This Result Keeps Me Up At Night
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- Опубликовано: 2 окт 2024
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What is the sum of all the natural numbers?! Of course the answer is obvious...right? Many have actually pointed to the result that the sum of natural numbers is negative 1/12 !
We're going to do a 1+2+3+...=-1/12 proof , then a false sum of all natural numbers proof. What can we make of this?!
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1+2+3+4+5+6+n=-1/12 proof
#math #brithemathguy #sumofallnaturalnumbers
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Video is 20 mins ago and you posted this 8 days ago. How?
Hey, Bri, have you heard?
@@TerryPlays Either it was unlisted or it was available only to members.
@@BambinaSaldana ok
Not gonna lie, the answer is wrong.
The reason the answer is wrong is because it doesn't use the correct definitions of the terms it uses.
The problem is that you do a sequence of + and - to INFINITY! infinity - infinity is still infinity. infinity contains all other forms of infinity , such as infinity+1 and infinity+infinity repeat to infinity.
If you minus infinity from infinity you still get infinity or 0, even if you minus 2 infinity from infinity, the answer is still infinity or 0.
Average complex math theory fan vs. Average "It clearly isn't, just look at it" enjoyer
After such rough comments on why the video is misleading/incorrect/incomplete, I really needed this comment lol
Based
me, an average “it clearly isn’t, just look at it” enjoyer
me a complex number theory fan(complex numbers apear in physics)
@@catalintimofti1117 they’re stupid (complex my ass)
I’m so happy with the context you provided and ended on “no the sum of all natural numbers is not -1/12.” Too many mathtubers don’t do that!
@@CellarDoor-rt8tt Also because of its relation to quantum mechanics.
@@CellarDoor-rt8tt The Riemann Zeta function isn't the ONLY reason. There are many reasons to associate this series with the number -1/12. Quantum mechanics (the Casimir effect uses this -1/12 in calculations and has been, IIRC, confirmed experimentally), Ramanujan summation, the standard naive series manipulation argument by way of "1+1-1+...=1/2" and "1-2+3-4+...=-1/4", and many more. It's just that conventional summation isn't one of them.
It is -1/12
Mathtubers?
First time I heard that term
@@nintendoswitchfan4953 cope
Subtracting Infinity is not a valid operation, so everytime you subtract S you rule out that S could be infinity
I like that argument. It's also pretty easy to prove that partial sums of S to n terms tend to infinity as n tends to infinity. Apparently this stupid result shows up in a textbook on string theory. What keeps me up at night is that people believe it.
@@simongross3122 it’s also used in Casimir effect. Physicists have a knack for using mathematically ill-defined operations…
@@FermionPhysics Like cancelling infinities. That's another one that has me scratching my head.
@@simongross3122 the renormalization thing is much less absurd (if that is what you’re referring to). It actually makes some sense if explained in a less hand wavy way…I might make a video on it sometime.
@@FermionPhysics Thank you, I'd appreciate that.
Thanks to Ramanujan for giving us this wonderful series
The problem isn't using the distributive law, it's grouping the series so that you are changing the "speed" at which the series grows.
There is another issue: he performs algebraic operations on infinity (for example S-9S = -8S), when in fact 9*inf=inf and inf-inf is undefined, so inf-9*inf is also undefined. The "mistake" is at the very beginning: treating infinity as a number (S)! INFINITY IS NOT A NUMBER! You cannot perform "normal" calculations on infinity.
@@lf9177 exactly. I think you are correct. if 1+2+...= infinity, then you can't say 1+2+... = X, then use that to do math, as infinity is not a number or constant.
So say for example the speed it converges is the speed of light so would there be a limit if there is a limit in physics?
@@emilemerten6535 well the speed of light is a physical limit by reality, numbers have no physical limit. There are rules for it to work, but there is no cap. Any rate at which you think a sequence could reach infinite quickly, I could make one go faster by just applying ^n or whatever
@@lf9177 but you are assuming S is infinity, we start out with no assumptions about S to find its value
Brian, this is probably my favorite video you've done. Outstanding, clear explanations. When Numberphile made a video on this result, I commented that it was at odds with what I'd learned in real analysis, and the other commenters chimed in with, "I can't believe your arrogance, contradicting the work of high-level mathematicians." Well, the professors at UW Madison ARE high-level mathematicians! Thanks for showing the elasticity of this approach to computing divergent infinite... just super fun and informative.
Really glad you enjoyed it! Thanks for watching and have a great day!
There are higher level mathematicians than UW Madison professors I believe. But the question about this series is more philosophical that mathematical.
I should add though that I'm a fan of public universities and in particular of UW Madison.
@🙂And I wasn't implying that their professors trump all others, just that I when I was claiming "but that isn't what I was taught," I wasn't referring to some goofball's blog or some random Tik Tok video.
@Glorified Truth There is indeed very great mathematicians at UW Madison and has been before. Like Georgia Benkart.
The main catch is assuming A is a number. While all the rules work for convergent series, in this case if we calculate A - A = 1 - (1 - 1) + (1 - 1) - (1 - 1) + (1 - 1) (with similar shift as we did with B), we get 0 = 1 and we just broke the math.
Yes! This right here.. A is the catch.. A is in itself non-convergent..
the alternating sum of -1 and +1 is the geometric series sum of (-1)^n which is very famously divergent because geometric series only converge if the base is absolutely smaller than 1.
Hmm that's correct besides if you take something from an infinite series, it just doesn't remain the same as in the case of series A. There is something that's been subtracted and we can't just ignore it no matter how small it is.
Furthermore what does this -1/12 imply? Does this show that the series is convergent? Perhaps it might have some uses but it's still a non conventional mathematics that doesn't apply accordingly to the regular mathematics
Thanks for this comment, this is a simple way to show some of the problems with these "proofs".
My question was how can we be sure that 1 = 2A, when we can very easily add another A for the same result to get 1 = 3A, and again for 1 = 4A, and so on.
@@lory3771simple, 1=A*X where X is any positive integer, as A is equal to both 0 and 1, thus 2A is the same as 0+0, 1+0, and 1+1. So as long as we can be certain that 1=1+0(X-1) we can be certain that 1=A*X for whatever valid X we choose.
This is the best explanation of this phenomenon I've ever seen and probably will ever see. Great job
Glad you enjoyed it! Have a great day.
Mathologer has a fantastic video on this topic as well. Well worth watching.
can you name the mathematician who had given this result??
@@aravindswami6243 Ramanujan
@@BriTheMathGuy infinity * any positive real number is still infinity
1:47
The only logical argument is when you're doing the cheap trick of moving B to the right by 1 digit, you're agreeing that right now Im adding the 1 to the -2, the +3 to the -2 and at the end of the string I will come back and add the last remaining digit of the lower B to the first 1 of upper B.
But since the string never ends you're never doing that despite it being a requirement, it can't be ignored just because the string is infinite. You CAN NOT leave the 1 like that, because some day you'd have to add something to the 1 but you're never doing that and saying hey it keeps on forever!
So B ≠ 1/4
No.
Thats not how addition works bro, he didnt move any number. He just add the second number with the first one thats is.
