factorial of sqrt(2)?
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- เผยแพร่เมื่อ 22 ม.ค. 2019
- What is the factorial of the square root of 2? Can we do factorial of a non-whole number input? Learn more fun math on Brilliant via brilliant.org/...
The sqrt(2) factorial is defined based on the extension of the factorial with the Gamma function. We actually get the improper integral from 0 to infinity of t^sqrt(2)*e^(-t). We can approximate this integral by using Simpson's Rule.
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Q: What's the best factorial so far?
(A) double factoreo
(B) subfactoreo
(C) hyperfactoreo
(D) sqrt(2) factoreo
Hyperfactoreo.... Because it sounds good 😅😅
I choose (E) Double Stuf Factoreo
(Ω) expofactorial
Please replied to my massege on instagram
misotanni That is simply equal to the product starting at s = 0 and ending at s = ceiling(x/SqRt(2)) of n - s•SqRt(2)
Γ(√2+1)
gamma(2.41...)÷2=sqrt(2)!
Pi(sqrt(2)) is elite
Wdym ?
@@samueljehanno Г(x+1)=x!
Oh woops, I thought ‘nah,ez’, but the factorial was outside the sqare root 😂
😂 haha
i! ?
see description.
@@blackpenredpen oh thanks!
6:07 I didn't know you swear when it's about mathematics
That sound clip of the kids cheering cracks me up every time.
Good lesson....I didnt think the integral would converge so quickly tho! So yeah 10 is def the better choice
: )))))
Cool that you are extending factorial beyond the natural numbers without resorting to the gamma function with its unnecessary and confusing minus one. BlackPenRedPen- tidying up mathematics one small step at a time. :-)
With all due respect, what's the difference between the definition of the gamma function and what he wrote on the blackboard?
If you don't like the name Gamma and you want to call it Susan, it's fine with me, but it is still the same function.
On the other hand, if you want to call it "factorial" when the factorial function is already well defined over and only over the positive integers, that's just stubborn nonsense.
Hi Sergio, For some weird reason youtube was interpreting minus one as a strikethrough so I've spelt it out instead. I'm not a mathematician but mostly when functions are extended from one domain into another, they keep the same name. Giving factorial the name gamma and requiring it's argument to be offset by one is confusing IMHO.
@@DeclanMBrennan
:-) Now your post is tidy and clear.
You are right. Analytical extensions do not change the name of a function. But these two are *different* functions. Gamma(z) is defined as the famous integral, while n! is defined as n*(n-1)*...*3*2*1, and you finish in 1. When do you finish if the number is real, or even worse, complex?
Let's assume z=3+2i.
Then z!=(3+2i)*(2+2i)*(1+2i)*(0+2i)*(-1+2i)*...
When do you finish??
On the other hand, to calculate Gamma(z) you just plug the complex number in the integral and you solve it. Can you see the difference? :-)
I acknowledge that in the special case of natural numbers
Gamma(n)=(n-1)!
(that's the minus one that bothers you!) but that is just a coincidence, it doesn't mean that Gamma is the analytical extension of Factorial... they are defined in a different way!
For example, let's say that you have a problem in physics which solution is the zeta function of Riemann. Zeta is defined for complex numbers with real part positive. In some circumstances it is valid to ask "what happens if I extend the domain to the complex numbers with real part negative"? And then you use the analytical extension. But if the nature of your problem demands to multiply integers in descending order, you can't just replace the Factorial for another function (which happens to give you the same result at natural points) and expect to obtain a reasonable result. That would be like replacing sin(x) by tan(x) just because they give you the same result at every pi*n... that would be simply wrong.
Sorry for wittering on so much :-) but the definition for gamma also makes other formulae a bit more messy than they need to be, for example the volume of an n ball with radius R is:
pi^(n/2)
------------------------ R^n
gamma(n/2+1)
Using extended factorials this is:
pi^(n/2)
--------------- R^n
(n/2) !
which seems more symmetric and easier to remember (at least for me).
