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Partitions - Numberphile
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- เผยแพร่เมื่อ 9 ส.ค. 2024
- Partitions are a major part of the Ramanujan story (as shown in the new film about his life) - but what are they?
More links & stuff in full description below ↓↓↓
Dr James Grime tackles the issue.
Extra footage: • Ramanujan and Partitio...
More James Grime on Numberphile: bit.ly/grimevideos
Young Tableau: • Shapes and Hook Number...
Shuffling: • The Best (and Worst) W...
James discusses Good Will Hunting: • The problem in Good Wi...
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What's even more impressive about Ramanujan's achievement is that he didn't have the benefit of having Duplo to hand.
This made my day :)
+SquareWaveHeaven just dont step on it
Pretty sure he must have. Come on, I know the guy was a genius but he was still human.
*Megablok, he is lying to you
@Jace Wright I have Euler in my heart, you can't get me devil spawn!
"How many Christmas cards does Ramanujan have to send?"
"Infinitely many, but at least he can count them all."
He sent -1/12 cards
He sent none, he was Hindu lol
@Alex ask Galileo who was killed
@@antrixsharma3476 lol
@@murtazahamid6141 I wonder where his heart was? I am sure he debated the 1 and5 card.
even if I dont understand 98% of things he talks about, i just love his enthusiasm
The language of mathematics is like listening to a beautiful Latin language. I like listening to French and Italian despite my ability to understand most of it.
Батрић Гарић I ja isto! Obožavam
Aye he really seems to loove his noombers.
I understand all that.
haha yes
college text books often have the best footnotes because you can really see how people slowly become delirious after spending hundreds of hours going through them and writing it.
+Azivegu I know right xD
Your profile picture is upside down and in a mirror
@@shadowbane7401 non-ironic fun fact, in mathematics an upside down capital A placed in front of a variable represents that the formula that follows applies *for all* values of that variable.
example: (∀x)(x+1>x) means "for all possible values of x, x plus one is greater than x."
(Opening lines of "States of Matter", by D.L. Goodstein).
Ludwig Boltzman, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics.
@@thomaskaldahl196 thank you I've wondered what that upside down A meant
I'm reading *The Man Who Knew Infinity* right now. The depth of mathematical exploration in the book is more complex than the superficial depiction in the film, and it provides the extra complexity of human relations between Hardy, Littlewood and others involved in Ramanujan's life.
#÷#^=
+2b Sirius Well, there's only SO much that can be fit in a movie...
The book also has a lot of boring and unnecessary detail, like a huge digression about the Tripos exam system in the UK.
what did the book say about ramanujan or his genius?
@Grian Brant Yeah, and Ramanujan neither fought in the war nor took the Tripos.
pi keeps showing up in the strangest places.
I know, pi is just magic
+That one Lich Did you just agree with yourself? O.o
+Relux the Relux As opposed to disagreeing with myself? Or did you not catch the comment I was replying to
LOL for some reason the other comment didn't loas when I commented, thought you had replied to yourself. XD
+Relux the Relux I bet it was pi again
my favorite textbook footnote is from UW Math 234: "This is known as the 'sushi principle' - Raw data is better than cooked data"
UW as in University of Waterloo?
TheSasukeOwl university of Wisconsin I imagine
Isaac Galang v
That has to be a jab at a colleague.
I highly disagree. Grilled eel sushi tastes sooo much better than raw sushi.
I find it interesting that Einstein and boss were working on a physics problem closely related to partitions around the same time as hardy and ramanujan.
Then who is Einstein here .
@Agraj AnujBose, but boss works just as well
@@shambosaha9727 "Bose, but boss works just as well" Not when you're working with Einstein in the 1910's.
@@EGarrett01 I would like to point out that Bose did most of the work, but, ya know... controversial issue.
@@shambosaha9727 Seems like everyone is convinced someone else actually did Einstein's work.
I'm a simple person who knows nothing about maths but watches numberphile. I see James Grime, Lego, and Ramanujan - I click.
"Don't do it - you'll be sorry" on a science textbook explaining reproductive systems
Jack! Draw me like one of your french partitions.
+Nae Dolor I was just about to post that and then decided to check if someone else already thought of it. :)
Jack? Who's Jack?
Upside down and in a mirror?
I guess in this version it'd be Jacques.