A = 1-2+3
B = 1-2+3
no matter how you move it A+B is still the same
Even if you randomly places them
A = 3+1-2
B = -2+3+1
the results is still the same
@@watcher1109
1 + 2 + 3 + 4
... + 1 + 2 + 3 + 4
= 1 + 3 + 5 + 7 + ...
= 16
So, 2(∑4) = 16
This isn't a simple addition if the series is not ending... You don't account for the last hanging digit that way
Edit : If you're having trouble understanding the language, read it again. I've used the movement logic for simplicity, in actuality I'm accounting for the same thing you wrote in the reply.
@@NGL_Shinhok1 4:00 if what you mean we cant assume "finite" and "infinite" works the same law, he literally stated in the big part of the vid, why do you pointing it out lol
@@watcher1109 I stated a different perspective which wasn't in the video, and I commented immediately when he was saying that, without watching it fully.
After watching it fully, my contribution as a new perspective still stays unique, didn't see anyone add it in comments.
bro you know what is logical meaning of this addiction apart from this mathematical steps , I want to know from my high school…
Just found this video, and you instantly got a subscription from me. This irks me so much, and you succinctly described why and gave me a good counter example why it's flawed. Thank you.
So glad to see a video about -1/12 that doesn't just stop at -1/12, but keeps going and demonstrates that no, it is not indisputably the case that the sum of all the natural numbers is -1/12. Yes, the series of natural numbers is closely linked to -1/12, but the conventional sum is not that connection.
"it is not indisputably the case that the sum of all the natural numbers is -1/12" it is tho. Every single regularization method that will give you a finite value for this sum gives you exactly -1/12.
@@ДмитроПрищепа-д3я But those regularization methods aren't what sum conventionally means. That's the entire issue here. Many popmath sources (including Numberphile, mind you) say more or less uncritically that the sum is -1/12, and that just isn't true.
But it isn't at all. You can't use cesaros sum like that. People like to just be like "hey, sum(-1^n) from n=0 to infinity is = 1/2, but that is the cesaros sum which is a separate arithmetic. It's more like saying csum(sum(-1^n)) instead. For instance, we can say if a = b, and a = 2, then b = 2, but we can't say that if a^2 = b^2 and a = 2 that b = 2. It goes against fundamental laws of mathematics.
This cesaros sum shit just annoys me because when I integrate cosine from 0 to infinity, it's undefined, however this would suggest it has a solution.
@@DinsAFK (-1)^n can be summed up to 1/2 using pretty much any other method aside from the classic one tho, it's not just Cesaro.
@@ДмитроПрищепа-д3я those methods ignore divergence laws then and I really don't see any point in following them
Thanks for putting a smile on my face for showing the Riemann functional equation 😊
The ruleset is sometimes not as unclear as presented here, both diverging and alternating sums don't have a general sum that can be simplified without implementing supersums or regarding a generalized function, e.g. Riemann-Zeta. Thereby neither shifting a sum "left" or "right" completely breaks the initial result, nor can the distributive law be utilized for an infinite sum whose identity isn't convergent. Just wanted to slightly highlight it more, bc if we were to allow all forms of algebraic operations on infinite sums, then we'd not only get proofs for -1 = 1 or N = R, but also break mathematical operations by applying mathematical operations.
The scorn of the goddess of infinity is eternal, infinite even.
Wonderful exposition of the 'why' for this result and others like it.
I am pointing at the culprit as being the "shifting the infinite series" and then adding them to infinity.
And that's because at any step, you are counting 1 element of the first infinite series while completely disregarding the same order element of the other infinite series, which the further out you are, the bigger that number gets.
Yes, which also means you are infinitely away from the true answer: infinity.
That step is fine, you are just adding a zero to the beginning of the series which cannot change its value.
@@ruce9269 there is no value as it doesnt converge, but the limit is infinity
@@ruce9269 approximately.
@@evanfox3136 You could also add 0 to any other position and use that to create whatever result you want
It was founded by great mathematician Ramanujan
Brian, youre the best! Thanks for the video😍
You're the best!
When I do my taxes later this week, I am going to use the analytic extension of my gross annual income to assess my tax burden.
I think a key issue is at 1:30. I don't think that is something you can just do like that.
You can
It's better and easy if you do A-B
A-B = (1-1+1-1....) - (1-2+3-4+5......)
Now on opening brackets
A-B = 1-1+1-1...... -1+2-3+4-5....
Now on making pairs (such that one number from first bracket and second no. from 2nd no.)
A-B = (1-1)+(-1+2)+(1-3)+(-1+4)+(1-5)......
A-B = 0 +1 -2 +3 -4...........
A-B = 1-2+3-4+5...........,
Notice that 1-2+3-4+5.... = B
So
A-B = B
A = 2B
1/2 = 2B (becoz A = 1/2, already proved above)
1/4 = B
That is a technique of adding special series
Ok, now prove now it isn't
Huge respect to great mathematicians Sir Ramanujan
*mathematician
U are 100% an indian lol
Almost every Indian is proud of him. But this theorem that the sum of positive numbers ends up being negative is BS.
@@12345ngb Exactly! This sum cannot equal -1/12. Ramanujan was wrong hehe
@@CriticSimoneven though it's hard to believe, ramanujan proved it using simple math.
The various other details about applicability of the distributive law and such are really important and I'm glad you went over that as well, but regarding the original proof, the point at which I immediately said "whoa, hold on there" (and everybody else with some critical thinking should too, IMHO) was the whole "2B = A" addition trick.
Because you're shifting the terms over a space, the resulting sum must logically have _one more term_ than the original ones did, which is not being accounted for anywhere. This is actually the "last" term of the second B sequence, which will be "dangling off the end" somewhere out there infinitely far down the way. If this is actually taken into account, then 2B does not equal A. 2B actually equals "A + (the final term of B)". However, there arguably _is no final term of B,_ (B is an infinite sequence, after all) so this effectively becomes "2B = A + (undefined)". That is, it's not an actual value you can substitute into other equations in the first place.
This is all a great lesson in the pitfalls of blindly applying mathematical operations without understanding their limitations or ramifications, though..
Thank fuck someone said this
Great and easy to understand video. I saw this on Numberphile a few years ago with the professional math guys who got it wrong. Don’t ask physics guys to do simple math. I disagreed as they didn’t take into account convergent vs. divergent. Comments weren’t kind. Mathologer did two very nice videos on it also.
Numberphile didn't got it wrong. FYI the exact calculation Numberphile made, was made by Ramanujan. So, it's not a problem of "physics guys" or whatever....and the videos of Mathologer were not accurate on all points...
There's no need to "take in account" convergent vs divergent series, because such calculation can be made with properties satisfied with convergent AND divergent series!
@@coursmaths138 I went back looked at the videos again. On Numberphile (which I enjoy) physicist Tony Padilla obviously doesn’t understand what is happening and even says, “you have do the mathematical ‘hocus focus’ to see it.” (ruclips.net/video/w-I6XTVZXww/видео.html 6:48) I’m not saying they aren’t good physicists, but obviously there are mathematically principals escaping them here very specially in what you can and cannot do with convergent and divergent series. He refers to Konrad Knopp’s book and string theory. In a referenced video in the description they discuss the Riemann Zeta function which is not a sum but a value at -1. Mathologer discusses Ramanujan’s note’s and infinite sums and how to sum them. Numberphile did not discuss this at all.
