13 reviews. http://amzn.to/1GhUUSH "Math Puzzles Volume 2" is a sequel book with more great problems. (rated 4.3/5 stars on 4 reviews) http://amzn.to/1NKbyCs "Math Puzzles Volume 3" is the third in the series. (rated 3.8/5 stars on 5 reviews) http://amzn.to/1NKbGlp "40 Paradoxes in Logic, Probability, and Game Theory" contains thought-provoking and counter-intuitive results. (rated 4.3/5 stars on 12 reviews) http://amzn.to/1LOCI4U "The Best Mental Math Tricks" teaches how you can look like a math genius by solving problems in your head (rated 4.7/5 stars on 4 reviews) http://amzn.to/18maAdo "Multiply Numbers By Drawing Lines" This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 5/5 stars on 3 reviews) http://amzn.to/XRm7M4
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Text Comments (874)
Jaden Parrish (4 hours ago)
(1/2)!=0.8862269254528 (1/2)!=0.89 rounded √pi/2=0.8862269254528 √pi/2=0.89 rounded (1/2)!=√pi/2
NULL (9 days ago)
3:22 if f(x) is representing the factorial function, then f(x+1) != x * f(x); e.g. f(3) = 6 != 4 = 2 * f(2) ... f(x+1)=(x+1)*f(x) perhaps?
EllaNutellaOMG -YT- (10 days ago)
It's Chinese to me. Lol🤑🤑🤑🙃🙃😛😛
a2333232332 (10 days ago)
How about d(x!)/dx?
BornToRunBarefoot (17 days ago)
Wow, I have not used factorials since I was in college. Amazingly, I have not found it necessary to use factorials in my daily life outside of school.
Luca Christensen (19 days ago)
What doe "t" and "dt" stand for?
Electronic Music Wave (24 days ago)
a little mistake f(x+1) = (x+1)*f(x) not x*f(x) eg 5! = 4!*5 not 4!*4 you probably got this wrong cause the gamma func does have this property Γ(x+1)=x*Γ(x) cause it is shifted 1 to the left
BadBoy 16 (28 days ago)
The factorial of a number don't be defined only for natural numbers?
Просто Я (1 month ago)
(√π/2)² = ((√π)²)/(2²) = (π/4)×4 = 4π/4 = π = 3,1415926535897932384626433832...
shivanshu mishra (1 month ago)
I was expecting some.kind of proof
Alessandro Marchetti (1 month ago)
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.
nicolas turek (1 month ago)
one thing... if you create graph, every number in range <0;1> will has it's factorial 1
mehul singla (1 month ago)
Absolutely meaningless....how can you arrange 1/2 objects? First there was that debate on infinite series now this...stop ruining everybody’s concepts...maths is not a subject where you can bend the rules
Allan Prince (1 month ago)
Small correction, u said n! Is product of all whole number upto n, which includes zero which will make the product 0.
wither (1 month ago)
Is there factorial of π?
lyndonkimthomas (1 month ago)
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Aditya S (1 month ago)
You could have simply pulled out a curve
William John (2 months ago)
Mind you desicion why don't you make videos on interpolation methods?
THE1GAME50 (2 months ago)
hi I've proved that gamma function is exactly equal to (x-1)! just by using LAPLACE TRANSFROM. is it something interesting
Cameron Gray (2 months ago)
Hi, what is the function used for the inverse factorial?
Mind Bend (2 months ago)
Sir, which software or application are you using?
aman juneja (2 months ago)
Hey hey presh At 0:36 you said products of all whole numbers upto n which means n! Should be equal to zero but in this video you have shown n!=n(n-1)....(1) Whole numbers include 0 and natural numbers do not include 0
John Nastrom (2 months ago)
I realize this video is 4 years old but honestly this is probably the worst video on your entire channel. The basic question of what it even means to take the factorial of a non-whole number was not even addressed.
Stephen Devaraj (2 months ago)
if factorial follows graph pattern then how can 0!=1 becz the graph goes up now ???
farefouse olly (2 months ago)
so there isn't an answer so we decided to use a similar function that has the same outputs at the integers?
Dantè Pillon (2 months ago)
I got lost after he mentioned log
TIMIR SHAH (2 months ago)
I m getting 0.00000000001 factorial = 1 How ..
Vaibhav Waghmare (2 months ago)
I am here for proof
MIKELIAN HAJRULLAI (2 months ago)
How is e! ?! And pi! without used the calculator ?! ...
bob53135 (2 months ago)
Using a serif x instead of the proper multiplication symbol × (or at least a non serif x) in a math video should be illegal.