@@ardenvarley-twyman8352 Jack from Titanic, man 😂
damn, Ramanujan died way too young :(
What could he have done if he lived longer?
@@vinayvekaria3400 he had a book of formulas that were not proven when he died so who knows what else....
@@vinayvekaria3400 we don’t know. He passed away too young.
A child genius may find that ata certain age the normal others caught up with him at 30. Who knows?
@@brendawilliams8062 how on earth can that happen
I have seen Ramanaujan handwritten notes and O dear this man freaked me out. He was living in a different parallel universe
i just finished watching the movie now about Ramanujan...i had to come here ..
same here
you liked it?
i loved it
same haha, i didn't find my phone so i used my sister's computer to look for it.
I'm now going to watch the movie lol
OMG I love Ramanujan so much
A hero is what he was to his cultural heritages.
I *do* have a favorite footnote, actually. :-)
Bernard Hoëcker began his first book with a footnote (even before the first word).
It said that this footnote only existed because he just learned he could do begin a book with a footnote. :-)
So even the book wasn't about mathematics at all it was a nice self-reference, which is always cool. :-)
Someone should write a book entirely consisting of footnotes. It would probably have a tree structure. :)
@@cube2fox i think you started watching numberphile like 3 days ago and fyi its a great channel :) also i like how your profile picture comes from mario maker 2
@@leo17921 Hmm, why do you think that? :D I also like your fire Mario profile picture. :)
@@cube2fox cause in most numberphile videos i watched i see a comment from you from a few days ago
@@leo17921 Haha, actually I know the channel for several years, but recently my TH-cam decided to start recommending them again.
There's a movie about this? Wow! Hope it won't be a Parker square of a movie!
888SpinR
yeah, truly.
Most movie about famous people are parker squared usually I guess. Let us hope for the best.
Next movie about a famous mathematician should be about Parker himself and his parker square.
Title suggestions: giving things a go- a parker square of an autobiography
An almost perfect prequel - the success of a parker square
Matt parker- the mascott of parker square.
+achu11th (parker)^2
the parcker square deserve more recognition
+Almujtaba Osama there should be a place called parker square square or something which shows the amount of recognition it deserves. It definitely needs more recognition, you are right.
The partition function, along with the sum of partitions and the permutations of partitions, is probably my favourite function in all of number theory. It's just so useful!
My favorite footnote was in a mathematics book in first year of my bachelor study. It was about shear matrices and showed a picture of a sheep and a deformed sheep, calling it a "sheared sheep". I found that pretty funny ._.
Was it Lang? I think I saw that picture too!
You need to post more videos with James Grime.
Epic video as usual btw.
+Priyadarshini M James Grimes is my favorite, not only is he brilliant, but his enthusiasm draws me in and makes me care about the things.
+Dr. Evil (Hjalte Hørsdal) Agreed! He definitely has a way of explaining things that people can appreciate.
6:07 imagine going to vacation with your infinite number of friends. What hotel would you choose ? I wonder if a mathematician thought about that ?
Hilbert's infinite hotel
ez
+Citizen Babao ...
+AAA That's the joke.
+Citizen Babao Do mathematicians have friends?
If you had all the positive integers as your friends you'd have to send infinite cards and you'd only get back -1/12
+yriafehtivan i see what you did there.. ;)
+yriafehtivan Only if your first friend replied with 1 card and the second with 2 cards and the 3rd with 3 french hens.
+Reckless Roges
Very true.
Only when their number of gifts equal to their ranks
Lol. Great one
I LOVE JAMES GRIME
Me too ❤
+BigMan Stan ALL GRIME ALL THE TIME
8:57 for anyone wondering exp(n)= e^n
Leo179 tyvm
Thanks man I was so confused
Yeah, but how many ways are there to partition Poland?
4
*coughs* ask its neighbours *coughs*
Ask gps
Probably my favourite story of Ramanujan is when G. H. Hardy went to see him, and I'll let Hardy tell the story: "I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.
I love how enthusiastically James is playing with Duplo in this video only to go on to play Pen&Paper Tetris
My favourite footnote is footnote 34 page 69 in the Griffiths Introduction to Quantum Mechanics textbook where the author points out that " If you are irritatingly observant you may have noticed that the general theorem ... doesn't really apply".