Mathologer video (ruclips.net/video/YuIIjLr6vUA/видео.html) I think is much on this topic and in greater detail on why the infinite sum of is not -1/12 and why it just isn’t mathematical “hocus pocus”. He discusses why this is the case, where in the Riemann Zeta function the and hypothesis there is a value. How Ramanujan fits into this (not in any sense via a proof or acceptance, but an initial note). ruclips.net/video/YuIIjLr6vUA/видео.html
And I’d be interested to know where you think Mathologer got it wrong. You mention videos plural, however I was referring this one singular video on the infinite sum equalling -1/12, so please confine yourself to that video, unless you’d like t open a different discuss on another topic.
You certainly do need to take into account divergent and convergent series or it’s math versus hocus pocus. Can you please show in detail how such a calculation can be made otherwise as you say, “because such calculation can be made with properties satisfied with convergent AND divergent series!”
Thanks for your engagement even if you disagree.
@@csb178 Thanks for your precise answer and your respectful tone 😊.
I totally agree with you on the fact that " _Tony P doesn't understand what is happening_ " . And i know you didn't question their physicists' habilities. But you basically said (1st comment; if i understood well) that " _their math was wrong, because they're physicists_ " and that was the point i wanted to invalidate. This point is invalid, because the calculation wasn't made by them, but by the mathematicians Ramanujan (last part, on his note) and Euler (the 2 first series). So, their misunderstanding (although true) of the subject isn't the cause of the weirdness of that computation. Because, it is not *from* them. Yes, they do not understand what they're doing; but they just did _exactly_ what was already made before them. And if someone just made the same (write all these calculations as they already were in the previous litterature (mainly Euler and Ramanujan)), the result would have been the same: a weird calculation we don't really understand (and it wouldn't have nothing to do with that person being a physicist, a nobel prize, a mathematician, a field medal, a plumber, a soccer player or anything else...).
" _the Riemann Zeta function which is not a sum but a value at -1_ " It is not a sum in itself. But, the value of -1/12 (as well as other values of zeta) can be obtained by using a summation operator (ie a "sum") acting on a subspace of sequences, and constructed with the zeta function. It is called the Dirichlet regularization (improperly known as " Zeta regularization"). And studying the properties of this summation method enlighten why the weird calculation using "sum properties" works well (as unexpected).
I mentionned "videos" because your 1st comment mentionned "two very nice videos on it". I thought you talked about the one with the link you posted (in your second answer) and another on Ramanujan summation (ruclips.net/video/jcKRGpMiVTw/видео.html). But, yes we can focus on the 1st one.
What was wrong about Mathologer's video then?
I think his video is very well supported, and yet paradoxically very bad. Because he missed the main point of all of this. And his explanations, though clear and (almost everywhere) mathematically right, didn't really clarify the subtlety of the topic. And you can see it in almost *every* (i said almost *every* ) comment refering to his video (including yours). So far, i have just met one single guy (english and others languages included), over all the hundreds of comments i read (litteraly), that understand well the subtlety of the topic. So clearly, something went wrong...and you'll soon get what and why.
Mathologer's clearly right about saying that the limit of partial sum of the sequence (1;2;3;4;....) is obvisoully not -1/12. And he is also right about saying that answering this, would have make you fail your exam (5:35). But here's the main problem: every time he talks about the limit of partial sum, he says "sum".
Why is it a problem? Because a limit of partial sum *isn't* a sum!
No, it is not a sum.
If you think the opposite, you're confusing the vocabulary and the mathematical meaning of math objets. Yes, we call the "limit of partial sum" the *sum of the serie* (the choice of word is questionable). But it is only a choice or *words*. And though this choice seems legitimate, it made forget that we're dealing with and operation other sequences that is not a sum.
A sum is an operator other finite sequences (2-lenghts sequences to be exact, but can be extended and generalized over n-lenght sequences because of associative property).
And the limit of partial sum clearly do not correspond to this definition. Moreover, limit of partial sums just doesn't have all the properties of sums: commutativity (Riemann Theorem), associativity, inversion with derivative and integral sign....and so on....
But, it is true that the limit of partial sums is an extension that has some properties in common with "sums". And that's why, i think the choice to call it "sum" isn't that "bad". But the problem is that almost every student fail to understand (ou memorize) that *this is not a sum* , and that it's often dangerous to think of it as it is (though it is often convenient). The problem is so bad that many teachers and students claim (with vigor and confidence) that *we cannot do anything* with divergent series. If the absurdity of this sentence doesn't shock you, let me explain:
We said you have an operation other sequences (the limit of partial sums). Naturally, like all functions/operators, this function (the limit of partial sums) has a domain. And this domain is exactly the set of convergent series (by definition). And, of course putting a divergent serie (by definition a sequence not in the domain) inside a limit of partial sum is absurd. But it is just *one particular* function. The fact that some sequences doesn't fit with that function, obviously *doesn't mean* that we cannot do *anything* with these sequences at all! It's like saying we cannot do anything with the numbers -1 or -2, because log(-1) or log (-2) doesn't exists (or doesn't have a sense in the reals).
It's obviously false and totally absurd....many functions can take -1 as an input...
And it is the same for divergent series. Many operators can take divergent series as inputs.
But, maybe you're asking "yes it's great, but it is not a sum".
I would say yes; but (again) it's the same with the limit of partial sums.
And we chose a function which has many properties (though not all) of actual sums and choose to view it as a kind of "infinite sum". That's why the limit of partial sums is called "sum" (when it exists). But again an infinite sum doesn't exists (mathematically). If we want to talk about "infinite sums" (that's some intuitive concept), we have to make a *choice* (with an actual definition of an operation we will see as corresponding to the intuitive notion).
We make a *choice* . Thus, it is arbitrary. Because, the recipe "limit of partial sum" isn't the only one to generalize the notion of sum (in the math sense; ie for finitely many numbers). There are many other...
And, knowing that we made *a choice* to generalize the actual sum, we can also make *another choice* to generalize the "infinite sum", by choosing some operator that can act on the sequences that weren't summable in respect with the limit of partial sum: such an operator is called a divergent serie operator (en.wikipedia.org/wiki/Divergent_series).
@@csb178 The argument of the exam is pretty weak (5:35). It's basically saying that "if you write something (the sum of 1+2+3...) that is not what we *chose* to call infinite sums (meaning the limit of partial sums), you're wrong". The same could've be said about complex numbers in elementary school or earlier (when it wasn't accepted). Write something as aparently stupid as x²+1=0 is false in all elementary school exams and was viewed as extremely suspicious during a long time....but does it makes it wrong mathematically? Or course not.
And that's one other missing point in his video. It surely talks about super sums later, but all the begining (and the video in general) is assuming that Numberphile calculation is (or must be) about the limit of partial sum (8:48 - 9:05), and then is "nonsense" (cause the limits of partial sums don't exist). But *the calculation absolutely doesn't say that* .
