But HOW did Euler do it?! A BEAUTIFUL Solution to the FAMOUS Basel Problem!
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- Опубликовано: 23 май 2019
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Sine taylor: • The Sine Function and ...
Sine Product: • Deriving EULER's INFIN...
Cotangent: • The Cotangent's Series...
Basel Problem: • The Basel Problem & it...
Today we are going to go bacc in time! Following in Euler's footsteps, we are going to solve the basel problem using the weierstraß factorization theorem. Decomposing the SIne into its linear factors and the comparing coefficients with its also established taylor series expansion is going to be the key in finding the peculiar value of pi^2/6 of zeta of 2/The sum of the reciprocals of all the natural numbers squared! Enjoy! =)
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Euler dont need rigor as he was born with a divine mathematical intuition
Intuition drives innovation, but rigour keeps the system working
"They say that effort breeds success, but that's a complete lie; the world is not that accommodating. People with talent don't become talented, they're just born with their abilities right from the start".
Why do you say that..how can I have the intuition he had?
@@Sir_Isaac_Newton_ says who? How do you know that's true?
@@Sir_Isaac_Newton_ absolute bullcrap
That last part is mind blowing. Makes me think that Euler just played around with the sin function and that his result was "just" a byproduct of his experimentation. Really amazing
fun fact: euler calculated the first 16 digits of pi^2/6, before developing his actual proof.
Teachers: only first graders can produce unrigorous proofs
Euler: hold my π²/6
I must promote the chap I chatted with who really did what he said he did, just like me. A fucking genius. Pi IS 3,16 or 16/9. He finished the history of human astrological observation and fixed so many things we take as true. I am rewriting math and physics for RATIONALITY in both as well as language. Stay tuned, and as long as channel metrics decide truth, get ready for aliens.
@@dsm5d723 Pi is 3.141592653 and 16/9 is 1.78
@@ryanjagpal9457 It is his thing. i will give his screen name. He does have the perfect explanation of pre-literate astronomical observation. It is the obverse of my thing, Dimensional Gauge Symmetry. Three rational "irrational" gauges, e (c), sq2 and Pi. I was working with the ERRORS of modern math, and I did find them. Add the three numbers to the third decimal and the first 1 from the prime sequence and you get a mathematical model of a dynamic dipole, plusy dynamical friction embed in the Euclidean Plane. Math with out a dynamic explanation of physics is not measuring anything but math.
1+0.577+1.414+3.141=6.132
And I am working on the last bit of rationality in the modern paradigm, and it has to do with the resolution hidden in the "irrational" decimal expansion of these three numbers. Repeating is not understanding. I finished Einstein and Poincare with the Tesla Identity Matrix Determinant.
gab.com/23andMe24andYou/posts/105477983888996278
@@ryanjagpal9457
DysonTorus Tesla Code is 3r 6r and 9r
5 days ago
Ed leedskalnin.
Coral castle
π is 3.16r
Tau is 6.3r
Everything is out by 1.1
1.1s
66.6s
66.6m
22.2h
333.3days
We are in 2222/3
Egyptians used 3.16
I recurred it and tested manually.
If there's missing math then there's missing km2 of earth
Eratosthenes was 10deg adrift lattitude as the magnetic equator is the real equator.
DysonTorus Tesla Code is 3r 6r and 9r
4 days ago
@DSM 5D best comment I've ever heard. Thanks buddy! Now find the sq root of 10. 3.16227766. That's just using my phone. 3.14159 is the error to hide 47m km2 in the south and is hidden in the north, or just missing full stop. 111.1 km as everything is a ratio of 1:1.1r mostly in my model. 10deg passed np from UK is let's say 111.1km, that offsets everything. Bit if there's 26666.66r km from true south to true north as 6666.66 is deci more than the famous 666 or 666r. It's all in plain site. The moon is the ruler. Look up first use of lunar calendar. It's way older than we've been worshipping the sun. Base the whole geo model onto the moonpole or monopol'y' as I call it. Gets really interesting. Please sub as all my videos are going into one amazing presentation noman has ever thought of since 4236BC
DysonTorus Tesla Code is 3r 6r and 9r
3 days ago
@DSM 5D Topman. I like Ur style. Babylonians. It's all about time line. 60 didn't fit in with lunar. Sumerians were 3000bc. Egyptians used lunar before 4236bc and Scotland found evidence of the lunar calendar 8000bc at least. So 60 base started 4236 by Egyptians. 365 calendar was 4236bc as well. I claim they could never figure it out. Without studying the world, they would have never have know the full path of the moon. We do. We should be using a 100 based system for time which is navigation. Because we don't, I have proven 10deg x 4 is missing at both poles. How did they hide 48m km2. Through assumption of 40000km. Well Magellan proved equator was way way shorter than 40000km. I proved it 100% in my day 2. It's all in plain sight. Nothing is hidden. We just aren't looking
@@ryanjagpal9457
DysonTorus Tesla Code is 3r 6r and 9r
3 days ago (edited)
@DSM 5D
I'm a cook with south African education. 86-94.