This needs to be reuploaded upside down and in a mirror
+Damien W With French voiceover.
+Damien W
Can it be a one way mirror?
+Ζήνων Ελεάτης We do not believe in things that don't exist around here. Now, let me quickly sum up all the naturals and get -1/12 as a result.
+Damien W So you mean into a server in Australia? Hey, stop booing me! You don't get puns like this every day!
Franz Luggin
Hahahahaha! Possibly, modulo-infinity!
8:30 Well, that escalated quickly
Thanks for making a new video with James Grime, it's been a while! He's my favorite.
Uploaded on my birthday and I love Dr James Grime. This is awesome.
1:50 Oh hey, he plays Tetris too
such a fascinating subject...I dont know much about maths, but when I read about Ramanujan I remember thinking this guy was one of, if not the most naturally gifted mathematicians in History. Had very little formal training, and so poor, he couldn't even afford notebooks, so had to use slates to do his calculations on, what mathematical marvels were lost on those slates, though he did of course, keep his best ideas in his three notebooks. Those slates are possibly in landfill somewhere.
Ramanujan's Christmas card list wasn't too bad. But it got bad when he tried to give all his friends presents by putting a candy in one sock belonging to each of them.
whenever I see Dr. Grime is in a video I get super excited. He just seems so excited and happy about math! Pretty much I'm saying put more Dr Grime on the channel.
Really nice explanation of partitions. I was struggling with a very dry math book that didn't really make things clear. Thank you!
Euler, Hardy, and Ramanujan are the mathematicians who inspire me to actually explore math. And Numberphile and 3blue1brown are the best TH-cam mathologists
Jacobi, Euclid, fermat were excellent too
second semester of undergrad I started studying partitions and q-series. I've fallen off since major surgeries and the COVID-19 pandemic. This video reminded me of how beautiful they are and how much I loved the maths involved. I swear to get back into it. So much left to be discovered and mulled over
Ramanujan was probably interested in this because he may have felt that this formula would explain what a number is. (The sum of its parts). In this case partitions.
That was a great video, thanks James!
Amazing video even after so many years his work powers the curiosity within me
Please do a few more videos on Ramanujan's ideas. Great video.
Many thanks Numberphile...Was looking for this for quite a while.... Cheers!
"Ramanujan had an amazing intuition for numbers; another Cambridge mathematician called John Littlewood said that all the positive integers were Ramanujan's personal friends, which sounds like another nightmare to me, having infinitely many personal friends. Imagine the Christmas card list. It sounds terrible".
I believe that he was deeply spiritually inclined and found pleasure in philosophical thoughts.
"Imagine the Christmas card list... sounds terrible" best phrase ever
A legendary, humble individual...
Your channel has helped me so much in my programming/math journey!
Happy to help 👍🏻
Thanks to Cambridge for recognizing his talents
11:00 - "Even in physics like eh, shuffling energy. So you know energy isn't created or destroyed it just gets moved around, right?" This is almost completely true. It starts to break down when you are working in the quantum range and having deal with virtual particles.
Could it be, that REALITY isn't, really, real? O_o
Actually Objective Reality is REAL.
Try the Solipsist Litmus (u)Test:
Pinch your self; TRUTH Exists
Pinch another sentient being; TRUTH WITH CONSEQUENCES...
Save your solipsism for your "DEATH" inversion - that's when you'd better KNOW NO HARM...
JUST say'n
124bel875ive The problem is the definition of real. Some of the words used to describe real are: not imaginary, objective, not artificial, and absolute. Imaginary implies a higher conscious, and I have no intention of weighing in on one side of that debate, but if one did exist, imaginary would be an apt term for something created by a being's mind. Objective and absolute completely breaks down at the quantum scale, it's only at larger frames of references things seem so tangible. And the number of peer-reviewed articles in favor of the idea that our universe is a simulation is immense. So, in summary, all four of those terms would have to be true for what we call reality to be real. There would have to be no higher power, we would have to not be in a simulation, and we will have to distill the probabilistic nature of subatomic particles away. Seems like that would be quite a feat to show that our reality, is actually real.
Arm4g3dd0nX
Pinch your self until it HURTS Brother. DO YOU EXIST?
Pinch ME (or a sentient being) with intent to Harm - and one or both will cease to exist.
People that deny objective reality are solipsists.