The only thing that the calculation says, is that this "sum" is an operation with some algebraic properties, and that these properties imply values for 1-1+1-1+1-...., 1-2+3-4+5-.... and 1+2+3+4+5+.... (1/2, 1/4 and -1/12). Nothing else.
So, it *can't be wrong* , unless you prove such an operation cannot exist (and not only that it doesn't fit with the particular operation "limit of partial sum")
....but here's the thing: we can prove that such a function exist.
And he is wrong on an other point (17:40): there exists a supersum on all 3 series (indeed over all series, with the axiom of choice; and only on a "big" set of sequences (including many divergent series) if you don't use AC).
He is also wrong saying that "they don't have a supersum" (19:49; 20:30). The last serie just doesn't fit with *one particular type* of supersum (he is talking about Cesaro). But the fact that *this particular supersum* doesn't work on this serie, doesn't mean it does'nt have a supersum at all. Moreover, his (false) reasonning tends to make believe that all supersums are only about Cesaro (and Holder) and averaging (you can see in the link above that it is obviously false).
And in 22:18, he is saying that inserting infinitely many 0 in the "sum" can change the value, and it's true. But in *that* timing, with *that* insertion of 0, the sum is unchanged. And his argument was that the insertion in the numberphile calculation was illegitimate, except....it is totally legitimate. That particular insertion of 0 is legitimate and can be justified with (almost) only the 3 basic properties of supersums (contrary to his claim in 22:42). Again, the calculation doesn't say that *any 0-insertion is okay* , but just use this one particular 0-insertion.
At 23:20, he claims that the 3 properties (mentioned earlier in the video) are contradictory with 1+2+3+4+.... and it's true (in fact only the first two properties are sufficient to exhibit the contradiction). But, by saying that, he also claims that 1+2+3+4+.... *doesn't have* a supersum; which again is false. 1+2+3+....can't be supersumed with these 3 properties, but again it doesn't mean it cannot be supersumed with other supersums *at all* . And actually, it can (cf Dirichlet summation).
At 25:40, he says that the zeta function is the "genuine, real, actual, connection between 1+2+3+....and -1/12" meaning that because supersums fail on this serie (according to what he said just before; again even if it is false ). And though, zeta function is clearly one aspect of that connection, it is not the only one. More: the connection between 1+2+3+...... and -1/12 can be made *without* any use of zeta function (and complexe analysis), in at least 2 different ways: one analytic made by Tao (terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/) and one with purely algebraic methods (sorry don't have the courage to develop, maybe an other day 😘). So no, zeta is not *the* connection between the serie and -1/12 (as many comments say in all video about this theme), but just an aspect of it. What is strange is that he put the link of Tao's article in the description. But this article specifically says that it is possible to give an interpretation of these computations _by purely real-variable methods, without recourse to complex analysis methods such as analytic continuation, thus giving an “elementary” interpretation of these sums that only requires undergraduate calculus_ .
And then, he is confusing supersuming and analytic continuing. Eta isn't extended by supersums, but by complex analysis methods, and then corresponds to some supersums values.
And in 35:20, the Numberphile computation isn't nonsense because there's a coherent set of rules which allow all the calculations made, and imply the values 1/2, 1/4 and -1/12.
And to answer your question, no we don't have to check wether the series are convergent or divergent, because the calculation is legitimate *if* such an operator (on these series) exists. So, we just have to *prove that it actually exists* , and then it immediatly follows that all the calculations are mathematically justified.
And guess what? Yes, you can prove it 😉.
For now, i'm not writing the details, but there's an excellent article on divergent series (arxiv.org/abs/0705.1578 ) which can help you to understand how we can construct this operator (left as an exercice to the reader 😝; maybe i'll post the solution if you can't find).
@@coursmaths138 Thank you for your quick and courteous reply. I think you saw an insult where more of tongue and check humor was meant. I would say math and physics are certainly overlapping disciplines, but each still have their own nuances and specialties that the other does not necessarily deal with. My point was not to invalidate the physicist’s math, but rather for the physicist to stay more in the physicist’s proper lane rather than in the mathematician’s lane, and they do as I say overlap much. My point here was in Numberphile bringing in the physicists’ credence as professionals and experts particularly as physicists in order to explain an especially nuanced part of math that is not part of physics (although can be used in physics) and that they certainly did not fully understand. Numberphile is watched by many, especially young aspiring mathematicians and the explanation was permitted and accepted as true but only via “hocus pocus” and not through the rigor for which math is done.
I also do think it also goes beyond semantics of simply what we want to (or chose) to call infinite sums. I do not think this is a language problem or an epistemological problem. In this instance, I think it is purely an understanding or comprehension problem. In elementary school I was never told x2+1=0 was false. Not being able to explain why it is not false does not make it false even if the explanation is unknown. We are both beyond that. And it wouldn’t not have satisfied the rigorous mind especially from an equation for graph point of view. However, I do think if it were explained as wrong, even if not understoof, there are not answers in the real numbers, that would have been acceptable. I just haven’t been taught imaginary numbers and that there exists it was just not taught to me until later so I could be taught and understand that ι is the √-1
Their reference to Ramanujan also I think was deceptive because it was not for his summation of divergent series but his note on it as an oddity, with no supplement, further confusing the matter for viewers.
I appreciate your in-depth discussion (the arxiv link does not work, it gives a 404 error) and also your polite tone and interaction.
Here is my biggest problem with Numberphile’s video on this. And I will give you much credit because I think you hit the nail on the head, you have a very in-depth knowledge of mathematics, and I would say myself to a degree. Numberphile usually presents a range of math from easy to difficult in a manner that most people can comprehend in order to further the desire to study and appreciate math. They also use well-known and established professors to do that.
In this video, under 8 minutes, they took and presented what in their own words is A POSITIVE INFINITE SUM OF REAL NUMBERS (not verbatim) and told their audience that it equals -1/12. At this level to present this as they have in the time they did at the very least THEY MUST discuss divergent and convergent series as Mathologer did. Mathologer had to do it in a very longer way in order to discuss the 7 minute and 50 second video which was not sufficient.
Numberphile did not talk about imaginary numbers. I brought up the Riemann Zeta function and hypothesis because I know these properties. Now, let me ask you, do you think anyone was thinking about Dirichlet regularization in this video? Or even understands it (I’m not saying they can’t understand it, but they currently did not, I would say. They talk about String Theory to back up -1/12 but not Dirichlet regularization).
I can see your issues with Mathologer and comments, perhaps even my own. However, to me it is very clear Numberphile set-up an 8 minute video talking about the infinite sum of real number equaling -1/12. I listened and watched the video, I saw the equations they wrote and referenced (math is universal, there was no word choice, and the equations they chose did not use limits, etc.) it is presented as simply the sum of all the real positive to infinity. Presented as such, and that is how I, and almost universally everyone else, interpret that particular Numberphile video, I cannot accept it. And on those terms, I think Mathologer did an objectively better job explaining how the sum of all the real positive numbers to infinity cannot equal -1/12.