Nicht von Bedeutung (3 months ago)
Hmm... so I know, what this is all about - like before. I also know, that Γ(1.5) is not the minimium of this function, beause this minimum is 0.885603... at Γ(1.461632...). So the value of sqrt(π)/2 (0.8862269...) have also to be reached by another value smaller than 1.461632... how can be determined, how big (or small) this value is (except by "higher or lower"-tests)?
Thom C (3 months ago)
So when does n! reach its lowest value? At like 0.6! ?
Minho Kim (3 months ago)
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?
Степан Репин (3 months ago)
Factorial is defined only for positive integers and 0.
Dakshi R (3 months ago)
f(x+1)=(x+1)f(x)
sayan dhar (3 months ago)
which software u use plz tell
Nathan Leonard (4 months ago)
I thought we were going to see you do the integral.
가우스카를 (4 months ago)
I know that factorial is only for N numbers. Is it posible to factorial for Q or R numbers?
Abhishek pandey (4 months ago)
worst explanation
Ashish Jha (4 months ago)
where is the proof?
Priyanshu Singh (4 months ago)
0:35 Not whole numbers Natural numbers
Chraman (5 months ago)
I don't get the f(x+1) = x(fx) part. It seems wrong when you enter numbers
Wangshuo Leon (5 months ago)
use the gamma function
Jai Sreedhar (5 months ago)
Wow man do more vedios
Darshil Shah (5 months ago)
F(x+1)=xf(x)!!!?wrong
Olivér Kovács (5 months ago)
In this formula, what is the t?
Anshul Jain (5 months ago)
How to solve m^3+m^2+1=0
rajbir sethi (5 months ago)
Factorial of 1/2 is less than factorial of 0?????
HYPISK - Simply splendid (5 months ago)
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?
baflah aissa (5 months ago)
Thank you. I would know about the program you make your interesting video.
Rakshith S (5 months ago)
hey that was really cool but why is f(x+1)=x*f(x) n not =(x+1)*f(x) like 9! is 9*(8!) ??
VJ Krish (5 months ago)
If fact(0.5) is 0.88 something, then how come fact(0) is 1?
News Boosters (5 months ago)
shouldn't f(x+1) be = (x+1) f(x)
lyndonkimthomas (5 months ago)
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9CBGamer (6 months ago)
So. means 0.5! = √(π÷4)
Topher TheTenth (6 months ago)
What is "t"?
Billy C (6 months ago)
γ(x) = ∫ 0→∞ t^x-1 e^-t dt x! = γ(x+1) (½)! = (√π)/2
Marcelius Martirosianas (6 months ago)
PROTESIME toliau ||||1/2!=1 |||| tada: tada atsakymas yra::: ||||||||| 12 |||||||| epistematFilosoficallogics analizes atlijejas yra MARCELIUS MARTIROSIANAS , LIETUVOS RESPUBLIKOS PILIETIS Prof. A. Aksuomaitio mokynis nuo 2oo3 m.========
Marcelius Martirosianas (6 months ago)
1/2!=(Pi)^1/2/2=[[1/2!=1/2Pi/2]=[[1/2!=Pi/4]]=[[1/2!=4/4]]={{{1/2!=1}}} ,kadangi Pi=4 atradejas ,kūrejas , MARCELIUS -------------- MARTIROSIANAS (TRIGONOMETRIJOS KLASIKINE MODELIO ATRADEJAS, 2006 , BIOMECHANIKOS kūrejas 2003-03-11------ LIETUVOS RESPUBLIKA. , MATEMATINE LOGIKOS kūrejas 2009: VILNIUS, LIETUVA autorius MARCELIUS MARTIROSIANAS,
KingMidaz Olam (6 months ago)
I get: sqrt(3.14) / 2 = 0.88622692545275794095971377828391268849372863769531
Craig Chamberlain (6 months ago)
What calculator are you using that gives 32 digit answers?? Is it available for free under most recent versions of Windows?? Can it do 32 digit answers for, say SIN, COS or TAN? How about Logs? If it uses the Gamma function to compute factorials, does this calculator use other advanced functions for most of it's calculations or just for some? Even my TI-Nspire CX CAS can't compute to 32 decimal places!!
Nikhil Nirmal (6 months ago)
Must watch channel Nikhil Nirmal Lot of geometries researches are there.Also see 18°72°90°Traingles Theorem.
MultiRobotnik (6 months ago)
Nice animated hand there.
Jan Lewandowski (6 months ago)
It triggers me as he uses "x" for multiplication xD
Vishwanath GK (6 months ago)
Brilliant !!