(Of course we could use pi=( (-1/2) ! )^2 to have a formula with no pi and just factorials and powers but that's a digression. )
That's pretty cool. I saw 3b1b's video about the exponential function as a power series and now I realised the gamma function is probably a way of rearranging that! Pretty cool.
How many ways to arrange sqrt(2) object...????
Brain lagging
Obvs 1.2538 ways
can you solve ¡i!
and written as that, should we read it as (¡i)! or ¡(i!) ? OR IT DOESNT MATTER? :OOOOO
Thanks!
of course it matters.
by your notation seems to be the second.
the first would be the same process here with e^(-π/2).
the second would be doing this process with i and then getting i to the poer of it
Can you derive the gamma function?
d/dx( gamma(x) )
your welcome
@@Tomaplen Dude, thats the derivative not the derivation...
Mathedidasko That's the joke. He took the derivative.
@@angelmendez-rivera351 Ha. very funny
The gamma function is not derived from anywhere... it is BY DEFINITION that integral.
Confirming that it's not worth trying 0 to 5 in 40 steps ... I get 1.168. So, quite right that 10 is 'just close enough' to infinity ...
Keith Brain
Yup! : ))
別忘了順便來這裡訂閱一下中文版的! (還有記得打開小鈴鐺)
What if you use the Gamma function?
You would end up with the same integral.
Both are same
Have you done i factorial?
I recorded and will post it this weekend or so.
@@blackpenredpen
I will be waiting💕💕
@@blackpenredpen I will be also waiting :D
Interesting question. Without seeing your answer, I'm thinking 1st try should be better. The integrand is largest around t = sqrt(2) so most of the quadrature error should be there. I expect that to be more significant than the missing tail since the integrand is already pretty small at t = 10.
Moral lesson: quality (n) over quantity (interval)
ClubstepDJ what is n ?
skalderman it's how many times you do the simpson's rule, the more the better approx.
Congratz with 6!k Subscribers!!!
Dont know If Im wrong, but t^sqrt(2)=(t^2)^(1/2)=t
Then you can do int. By parts, letting u=t, dv=exp(-t) and in the old manner?
Edit: I see the problem with \infty now in The first term of int by parts..
You may have already figured, but there is no rule that a^(b^c) equals (a^b)^c.
Yay! Another bprp video!
(√2×1)(2-1)
"Did you figure it out?"
Do a video on the euler sumattion formula!!!
What if you took the Taylor series of t^√2 to create a weighted sum of factorials? How well would that converge?
Taylor series centered where? Not at 0 since the second derivative of t^√2 has a pole at 0. The convergence of any Taylor series will be pretty bad for that reason.
Would you be able to find the gradient at any point on the factorial curve by taking the derivative with respect to x of the Pi function?
Shoot a video about what is t:a^b=b^a*t
I am a vet ut watching for fun and ı like math, thank you.
: )))))
Its of course the 1rst one. The function converges very quickly, and a bigger n in comparisson to the interval gives a better aprox.
Well i have this question that plays in my mind... youhave gamma function for factorial BUT is there any inverse approcimation for factorial?? Gamma function does not have any inverse function but is there some function that does an approximation at least??
Okay, can you show us, if we make x = 3 or any integer, using that intergal we get 6 which is 3 factorial?
This one's best
Factorial(√[-1]). OR
Factorial(i)
I'm interested in what you had hidden behind the beep sound.
Your channel is F A N T A S T I C
You could See if pi factorial is related to pi I’m any way
I know this one. It's equal to pi factorial, so yes.
that beep freaked me out
√(2) > √(2)!
Can make a video on a factorial of a really small number?
Can you make universal solution to the integral: exp(f(x)) ?
When I saw this I was like
Wait that's illegal
Please solve: A(subscript n) =[a(subscript ij) ] , where a ( subscript ij) = (3i - j) /2^2n for all i, j , both i and j greater than equal to 1 and less than equal to n. Then value of lim n tending to infinity trace { 2A1+2^2 A2+ 2^3A3+ .......2^n An}.