Solipsists justify Violence and HARM sentient beings by pretending they are imaginary.
Solipsists are THE reason Earth is in trouble today.
150 billion animals per year killed and eaten by carnists (frugivores eating corpses!) is NOT SUSTAINABLE.
It's a Death Cult.
Again, Objective Reality.
To deny Objective Reality in insanity.
JUST say'n
Arm4g3dd0nX
Beware YOU are in MONSTER Territory.
Some of us OWN the Partitions...
Madness? THIS IS MATHEMATICAL!
Ah, James! He is a special one of my favorites here on Numberphile!
James is back! Happy days.
I am very excited about The Man Who Knew Infinity :)
Having a favorite textbook footnote is so unbelievably nerdy and I love it.
So much respect for this host so smart
Ramanujan forgot two of the most important Tetris pieces:
The Squiggly: ◘◘
◘◘
The Reverse Squiggly: ◘◘
◘◘
+L00NGB00W
Line piece...
*Line piece.*
LINE PIECE
*LINE PIECE*
*LINE PIECE!!!!!*
They’re called the S and Z.
He forgot T as well
whoa whoa whoa
Fun fact, if you ignore mirrors, there are 5 Tetrominos (Tetris pieces) and 14 pentaminos. It's Catalan numbers.
Ramanujan was lego genius
The man who infinity is just unpredictable and contains the mind at the level more than the infinity....
Respect Shri Shrinivasan Ramanujan
Dude your a real math nerd. I really enjoy your teaching. I bet you make a great teacher / Professor .
Thanks.
We're going to need some more Duplo's
+Pandadefoggi or Tetris
Thank u so much for showing us how beautiful math is and the supreme beauty it holds. I was always inspired deeply by Srinivasa Ramanujan. And it’s such a honour for India for the teachers like you are expanding his vision. In India we need teachers like you who can not only teach maths but show us that it is at the deepest in the heart of the cosmos. Thank for ur help. HAPPY TEACHERS DAY!
That looks like a #ParkerSquare formula
lol
+Subin Mdr My name is Jesse Parker, and I approve this message.
My name is Parker Mowery and I too approve this message.
My name is Barry Allen and I'm the fastest man alive.
+Murariu Ciprian
Hi, my name is, what?
My name is, who?
My name is, chka-chka Slim Shady
although I didn't understand it that well, thanks for trying to elaborate it. u r putting a great effort and enthusiasm.
1:55 As he is drawing out the various partitions of 4, I start hearing the Tetris theme going through my head.
Could you talk more about the applications of it and show us how the full formula looks like?
+Wood Croft I just did a module on Thermodynamics in College, I think its the same idea for partition functions they had the same sum of the exponent of a variable form anyway, they're essential in that field, look up the partition function in thermodynamics :)
James!!!! Your back!!!!
You're
What about his back?
+Zac Lee hahaha nice one
Zac Lee Yeeeessssss thank you!!!
I legiiamately thought you meant that something was wrong with James' back and started looking through the video until I suddenly realized xD
James Grimes! :D You are one of my favourite people in the world, sir!!!
Fantastic video thank you very much for that !
ramanujan roll up the partition please
Finals in less than a week? Screw it, JAMES GRIME!!
At 10:04 Dr. Grimes says of the classic Hardy-Ramanujan series approximation to the partition function: "in fact it becomes equal [as you include more terms of the series]." However, this is incorrect, as the Hardy-Ramanujan formula is only an asymptotic approximation (i.e., the value of the kth partial sum converges to p(n) only as n --> inf for fixed k, but *not* as k --> inf for fixed n). It was actually Radamacher who, in 1937 (some twenty years after the publication of H. and R.'s original result), was able to modify their formula to make it absolutely convergent.
thanks for explaining that partitions is about shuffling. that rang a bell!
When I showed my friend OEIS for the first time he randomly entered some numbers and this sequence came up.
oeis?
Online Encyclopedia of Integer Sequences
You're missing the Z and S tetris shapes. Oh, this isn't a video about tetris.
They also forgot the T...
oh.. it's not tetris? nevermind..
This is just a Parker Square video about tetris.
That was fascinating.
What a wonderful thing to have started.
Nothing is more misterious than that brown paper.
Full formula on wikipedia, you'll understand why he didn't give it to us when you'll see it...