19/6 is 3.16r
Manually 3.16r was bang on
π was .08% out. I used a plate and tailors measure.
I'm stuck on why perimetres change with shape change but not area?
Given it a rest for day.
Eric verlande talked of entrophic gravity. Will reply more later on. Thx bud. Heads going wild with numbers. What's these prizes? Clay math is who I emailed.
Other mathematicians: QED
Flammable: its pretty f*cking dope
Everyone is scared of swearing on youtube except math channel wtf?
well this guy was once FAPPABLE maths if i recall correct you, so yeah Jens isn´t the guy with the best filter^^
@@Metalhammer1993 He is accurate, which is the most important thing. This IS pretty fucking dope :)
It's because he found a proof to get away with it.
@@Metalhammer1993
Well that name isn't wrong. This shit gives math boners.
@@thomasrad6296 Don't you mean he found a "proof way" to get away with it?
Just a math puns!
I thought you were going to write sin(x) = x at the beginning, I think I'm too involved in my physics degree it's becoming an issue
Haha
x is a perfectly fine approximation of sin(x) for values of x close enough to zero.
Sionae 😂😂😂😂
I had to stop and rewind to 15:35. My brain was automatically rounding. It took me more than a few seconds to shift gears.
Well, for a sufficient small enough interval of x values around x=0, you can replace the "=" sign with a "~" sign
Euler? Nah...
Wheeler? Perfection...
1:04 hoyristically
*You* ler? Nah...
*We* ler? Perfection
*_ussr intensify_*
@@arnavanand8037
Oil er
USA intensifies
"If two functions have the same zeros, they are basically the same". Amazing. New theorem for engineers! (Notice: x = x^2 :)
LMAO
"If two polynomial functions have the same zeros, then they are basically the same, if and only if their coefficients and degrees are the same."
@@kuronekonova3698 actually, if two polynomials have the same zeros and their degree is the same, they are the same polynomial, hehe
@@ernestomamedaliev4253 not necessarily cuz you can multiply any polynomial by a constant to get a new polynomial with the same degree, same zeroes, yet different. I think if two polynomials have the same degree and (complex) zeroes, they are proportional to each other by some constant.
@@snootiermoon yeah, thinking abou that, I guess you are right. We need to specify that the coefficient of the maximum degree term is 1 in order to establish what I said earlier. Thank you for the correction! 😉
I love how u were so rigurous at the end with the Peano axioms and stuff to compensate for the cancer and headache that the unrigurously pi^2/6 proof gave me
Lmaoo
LMAO 4:43 AM I THE ONLY PERSON WHO NOTICED PAPA FLAMMY WAS USING THE FUNDAMENTAL THEOREM OF ENGINEERING? XD
Mathematicians: "This expression isn't well-defined."
Euler: "But what if it was?"
Physicists: "No biggie. All we have to do is multiply and divide this by infinity (because it's not equal to zero) and we get the charge of an electron."
Pretty much, ever heard about renormalization? Essentially you just "hide away" some term that blows up to infinity and the leftover is your answer :D
Luigi T. Sousa
In the words of Andrew Dotson: Ree-normielization
@@MessedUpSystem Yeah! We use this idea of renormalization in Asymptotic Methods, one of my modules. More specifically, finding solutions to small perturbations of Duffing's equation in which a straightforward expansion ansatz gives rise to a non-uniform solution.
Papa Euler was truly a genius. Just a comment for all of you boyz and girlz watching this. By using the exact method shown here you can derive what the values of zeta(2n) are i.e. zeta(4), zeta(6) etc. by comparing the coefficents of the part of x^5, x^7, x^9 etc.
Or you can use papa fourier's series.
You can still compare but the results require further insight to get. Try.
Wait what is zeta?
@@ryanjagpal9457 C'mon, every body knows the zeta function! Even a 3rd grader!
@@jkstudyroom How can a third grader know that?
Pretty sure they should be learning how to write by then
Idk where you go to?
Using 1/n^2 for thumbnail but 1/k^2 for video? Disliked, don’t need unreliable people in my life rn
Variables vary too much, so unreliable.
LMAO this made me laugh harder than what I thought
n, k, all much the same just letter placeholders for some variable. Get used to it, or you'll end up exploding in flames in your life.
Easy... Good that he didn’t use x or y instead of k, then n to solve it😂😂😂
K
Lol at the end i was like: wait.. that's it?THAT'S IT?¿?? ¿?
THAT'S AMAZING
I said, "you're shitting me?!" My 9 year old says, "Dad, where did you think he was headed?"