Goldstein Gaming (7 months ago)
Alternate definition of gamma :- 2 times the integral from "0" to "infinity" {(pow(u, x^2))(pow(x,(2n-1))(dx)}
Nightmare RO (7 months ago)
I'll wait a few years to understand this
the Kashmiri (7 months ago)
😱😱...what is this gamma function?
Benjamin Karazi (7 months ago)
Again, Pi is a standardization value (1/2! ...!!!?) and cannot be used to solve any problem or equation that contains a curve, circle, sphere, torus, etc. With a value that does not end and →∞, your scientists-science is very ignorant and foolish, PERIOD. There are infinite ways to get the Pi number!
Tim Tran (7 months ago)
Que?
Billy C (7 months ago)
0.5=0.88622692545275801364901364908374167057
Billy C (7 months ago)
3!=6
Billy C (7 months ago)
1!=1
Choi Yat Lam (7 months ago)
I would like to ask that in the gamma function, is f(0) still equal to 1. As 0!=1 by definition. If so, how would the curve look like. Just curious as you haven’t mentioned the case of 0!
John Tate (7 months ago)
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.
Ady73 (7 months ago)
I was waiting for the connection between Pi and factorials and bang, video end. The connection is only at 0.5! or it happens at any R! ? I'm waiting for more details please.
Alexandre Rocha (7 months ago)
i´d like to have seen some answer...
krishnendu Roy (7 months ago)
😵😵😵😵😵😵😵😵😵😵
Venkatesh babu (7 months ago)
Halving things lead to that number and if you take four times it is the total amount of the value of full factorials.
Subscribed!
Bana na (8 months ago)
00:00~3:35 hmm okay i get it 3:38 what
VENOM 86 (8 months ago)
Hey f(x+1)=(x+1) fx You are wrong
Hùng THV Trần (8 months ago)
1:25 and 1:27 it isn't the result of half Pi because you has the result of 0,8862269...057 to the power of 2 and at 1:35 and 1:37 you multiply it by 4 so the factorial of one half should be (Pi divided by 4 and divided by 0,8862269...057 because 0,8862269...057 multiply itself equals 0,7... And multiply by 4 equal Pi)
Anita Gofradump (9 months ago)
I recently learned about taylor/maclaurin series in calculus to make power series, is this sort of like a similar idea except obviously you cant derive the n! 'Function' we start out with
Carlos M (9 months ago)
1! = 1 (3/4)! = 0.919 (1/2)! = 0.886 (1/4)! = 0.906 0! = 1 -(1/10)! = 1.068 -(1/5)! = 1.164 -(3/10)! = 1.298 -(2/5)! = 1.489 -(1/2)! = 1.772 -(3/5)! = 2.218 -(7/10)! = 2.991 -(4/5)! = 4.591 -(9/10)! = 9.513 -(19/20)! = 19.47 -(199/200)! = 199.427 -(1999/2000)! = 1999.423 ... -0.99999999999999999999999999999999! (thirty-two 9's) = 99999999999999999999999999999999 (thirty two 9's) -0.999... = 9999... -1 = Infinite? Nope, just Error Well, that was math logic! (I checked all factorials in my computer's calculator)
Wassup Yo (9 months ago)
Wait... but you multiplied it by 4
Bill Russell (9 months ago)
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.
DaJackob (9 months ago)
i am exactly the 7000th like.
Nickjc1999 (9 months ago)
one fatal flaw with this. Its the factorial SEQUENCE, not the factorial FUNCTION. Therefore, since it is only defined by the natural numbers, and sqrt(pi)/2 isn't a natural number, the correct answer to (1/2)! is NA.
Hassan Mohamad (9 months ago)
pressure locker ??
Casper YC (9 months ago)
This video is a waste of time because I can just look it up on wiki, which has MORE explanations. You practically has done nothing here other than Giving The Formula.
VAT1N (9 months ago)
There are 0,00000000000002… difference between pi and the result of the equation
MotorMusic (9 months ago)
Shouldn’t the constraint be f(x+1) = x+1(f(x))
Algy Cuber (9 months ago)
i think the pi function is better
studio preesha (9 months ago)
Ok thanks
James Clark (9 months ago)
So the factorial of 3/2 is 3✅pi/4. Good to know.
Steve Applegate (9 months ago)
The value of 0! is 1, but the value of (1/2)! apparently is less than 1. Could you explain this?
Mako Duka (10 months ago)
Mali bob
MrHatoi (10 months ago)
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.
ophello (10 months ago)
Thanks for nothing, Presh.

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