Notwithstanding higher math and the wonderful work you did, and that I agree there is a place for imaginary numbers, axioms (not particularly in math I would say generally), but this video was not the place for it. In the end, I will be glad to say it was poorly explained on their end. I do not think, even with your explanation, that your explanation is what THEY meant. 🙂
its not a paradox its a mistake
yeah the indian who “proved” it was a fraud
That is what a paradox is !!!!
@sakshamraj1566 i dont think so. I consider a Paradoxon as a problem which is solved logical but the solution isnt logical. But in thos Case the way of solving the Problem is the mistake . Because 1-A=A is only correct If one A ends with +1 anda so A=1 and one A ends with -1 and so A=0 . The way used for B is based on the same phenomen. The result for a + as last sign is not the same as if the last sign would be a - . Even the length is Not defined and so you Just Take an average of the results what is a good way ti simplify the Problem but technically Not absolutely correct the way its written. And so even the S=-1/12 is Not a paradox but based on the simplifications the only result which could come out . I would declare It this way: S IS the sum of all Numbers until n So S=Sum(x=1,n)=(n+1)*(n/2)
But I havent studied Maths at University so I might be wrong
well a paradox is not something that can be proven till now obviously but it can be applied to prove others things .
It is not. This is used as a regularization method in Physics. There is a paper using this to explain zero-point energy. There are also other methods giving meaningful finite values to divergent quantities. An example is the analytic continuation of the Mellin transform or finite-part integration. It might seem weird to Mathematicians, but these kinds of methods were found to be useful in physics with EXPERIMENTAL evidence. People used to mock the notion of imaginary numbers, and then boom: complex analysis.
Fascinating. I like this. Looking forward to learning calculus.
Overall a great video. However, there is a bit of an error. The Riemann Zeta function is not defined to be the sum, but instead the analytic continuation of the sum and thus zeta(-1)=-1/12 is true. Whereas the video says it's defined to be the sum, and the analytic continuation of the sum at -1 is -1/12.
The Riemann Zeta function may not be defined as the series, but zeta(-1) coincides with -- is the same thing as -- the series. So the sum is -1/12. As for the value -- the (part of) mathematics connecting to this part determines it numerically. This is valid because these parts has other -- from this matter independent -- connections.
@ No, the Riemann zeta function coincides with the series only in the region of the complex plane where the real part is greater than 1. Otherwise, that series does not make sense. Usually mathematicians call Riemann zeta function the analytic continuation. ζ(-1) is NOT the series evaluated in -1, but the value that the analytic continuation assumes in -1 (and cannot be expressed by the usual series that defines the function in the domain I referred to previously)
Lorenzo Barbato The Riemann zeta function is much older than Riemann but had another name of course. The treatment for complex values are due to Riemann and followers. The important thing is that the series has also a multiplicative definition. Euler was investigating the structure rather than particular values. The modern view of the zeta function may not easily correspond to the series but the original zeta function does. So this is a matter of interpretation. The important thing is that the value is -1/12. A value which is determined by methods that ultimately arose from the studies of the structure. Analytic continuation didn't come out of nothing.
@ No, the Riemann zeta function is not the same thing as the series, it is a meromorphic continuation of the series using its functional equation. It does not evaluate to the same value when Re(z) = 0. This gives 1/(1-x) when |x|=1, the series diverges. This means that 1/(1-x) is a meromorphic continuation of the series. A continuation does not mean that you can now evaluate the series outside its original domain, it is something completely new.
@@ethanbottomley-mason8447 No. It is not something completely new, The functional equation is not arbitrary and can be obtained from both "starting points". Via Abel, Malmsten, Riemann, Mittag-Leffler, Weil for example. But you are right that considered only in a narrowly defined series setting it is a a new thing. Of course the very goal of generalising is to unify.
Infinity - Infinity is undefined, when you subtract you didn't consider about the last terms probably +infinity or -infinity. Then we can't do it.
3:55.
Ahh, it's only undefined when we don't know anything about the infinities. Lim (x->inf) (x+1)-x is always 1, even though in the limit it is inf-inf
Infinity - infinity = infinity, not undefined. Check out Hilbert's hotel paradox
@@rykehuss3435 that's just wrong, you can't just compare infinities like that, it's undefined. Infinity minus infinity can be anything.
@@AbelShields Hilbert's hotel paradox proves infinity - infinity = infinity. Go read about it.
thanks for actually showing *how* the maths breaks down when you just assume things that don't hold
Ramanujan found it right?
Ramanujan summation is a beautiful tool, anytime I see -1/12 I can't help but think about it
Ikr! Same
It’s nonsense
@@FermionPhysics ur mom
@@FermionPhysics shît up
@@FermionPhysics It isn't nonsense.
I guess one of the issues that I've always had with these sums is that it looks like a simple proof by contradiction. Where the base assumption we're making is that we can assign a value to a divergent series at all. It also appears that depending on how you work with your divergent series you can derive other values that are not necessarily equal to each other. Sounds quite similar to how if you're not careful you may assume that 1/0=1=2=..., which is obviously a contradiction because 1/0 is undefined. However, people are free to define their own 'rules of the game' so to speak.
yep that's what it looked like to me, If I say that 4x= 5x and then divide the entire thing by x then 4 = 5. Since you don't know what the divergent series are and you are doing something to estimate the result no shit it winds up wonky.
My explanation for this is not that distributive law does not work here. I think it does, but the problem is in the step of converting "S = 1 + 9S" to "-8S = 1". You effectively subtract infinity from another infinity, that's indefinite, i.e. it can be any number (or infinity as well).
And where'd you get that S is infinity
@@mbrusyda9437 1+2+3... = infinity. From there.
Hilbert's hotel paradox proves that infinity - infinity = infinity
@@rykehuss3435 that explains nothing...
@@mbrusyda9437 Then you are really stupid. You asked where he got that S = infinity, and I just literally showed you very clearly where from. Try using your brain
Partial sum formula: y = x(x+1)/2, area under this curve from -1 to 0 is -1/12
Completely counterintuitive. Crazy.
My opinion is: S = 1+9S & S-0.25 = 4S could be equal to two solutions: one is the algebraic usual way -1/12 & -1/8, two is the way that says S could be ∞ (infinity).
What should keep people up at night is that there a few who believe this wrong result.
I am glad how ramanujan ( indian mathematician and discoverer of this series) discovered this while being in 12 class and blew the worlds mind, true genius
6:36 is where it forces divergence and gives a wrong answer otherwise.
An excellent illustration of the pointlessness of wondering about divergent infinite series.
Even my 15 year old brain could figure out that this doesn't work because, say you take S-S, but shift the terms of the second S one space. That gives you S-S = 0 = 1+1+1+1+1+1+1+....
The normal summation doesn't work in these type
We're just playing with infinity in ways that we would never know because... infinity
Your method gives us: 0 × S = 1+1+1+1+1+...
0 × infinity is undefined.
@@j.r.8176 no, we are not talking about the product of limits there
@@j.r.8176 we are not multiplying by 0?