Could you do videos about Functions of Several Variables and more fun stuff? Absolutely loving your videos
4:45 glad to see Papa Euler knew the facts
Omg that Taylor Swift meme i'm crying
Was that some math joke that my dumbass producer mind won't get
@@williamrichmond814 Taylor series expansion
I think Euler used the sinc function (sin(x)/x) to reason about the constant multiple in each root in the infinite product (i.e. (1-(x/kpi)^2) vs (x^2 - (kpi)^2) vs all other constant multiples) which sort of justified why each term in the product looks the way it does.
Euler 'used' the "sinc-function" 'quite often' , e.g. : cos (na) + i * sin (na) = [ cos(a) + i*sin(a) ]^n --> set here a = x/n , with a fixed & real x --> cos(x) + i*sin(x) = [cos(x/n) + i*sin(x/n]^n --->> sin(x/n) / (x/n) --> 1 , for x/n --> 0 , i.e. for n --> inf --->> so asymtotically 'we' have sin(x/n) ~ x/n , moreover 'we' see / "know" that cos(x/n) ~ 1= cos(0) , for large n --->>> ; so it's "plausible" to write : cos(x) + i*sin(x) = lim [ 1 + ix/n]^n , for n --> inf , thus 'we' get a 'definition' for the exponential function [ on which the "Euler method" for solving ODE's numerically is based ! ] : e^x = lim [ 1 + x/n ]^n , n --> inf , according to the last "well known" limit ... !!! { exercise : show that lim [ 1 + x/n ]^n = sum(k = 0 to inf) x^k/k! , n --> inf }
YoU dOnT nEeD rIgOuR wHeN yOu'Ve GoT aUtHoRiTy
BISS
When you’re Euler, you tell both the steak and the eater what’s up.
What's going on smart people, today we start a meme war with 3 competitors including 'tis boi, send him some love for power
This is my first time seeing the product function in action. I knew what it was, but I haven't necessarily used it much. You made it very easy to understand. So thank you for that. :)
Euler did this whole thing in his head for sure :DDD Truly a mathematical genius
There are many other solutions as well but this is possibly the simplest solution of this problem! Nicely done!
This is so great to have freely access to such content. Thank you very much, this is really interesting.
My grandfather told me about the difference between two squares when I was about 11 or 12 - while I was helping in my grandparents' garden, actually while making the bonfire for the garden rubbish! It is a very useful tool.
Euler knows how to use ultra instinct in mathematics.
Never knew it was that 'easy'. Thank you for your work. Even though I am passionated about maths I do not study it and videos like this are pure gold for me.❤
Finally...
The video I was waiting for...
I thought when you share about sine product I always thinking about when this video realise
Beautiful proof by polynomial coefficients comparison. Very neat and doesn't require any geometrical construction. Thank you for this lecture.
pi^2/6 : exists*
Oiler: hmmmmmm
i'm studying to start undergrad Maths this year and this video made so many things click into place I'm a little blown away
Euler flaming past the screen never fails to make the highlight of my day. WHOOOSSSSHHHHHH!!!!!
Sehr schön und verständlich erklärt. Gratulation.
Nice presentation of a truly beautiful derivation. Thank you.
From a US HS tutor's point of view, I've noticed that many Asians as well as students from Europe write their "x" by writing a backward "c" then a "c". Also, noticed that the integer set is written as a "7" then an upside down "7". I will have to use this notation for the integer set next time!
Hooooooooly shit!
That was absolutely stunning. I've got goosebumps now. Good job!
Du hast mir geholfen bei meiner W-Seminararbeit über das Basler-Problem. Danke!
What's nice is that the same approach for the zeros of the cosine function can be used to get that the sum of 1/(2k+1)² from 0 to infinity is Pi²/8. Then it's easy to realize that the even squares are 1/4 of the sum of 1/k². From that it follows that sum of 1/k² is (4/3) of Pi²/8= Pi²/6.
I found a French article which showed a method that allows one to calculate zeta(2) when one knows what zeta(4) is, and vice versa. It came up when I was trying to integrate Planck's Law, and did not just want to simply write down the value of zeta(4) written in the book. So... now that when people ask me to calculate the value of zeta(2) or zeta(4), I just claim that I know the other one, and use the method in the article.
could you send me that article please?
@@eliaschavez364 Glad to, it is "Quelques conséquences surprenantes de la cohomologie de 𝑆𝐿_2(ℤ)" by Don Zagier.
Never seen before. That's beautiful!
lol "a regular third grader can do that", don't know where tf ur living my man
Germany switzerland
Nah that would be impossible to understand it at that degree and plus how are they gonna reach the blackboard
I solved it in 10-th form at school
@@firi4737 is that basically year 10?