This took me ten minutes to type. I think there is something wrong with the problem because you can't add matrices of different orders. So the traces must be added separately . It should be trace of 2A1+ trace of 4A2 and so on. Please please please do this.
Just one question. What is t?
Trying to learn how to do:
16!17!
---- = perfect root?
2
Is there a shortcut to find if that is a perfect root or not?
ik this is a bit late but:
16!*17!/2
17! = 16!*17
16!*16!*17/2
16!^2*17/2
17/2 is not a perfect square, so 16!*17!/2 cant be either.
Why do you use Simpson's rule, why can't you evaluate the improper integral as a limit as R approaches infinity? Pls don't roast me I just started Calc 2
Because we can't integrate that function with our usual techniques. That's why I chose to just approximate.
blackpenredpen thx for the reply ily daddy
You CANNOT evaluate such an improper integral without numerical approximations. It is impossible.
Hi can you integrate (tanx)^1/n
Find the a number so that the factorial of it yields pi!!!! Make a video on x!=pi
I tried to find i factorial in internet but didn't understand them well so plz make a video on i!
My calculator has an maths error with doing this.
Nice... use the puzzling "sqrt(2)!" as the hook, then use it to deliver an unexpected lesson on the pitfalls of numerical integration.
Hello sir, I have a challenge for you: integral of (1/(a^3-x^3))dx
question: how to solve the infinite series of (-1)^n sin(n)/n ?
Is there any physical significance to this? Does this apply anywhere for practical purposes? Don't get me wrong I was curious to know.
Of course. Integrals that arise science are often difficult or impossible to evaluate analytically. You (or more likely your software) use numerical estimation all the time, and there's always a question of what's the most accurate and efficient way to do it.
What is i!, !i, i!! ?
Anton da Fuchs In the description, he has a link to a video on i!. We have yet to do !i
...!i!﹏i!i...
i terms
Plz do videos on PnC
I am not sure what it is.
@@blackpenredpen permutations and combinations(in other words good probability and game theory questions
Indian preparing for jee ,right?
@@anweshaguha7366 yes (lol how did you come to know)
Brindaworg
What about e!, pi! or phi! ?
Great video love the factorial work
Ah, back when the shamen of calculus's beard was still but a baby 🥲
first variant. because e^(-t) fast limit 0.
I think the first try is better. I've seen a graph of the inside function and it has little area under the 'righter' part of the curve
Yep, as I expected
# Copyright 2020.11.04. 신촌우왕TV수학자.천재작곡가 All rights reserved.
# Python Code: 임의의 실수(Real Number)에 대한 SQUARE ROOT 값을 구하기
# sqrt("207.109", 50) = "14.39128208326137980853764912599305145192129266880843"
# sNUM: 문자열 형식의 입력 숫자 (예: sNUM = '207.109')
# nDIGIT: 계산결과에서 원하는 소수점 이하의 자리수 (예: nDIGIT = 50)
def sqrt_by_WooKing(sNUM, nDIGIT):
u = ''
v = sNUM.split('.')
# sNUM에 '.'이 없으면
if len(v)==1:
u = v[0] + '.' + '0'*(nDIGIT*2)
elif len(v)==2:
if len(v[1])
This one pls
x+x^2+x^4+x^8+x^16+x^32 ...
|x|
What is d/dx x! ?
Is it possible to find a derivative?
niema nd It depends on how you define the factorial. If you simply define it as the map !: R -> R, then x! = Integral shown in the video. Then you can obtain the derivative and get that d/dx[x!] = x!(-γ + H(x)), where H(x) is the Harmonic number function, and γ is the Euler-Mascheroni constant.
You can take the derivative of x! if you can define the digamma function. For more information, here is a Wikipedia article about the polygamma functions: en.m.wikipedia.org/wiki/Polygamma_function . The derivative of x! (technically its the derivative of the pi function since x! is only defined for the integers, which makes it non-continuous and have no derivative, but the pi function is a continuous version of x! so we can take the derivative of x! in that sense) is π(x)ψ(x + 1), where π(x) is the pi function and ψ(x) is the digamma function.