An interesting way to get the initial result for Grandi's series is to use S = 1 + p + p^2 + p^3 + ... + p^{N-1} = (1 - p^N)/(1 - p), which is pretty straightforward to derive geometrically, and setting p = -1. When 0 < p < 1, (1 - p^N)/(1 - p) = 1/(1 - p) in the limit as N -> infinity, but when p > 1, the typically quoted sum formula of 1/(1 - p) actually gives the result of S' = S + p^N/(1 - p). This means it's no longer an infinite sum as N -> infinity, but the output is still "correct"; evaluating 1/(1 - p) at p=-1 gives 1/2 for any value of N, because the p^N/(1 - p) term in S' ensures that the result is always 1/2 without any magic or faith required. We can also get 1 + 2 + 4 + 8 + ... = -1 for p = 2, which is another famous result, and again, the p^N/(1 - p) term ensures that this is true for any value of N. Similarly, plugging in the manipulations in your video gives -1/12 for K = 1 + 2 + 3 + ..., but you end up with the unfortunate situation that K + (-1)^N/(-12) = -1/12. The issue seems to be that the manipulations required to get from Grandi's series to 1 + 2 + 3 + ... are invalid.
It doesn't even have to be valid. Z(-1) and the series discussed are litteraly the same and Z(-1)=-1/12.
It is the direct result of the Unicity of Analytical Continuation.
Eh, I have issues with the way this was explained. For one, there is this inherent assumption that it only makes sense to talk about series in the context of convergence, all without even an attempt for an explanation for why that must be the case. I think that is flaw. Also, the counterargument presented in the video, regarding the distributive property, does not work. The video, at one point, explained that if we take terms of 1 + 2 + 3 + ••• by groups of 3s, we end up getting 1 + 9 + 18 + 27 + •••. But the issue is that this is not allowed by the distributive property. Instead, it assumes the associative property, which no one claims holds for series, not even for convergent ones (see the Riemann rearrangement theorem). The series 1 + 0 + 0 + 0 + ••• = 1 + (- 1 + 1) + (- 1 + 1) + ••• = 1 and 0 + 0 + 0 + ••• = (1 - 1) + (1 - 1) + (1 - 1) + ••• = 0 are both convergent, and so this demonstrates already how associativity does not work with convergent series. The series are different, and so no one expects divergent series to be the same under a regrouping like that. And both of the series mentioned above are different from the series 1 - 1 + 1 - 1 + •••, which is divergent. It has nothing to do with the distributive property failing.
And on that note, that leads to me the biggest issue with the video, which is the treatment of series as a kind of summation of numbers. To be fair, the actual mathematical terminology we use does make it sound like we are just adding numbers together. But this is inaccurate, and that is revealed by the fact that we talk about convergence and divergence of series in terms of their partial sum sequence. Series are just that: sequences. We can add sequences, multiply them, and we can apply operations to them that will output a real number. Evaluation of a series is just that: applying some combination of operations on a sequence. Addition is involved in some of the operations, but describing the entirety of the operations as just being "summation" is misleading at best, and it leads to the kind of misunderstanding that led to the infamous Numberphile video in the first place. And this actually explains why the Riemann rearrangement theorem is true, and why "associativity" does not work for series: because in reality, we are not merely adding quantities, we are applying some kind of transformation to a sequence, and while addition is somewhat involved, there are also other things involved exclusive to sequences, that have nothing to do with adding numbers, and that is what makes this whole "associativity" issue not work. It does not invalidate a rigorous treatment of divergent series, which does exist, it just means we have to be more careful with our understanding of series. That is also why I find the insistence on focusing on the limit of partial sums to be a bad argument to use in this context: it is self-defeating. It misses the point of how series work, and it fails to appropriately bridge the gap between how series work, and how they are portrayed.
But, the video does a good job at explaining the role analytic continuation plays in all of this, and the video does clarify that, ultimately, this is a matter of context. I just wish that explanation was better supplemented by providing a conceptual understanding of what a series really is, and why it is a conceptual mistake to just think of these manipulations as summation, and of these properties as analogous to distributivity and associativity.
Thanks for the great insights Angel! Hope you're doing well.
I did not read your blog post so it seems you forgot the issue of being brief.
k
@@kazedcat never said that
learn to read bozo
Yes, it withholds the information that product of divergent sums sometimes converge conventionally, so using the convergence as some kind of test or filter is a red herring at best and misdirection at worst.
If it’s beautiful it’s true
thanks for clearing that up.
Why we feel it isn't -1/12 is because it seems to us that how can adding so many things equal negative number, but it is whwt it is, math is defined that way, and nothing to be surprised, because that's the truth! , its ok to surprise on a thing different from truth, but we have no right to decide by ourselves that it is infinity, it is that way!
I think it’s because 1-1+1-1… is never actually 1/2, it averages out that way but the result is always either 0 or 1
Its a summation of an infinite series. In calculus, it is undefined
To me its like an excuse to not being able to find out whether its gonna land on 1 or 0
Yes you are right indeed, in that case, since we have two results(0,1) for a single question, we would take the average of the result which would be 0+1/2 = 1/2…
I'm curious as to whether there exists some alternate metric on the natural numbers under which this result converges, or at least a partial proof...
I can't believe this topic is a thing. Many videos on YT explaining the issue.
I know. This sum cannot equal -1/12. Ramanujan was wrong hehe
wow!!!
how do you even get to think about this?
I love how he explains this to us like we understand right away
I mean, you should
My favorite thing is from string theory where I was asked to compute the sum of all ODD numbers.
sum_n>0 = sum odd + sum_n>0 (2n) so you split all numbers in even and odd. But then , sum odd = -1/12 - sum_n>0 (2n) = -1/12 - 2 sum_n>0 = -1/12 - 2*(-1/12) = -1/12 + 2/12 = +1/12 lol.
So even if the odds are clearly a subset of all integers, their sum is "bigger" than all integers. Peace
That's an odd result :)
Let
S=1+2+3+4+5+6+7+8+9+10
S=1+(2+3+4) +(5+6+7) +(8+9+10)
S=1+(9)+(18)+(27)
S=1+9(1+2+3)
S=1+9S( now this is where all mathematician doing mistakes
Because 1+2+3 not equal to S)
But during summing infinite no. They are taking taking it as S , you are summing infinite term doesn't mean u can ignore initial term so don't waste your time doing this its never a -ve.
You r right bro
Using the equals sign with something with no value is the equivalent of using divide by zero to make anything equal anything.
It's just a matter of convincing. You can arrange & group the digits differently to get a different result and convince otherwise.
For people who want to know, this is also called the Ramanujan Paradox
I know you can't just treat the end of the infinite series 1+2+3+4... as an actual number but lets say its k (even though it doesn't actual exist). So when he shifted over all those values and took the sum the very "last" number k would have no pair to be added to. This would mean it also is being added to 0 just like the initial 1 on the row above (because of the shift). Taking the limit, k = infinity, the sum is therefore equal to infinity +1 which is also infinity.
it would be like saying which is larger -1+0+1+2+3+4... or 1+2+3+4....? Both are approaching infinity.
I know there is a high chance I am violating some sort of rule in math but I am no educated mathematician. I understand limits but when they are being used in this way my confidence in my logic is very low. But if someone notices an obvious flaw let me know because i like to know if I made a valid point or learn where i got it wrong.
infinity + 1 is a different "infinity" to the that which you added 1 to. There are an infinite number of different infinities.