Pretty late, but I‘m from switzerland and in 11th grade... that stuff‘s pretty simple
Hey this was posted on my birthday! Love this proof :)
What a great job, guys!
Taylor joined the video
My calc teacher showed my class this back in the day. Still cool to this day.
Thank you, I was looking for this proof, you explained it clearly
Thank you so much for this! Really helped with my research
:)
This guy is my favorite mathematician 😊💙
This was recommended to me and I just watched it in the middle of the night :D University has been a few years, so I had to give you the benefit of doubt regarding the Taylor series, but the rest of it made perfect sense to me.
Gruß aus Deutschland =)
Why can't I find a nice guy who calls him Daddy Euler in my life?
;_;
Man..... I just love his energy
"It's very simple" euler just died here
4:43 - 4:55 when u are possessed by a ghost who was an engineer
5:05 "the same spiel.." sehr schön :)
Sehr interessantes Video weiter so.
This has been one of your best videos and I have been watching them for a while. This was super fun to watch clear and easy to understand. Definitely do some more og Euler heuristic stuff!
Bravo!Esti bun. Competent!
Mathologer has made an (and several other) amazing Video about π²/6 and Eulers sine formula!
I always confuse your Xs with lambdas
That T-shirt, lol, very cool!
That's how I love QED! Not overly rigorous, but right nonetheless. You have earned an ardent follower!
Awesome video....!!! I am watching this at 2AM because I don't need sleep I need answers.
Oh, yes. I first did it when studying classic Fourier transform in 1st year of undergraduate. Actually many basic equations and formulations follow Eular.
Good video! It’s really fucking dope.
Encountered this series as part of a homework problem
I'm so glad you exist Flammy ;_;
I recommended this video to friends cause of it xD
I just love this.
My favorite math teacher 😁
heuristic analysis better than malwarebytes 🔥😍
Man, that’s crazy my man!
:)
That proof was just breathtaking. Half way through, I was really questioning whether this was supposed to solve the basel problem in the end. But you beautifully showed it. Also, tell me the name of that third grader who can do this. 😂😂
16:55 That was the smoothest fucking thing I have ever seen in a maths video.
Mad props for making such a digestible video on such an intricate subject
Great job bro !!!! Wow
Thank you, daddy. I was using ^0 and ^2 together and got confused. I got the answer now.
I love your videos
Papa Euler 🤣🤣
I like how you say this is very easy
Just as I was looking up summation techniques
Boy I love your t-shirt! 😍
6:45 You're actually right! 😊
Leonard Euler was really great!
Hast Du gut gemacht! Daumen hoch! ;-)
On a good old fashioned chalk board. Euler approves this message.
Awww... I was about to recommend this lovely explanation to my 6 year old niece, but then you swore at the end!
liking for the Taylor meme
Good job! Excellent🤯🤕
Fabulous fabulous fabulous 👌👌👌👌
Early 50k congrats
Fantastic videos!
Love watching at work through lunch.
I'm using a formula in an app. Just come up with it. It does what I want it to do. But how do I know what's it called ??
(Sure you would know! ?)
This is great... this give us information about the relationship between that summation and pi but still the solution is unknown as nobody know the exact value of pi
Loved this simple explanation. Thanks!!👏
:)
awesome video!
„We can do the same Spiel for the next...“ :D
Pretty *ucking dope
And this approach works for all positive and even values of zeta
cool.. After almost 1 year following the channel, I realized just now that you are german-speaker =) (the hint was at 5:30: 'we can do the same 'Spiel'' for the next zeros).
Sauber! Das Channel ist ja Hammer! =).
Thank you, super 👌
My favourite thing to write when solving a math problem: "By inspection"
Impressing presentation becoming harder as well as you were in progression let me not to clearly catch how it turned out!
You deserve hats off; in the contrary the way of factoring sine would be wrong in polynomial case.
Notice that pi-x equals pi ( 1-x/pi ).
Keep up the great job!
PS Not understood how and we inserted factorial!
0:25
Thumbnail: n
Chalkboard: k
You've been click baited, and you know it!
Beautiful proof.
"Daddy Euler" schön, den Namen Leonhard Euler richtig ausgesprochen zu hören. Ich belehre Mathematik hier in Australien und ich ziehe meine Schüler ständig dazu, Euler nicht als "you-ler" auszusprechen. Eine kleine Anfrage, produzierst Du Videos in der deutschen Sprache? Beide Deutsch u. die Mathematik sind meine Leidenschaften seit Jahrzehnten gewesen aber mir sind die mathematische Begriffe nicht so wohl bekannt. Es waere schön wenn die beiden vermischt werden können. Ausserdem wäre es toll dass die Werke von Euler, Leibnitz u Gauss unter anderen auf der eigenen Sprache erklärt und verwendet werden.