Can you make a video about the Fibonacci sequence? (in a complex way) i don't know kk. Just talk about Fibonacci sequence, please.
Thought you’d prefer the Lucas numbers
6:07
So is 2=2!
Great. Really Great!!! :D
sqrt(2)sqrt(2)=2
I actually like your channel, but common. Seriously? Just plugging in in WolframAlpha? „Today we’re going to calculate the integral from 2 to infinity of sin(x^5)/(xln(x-1))dx !!!!“ „Now plunging into WolframAlpha we get ≈0.076 and we are done“ You didn’t do any math in this video. Just plugged the question into WolframAlpha, still there are many good videos on this channel.
Udv method
π! And e! ??
or do the 3rd try and you'd belive you'r going in a increasing way or decresing one
oh yeah yeah
In Gamma(x) the power on the " t " is x -1 isn't it ??
Yes it is
second one
DONT ASK ME WHY
MY SUBCONSCIOUS SAYS ME SO
1
Shouldn't it be `t^(x-1)` instead of `t^x` for the integral to evaluate to `x!`?
That evaluates (x-1)!
That evaluates Γ(x), not x!.
Angel Mendez-Rivera Yeah but my phone (annoyingly) doesn’t have a “math symbols” keyboard.
Dos!
Product formula works best, need use outside calculator or computer bet is till best algorithm to calculate non integers factorials.
Arturas Karbocius
what do you mean by product formula?
do you mean x!=x (x-1) (x-2)... ?
if so, that algorithm needs to end before you get negative terms. In the case of natural numbers, 2!=2 and the algorithm finishes with the exact solution, but with real x, how do you get the final approximation?
@@sergiokorochinsky49 m - any positive number from zero to infinity, m!=Product (range n=1 till infinity)=[(1+1/n)^m*(1+m/n)^-1],
Is a series a poor integral approximation?
The series 1.2538 + 0 + 0 + 0 +... gives a pretty good approximation.
Wait don't factorials work by multiplying it by the previous number
So wouldn't it just be root 2? LMAO
#beginnerproblems
"don't factorials work by multiplying it by the previous number"
Not quite. You multiply by the factorial of the previous number. The way BPRP does it we do in fact have sqrt(2)! = sqrt(2)*(sqrt(2)-1)!
Why can't we use integration by parts?
Bandana Shrivastava because sqrt(2) is approximately 1.4 so if we set t^1.4 as f(t) then f'(t) will be t^0,4 if we integrate by parts one time more we have f''(t) = t^-0,6 and so on...
At n=10 yousing 20 and n=20,10😇
I’m gonna guess that the first one is the better approximation. After all, e^t gets really small really fast, whereas t^sqrt(2) is not even at 100 when t = 10 (compared to e^10, which is probably already in the 10 000s), so only considering inputs near the origin will probably not affect the result as much as using only one parabola for every input step of size two.
What about ((1+sqroot5)/2)!
Don't tell me you have changed the way of presenting videos!
?
I think he is talking about you not showing your work but just approaches and results
@@blackpenredpen it's just that I have been following you for quite some time, but today it felt different.
first one because the exponential is decaying so fast.
What about (-sqrt(2))!?
my guess is 2nd try is better
oh well. i guess i should have known that it converges quickly
Do i!
guess the second one...
I like it .
BLACKPENRED PEN .YAY!!
I think 1st is better
Yay, I was right xD
So, my logic was that the interval being bigger but the number of sub-intervals being smaller would lead to bigger error. Simpson's rule would definitely have some error when approximating that function, and the error would be made smaller if your n is bigger.
I want i! i=sqrt(-1)
Vous pouvez traduits vos vidéos en français s’il vous plaît prof ? Ça sera mieux merci
Idk why i watch this videos if i dont understand abything
fun fact oreo
But why tho
Tatatata
1st one