1:33 B + B is 2B
me: 2B… or not 2B?
Adding a positive number to another positive number will always make a larger number.
At the very instant you arrived at 1-A=A, I pictured Fonzy, on water skis, jumping a shark.
Let's suppose s=♾️. In 2:51, you get 4s-s, which is 4♾️-♾️=♾️-♾️. That is an indefinite expression, it cannot be calculated and creates paradoxes
I think the very first step where we prove (1+1-1+1...)=1/2 is wrong. Actually, the (1-A) series will always have an extra 1 or -1 at the end and thus it cannot be equal to A. The only question that remains is whether we can assume the end of an infinite series, but i guess infinity is a theoretical concept and so it doesn't make sense to think that this series would go on forever without assuming the final number in the series.
Hi I'm french and I love maths I can't do maths in English in my high school so I do maths in English with your videos and I understand all you say thank you so much for these ❤️💪🏻
This is just a very convoluted way to divide by zero to hide the fact that you’re dividing by zero.
The series A is indeterminant, it doesn't equal 1/2 since it doesn't have a solution in the first place.
0:19 it equals to א0
The positive summands apparently adding up to a negative number made me kind of anxious, to be honest.
the way I always say it is "1+2+3+4+5... doesn't equal anything, it's a divergent series thus does not converge, but if we assume it does converge to a finite value, you get -1/12"
But that is a categorical mistake. Equality and convergence are not the same thing, so saying the series is not equal to anything because the partial sums do not converge is a mistake.
Convergence IN THE LIMIT and equality is the same thing. That what it is what is made/used for. There are infinite series that are truly divergent such as the harmonic series. The actual series has the numerical value -1/12 and the non-numerical value infinity.
@ The harmonic series is regularized to γ, so that defeats your point.
You can regularize any infinite series or infinite product. In particular you can show that the series sum to -1/12. The regularized value for the harmonic series is the difference between the harmonic series and the natural logarithm at infinity.
@ Both are intimately related by the Euler-Maclaurin formula, which is what Ramanujan summation is defined in terms of and is formally based on. Essentially, if we have some function f : R -> R, we can restrict it to f* : N -> R, such that we can ask about the summation of f*, the integral of f, and then analyze the difference between the two. This is what leads to Ramanujan summation and regularization.
Moral of the story kids: infinity, and our necessary assumptions on the behavior of the natural world, don't play nicely together.
The biggest issue is the use of the equals "=" symbol. Equals means "exactly the same as" - at least it does in engineering. So all you can say about a divergent series is that it equals a divergent series.
Correct me if I’m wrong but it seems like you can make a lot of weird answers to stuff like this because you are dividing infinity by a finite answer which is still infinity. So you can say s = inf and if we manipulate it we can say it’s equal to 1+ 9s so that makes s equal to -1/8. But in reality both are infinity. Love to here some insight on this and I loved the video
*"I applied the distributive law to a divergent series and proved that infinity equals -1/12. CONSEQUENCES!"* - BriTheMathGuy (2022), channeling his inner Key & Peele.
Not really, he applied a pretty crude regularization to this sum and got a correct answer. You can try another regularization, like exponential one, for example, and get the exact same result.
@@ДмитроПрищепа-д3я And what if I apply this regularization and got a correct answer. S = 1+1+1+1... there fore S = 1+S, which can be reduced to 0 = 1. I just proved with math that 0 = 1.
This is why you cant treat infinity as a number. S = 1+2+3+4... = infinity. Not a number, and definitely not -1/12.
@@rykehuss3435 "S = 1+1+1+1... there fore S = 1+S" prove that.
You can't even add a zero to the beginning of an infinite sum and expect it to be regularized in the same way always. Indexing matters in some cases.
"This is why you cant treat infinity as a number" nobody does that here tho.
"1+2+3+4... = infinity" only under weak summation methods.
"Not a number, and definitely not -1/12."
you've been proven wrong and yet you insist on this like your life depends on this. Why?
@@ДмитроПрищепа-д3я Here's proof that S = 1+S. You cannot say where in that line of +1's you cant fit another one. Therefore S = 1+S. Also S= 1+1+1+1+S.
And with that logic S = S+S.
Because infinity + infinity = infinity. Also infinity +1 = infinity.
Proven.
"This is why you cant treat infinity as a number" nobody does that here tho. "
Everyone who thinks that 1+2+3+4... = -1/12 is treating infinity as a number. Including yourself. And you area dead wrong if you do that. Ask any real mathematician.
"1+2+3+4... = infinity" only under weak summation methods."
I dont care if its weak or strong, and neither does mathematical logic. You cant sum all natural numbers and say its some number S. Its infinity, and nothing else. And infinity is not a number.
""Not a number, and definitely not -1/12."
you've been proven wrong and yet you insist on this like your life depends on this. Why?"
I just proved you wrong pretty easily. Maybe you should start learning some math and logic before you speak next time.
@@rykehuss3435 "You cannot say where in that line of +1's you cant fit another one. " I can say that and you just proved why you indeed can't just add new members in front of a series and expect it to have the same value. Can't you really see how you contradict yourself? It's amazing, honestly.
"Everyone who thinks that 1+2+3+4... = -1/12 is treating infinity as a number" wrong. They just treat equals sign differently. In case of infinite series, even convirgent ones, it doesn't mean that you literally add up all the members of the series one by one, if you didn't know. It just means you used some method to assign a value to this entire series.
"I dont care if its weak or strong" once again you show that you don't have any idea about the things you try to disprove so hard. It's really odd behavior, honestly. It's like if you heard about Riemann integration and then learned that there's extensions to it, like Lebesgue integration and started screaming about it being wrong because it allows one to integrate functions that aren't Riemann-integrable.
"and neither does mathematical logic" wrong, it cares a lot about it. Though here we don't exactly discuss logic, we discuss series and their regularization. Logic comes from the way we structure proofs from axioms we pick.
"You cant sum all natural numbers and say its some number S" I can tho. You just don't wanna learn how to do it and why it's useful.
"I just proved you wrong pretty easily" you've proven *yourself* wrong very easily. I said that you can't just add random stuff to the beginning of a series and expect it to be regularized in the same way under all methods. You then added random stuff in the beginning and then showed that it gives you a different result, proving my claim. Then you perform some exceptional mental gymnastics and conclude that I was wrong. It's really incredible how you managed that.
Technically your proof goes in the wrong direction even earlier. You assumed that a value assigned to a series directly corresponds to adding all the numbers in it using regular summation rules(or in other words S represents a whole series as a sum of its members as an arithmetic expression and can be interchanged with a value assigned to it). That's wrong even for converging series and you should know that.
The equations A and B are not convergent series. You cannot do these operations like shifting etc. this proof isn’t the right proof.
A is always between 1 and 0, maybe 1/2 makes more sense, but B just keeps getting bigger gaps, it isnt convergent at all
i like assigning divergent series to variables
Straight away it doesnt seem right to do regular algebra with an infinite sequence as if it was a regular number.