+Flandre Scarlet fuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu
+Flandre Scarlet yea my professor found it just a couple years ago..
i didn't found it
+Skxawng en.wikipedia.org/wiki/Partition_(number_theory)#Approximation_formulas
+Skxawng Of course you didn't. Ramanujan founded it.
It's rather ironic that Ramanujan should be particularly well known for his work on partitions. He was an Indian, and that's exactly what they did to India in 1947. And they considered how many ways there were to do it beforehand.(BTW, Ramanujan wouldn't have had a problem with his Christmas card list if all the positive integers were his close friends. As a Hindu he almost certainly wouldn't have had one.)What, too soon?
YES JAMES IS BACK!!!!
Ikr? I'm ecstatic
Finally, he's back!
Awesome video! Those figures remind me of tetrominos - all the ways you can arrange four squares in a plane where each square has at least one edge connect to the whole figure. Oooh! You should do a video on polyominos! The sequence of possible configuration with one square is 1, two squares = 1, three = 2, four = 5, five = 12, six = 35, and so on. :-)
James is the most interesting person on numberphile. Prove me wrong
I'd love to see more vids like this one😍
Oh man I missed James so much!
Merci pour le conseil, mais je vais regarder la vidéo à l'endroit quand même :D
Hi, thanks for this video, I appreciated your explanations. But after watching this I already not convinced by the fact this formula is exact for n small.
Could you explain what's the real formula to find partitions of n ? Because the formula do you explain in this video become more accurate when n is large, like you said... It is an asymptotic development. I would like you describe the Rademacher's serie if it's possible, because I read this serie is more or even accurate than the first. Thanks again!
That footnote really sounds like the author went "can't be bothered with the french mate"
I just watched the movie !!! ... thanks for explaining this partition stuff !!!
Every day I'm shufflin'
Ramanujan was the most intelligent known man in history.
0:52
2:02: so u shift it, and it gets easier?
2:22: The french way is redundant and more complicated.
2:57
3:34: gimme now.
3:57-4:43: Euler's method
5:23
6:50
I believe Einstein once said something like "If you can't explain it simple, then you don't understand it well enough" - YOU just nailed my friend 👍
I'm going to watch this upside down in a mirror.
Guard13007, How did it go?
Perhaps - Take a PEEK thru the Direct Opposite Reverse - The Prime Spiral Stills the Chaos.
BOO!,
Monster :-0
124bel875ive g
KupoForLife
triacontahedron pawns the monster and bloch walls parasites :-)
How many partitions of a parker square, are there? Probably almost infinite.
+achu11th memes are funny. even when they're not.
jk
+dammit dan no, you are wrong. They are almost funny, but they are never really funny. I am not making fun of Parker. I am just making fun of the meme itself.
+achu11th There is no such thing as almost infinite.
+Chad M but a parker square is almost complete. The series of partitions are infinite as I interpreted this video. So logically you could apply that there is something like "almost infinite". But it was for the sake of a joke and was meant to be unlogical. A joke is usually meant to make no sense. So yeah, you are right. That is my kind of humour, not everybody has to agree on it. Maybe you could make a version of my joke, which makes more sense. Thank you anyway
achu11th Yeah I don't parker square is, it just really dubs my anime when people say things such as nigh omnipotent and the like.
We do not know how many more formulas Ramanujan would have discovered had he lived longer. But, one thing is for certain and that is he would have been a Numberphile fan :)
+Ehsan Azarnasab I think Numberphile should interview Dev Patel (who played Ramanujan in the movie they mention) about that.
‘imagine the christmas card list’ -> the best tangent of all time
intuitively, i would have thought, that the pattern would look like this : 5 -> 7 is +2, 7 -> 11 is+4 and then either it would be 11 -> 17, because we always add 2 or it would be 11 -> 19 because we always multiple by 2 of what we add, so the next one in pattern should look like: P(19k +7). Thats what i would think, but of course it does not works, but i still wanted to share this ;)
6:08 that would be a christmas card list with cardinality aleph null! :D
+Meijke Balay Thanks to VSauce, I understood that comment :)
+VladVladislav VSauce is amazing. Micheal always comes up with something that I'd never even heard of and makes it easy to understand
I see that prime number generator in the background 🤣
Thank you. 🙏