Easiest way to add consecutive numbers starting with 1 is that if the number ends in an even number, just divide that number by 2 and multiply by the next number. If the last number ends in an odd number, divide the next number by 2 and multiply by the last number. Example: 1+2+3+4+5+6+7+8= 8/2=4, 4*9=36. 1+2+3+4+5+6+7+8+9=, 10/2=5, 5*9=45
So?
I gauss that would work (pun fully intended)
_You will get the S when you fix this damn door_
~Maguire the Bully
Honestly, what if this is real? What if our universe is a simulation and adding up all natural numbers causes an overflow?
You cant add up all the natural numbers because there is no end to them. Infinity is not a number
Don’t jump to a conclusion so fast. This is basically as thoughtful as flat earth, it’s really just infinite numbers working weirdly, probably.
Agree with you as I think what about the concept sum of positive is again positive and here result is negative. Strange or should I call it wrong.
Ramanujan sir great mathematician from India 🇮🇳
Yes
This sum cannot equal -1/12. Ramanujan was wrong hehe
We first have to prove that S is a number... Or else what just happened is: infinity - 1/4 = 4*infinity which is true
Problem is at 1:05, i dont get it. If 1-1+1-1+1... has even terms it get zero and odd terms it gets 1 how can we assume them as same we added one later and wrote it same as A .There are many flaws thats why i am never comfortable with this idea of summation of natural number or any increasing AP to infinity.
05:45
You could also recursively define it.
S = 1 + 9 ( 1 + 2 + 3 + ...) = 1 + 9S = 1 + 9 (1 + 9S) = …
You could say S = 1 + 9S then 1 = -8S and then S = -1/8
Or you could continue with the recursive definition to say S = 1 + 9(1 + 9S)
1 = S - 9(1 + 9S) = S - 9 - 81S = -80S - 9
10 = 80S
S = 1/8
damn
I was looking for this comment
He the second ine also equels -1/8
You forgot the minus in last one
I'm in University studying Engineering. Part of my degree is learning about sequences and series. What I just saw here was the mathematical equivalent of snake oil.
What you mean? that the sum is indeed -1/12 or that it is not? 🤔
Infinite sequences and series have their own rules.
Hey bro, i have a doubt that,
A=1-1+1-1+1-1+....... Here we don't know the last term's sign, so let A=1-1+1-1+1-1+......±1,so when take -1 as common, we will get A=1-(1-1+1-1+......∓1) so we can't put A and get A=1/2.
=> A = 1-1+1-1+1-..... ±1
A = 1-(1-1+1-1+......∓1)
A = 1-A
Given by legendary Ramanujan sir and credit by this man 😅
Legendary person in the history of mathematics who shook the entire West
Great Indian and Mathematician SIR RAMANUJAN ❤
At 1:54, you're leaving out a term in the second B. If you include the -6 on the bottom, then you have to include the +7 on top, which requires the matching +7 added to the bottom... so this trick of moving the numbers when adding uses an inequal number of terms on the top and bottom. So then why couldn't you rearrange the numbers as you please and create any value that you like? I think this violates some basic rules of algebra.
In order to remove the -6 from the right, you need to add it to the left, so your result would be 2B + 6 = 1 -1 +1 -1..
Then if you instead stopped at 200, you would get 2B +200 = 1 -1 +1...
I think the real issue here is that it's dangerous to do concrete algebraic operations on abstractions such as infinite series.
Yep, that was also my question while watching the video
I think the same
This is the main problem
i know whats going on here, the set of s gets so big that it encounters an integer overflow and defaults to -1/12!
12 factorial?
@@napalm5 nah -1/12 factorial
Best answer!
@2:03
B1 + B2 = 2 B2 + nth number from B1
i feel this is as math shenanagins as getting 1 to = 2, but just dealing with infinites, of a type. which sucks because we do have to deal with infinity, in a weird way, at least due to pi
We can also use divergence/convergence tests to test whether the series actually come to a point, and we find that “1 + 2 + 3 + …” is divergent by the nth term test, and we find “1 - 1 + 1 - …” and “1 - 2 + 3 - …” are divergent by the alternating series test. Thus, you can’t find an actual solution for these
This only prove that the limit of partial sums doesn't exist. Not that it has no "actual solution".
@@coursmaths138 Ok but then you have to define what an "solution" would actually be. The epsilon definition of convergent sequences (along with the extension of this definition to series in terms of its partial sums) is what people generally are referring to when using the equals sign next to a series. The -1/12 comes from the analytic continuation of the Riemann Zeta function evaluated at -1, which is not the same as saying that 1+2+3+... actually converges to -1/12.
@@garethreynolds557 i mean, people do use the -1/12, as in the often mentionend casimir effect. So, in a way physics confirm the result experimentally. It seems like "sums" aren't defined clearly enough to concluce that the natural numbers do not converge at -1/12 or go to infinity depending on where you use it.
3:40 "You can't prove the distributive law." Yeah you can? In a general ring-theoretic setting, sure, it is a proper axiom that you don't prove, but to show that the integers, rationals, reals, etc. form a ring, you need to prove the distributive law for that ring. It's not even especially difficult to prove the distributive law, you just prove it for the naturals with recursion, then lift it to the integers, rationals, and reals by unpacking the definitions and applying the natural number version.
I have always visualised the distributive law as the sums of areas of rectangles that form a larger rectangle, where the larger rectangle has the length (a + b) and the height (c + d);
in my opinion, this is by far the most intuitive way to teach the distributive law.
It would be nice if you mentioned the relationship between Ramanujan's result and the Casimir effect in physics.
The way I look at it, this result proves Goedel's incompleteness theorem.
A key way that has helped my understanding of these ramanujan summations is that one should not allow themselves to "combine" terms into one term. As long as you do that, results tend to be fairly consistent
I tried showing this result to my Calculus teacher but couldn’t remember the whole process of getting to that -1/12. Now this result really does keep me up at night.
you can't put the S there for the infinity
One of these sums involves the partial sum of an alternating series which is conditionally convergent at best. In a conditional convergent sum, the order of addition and subtraction DOES matter. So 1-1+2 = 2 + 1 - 1 for a finite set of numbers, but not for infinite
Well when you denoted the sum as B the first term of the series was 1 and the common difference was also 1
But in the second case when you added the 2nd term of the first B with the first term of the second B, the second sequence cant be termed as B since the 1st term is now zero
So basically you added there A+B and said the sum to be 2B which is definitely not true as the first term of both the sequences differ
At least this was what i could understand. Please correct me if I am wrong. Also please ignore the difference of the term sequence and series. I get confused between both of them frequently. At least thats what you can expect from a student of 9th standard
Power of Ramanujan
It is worth clarifying that this -1/12 result IS ACTUALLY used in a lot of physics (and probably elsewhere too).
Thanks to Ramanujan for all of this.
May his soul rest in peace 😔.
What is the application of his discoveries?
@Occ881 Until very recently, his formula was used as the basis for the fastest approximation of pi in computers.
Also, his number partition formulas are used to study black holes.
i suggest to read works by Ken Ono, who explains Ramanujan's contributions in detail
This sum cannot equal -1/12. Ramanujan was wrong hehe
@@CriticSimonhow?