so basically if someone wanted to give you $1 today, $2 tomorrow, and so on, you shouldn't accept the deal because they would be trying to steal from you.
@@MrPacoHamers Well no, because the value of an infinite stream of cash flows isn't adjusted based on the lifespan of the recipient, as the cash flows can be sold to another party at any point in time. That's why the theory of a the price of a stock being equal to the present value of all its future cash flows is rooted in infinite time, not a human lifespan: you can SELL the stock at any time for immediate cash.
"It could be wrong, that's fine. Mathematicians are wrong all the time, that's why they're discovering cool new stuff." And this is what makes a great teacher! None of that "you have to be right or you're failing and it's your fault" nonsense we often see (bad) teachers and educational systems do, which only discourages people from wanting to learn more and learn from their mistakes.
People are wrong all the time because they are just wrong, though, like in this video. Setting the sum of a divergent series to a discrete number is flat out wrong in mathematics. Being wrong doesn't always mean you're on the path to discovering something new.
@@eXJonSnow that's not the point though, it's about trying and proofing you're either right or wrong. That's how you learn and improve yourself. That's why it's allright to be wrong and how you discover cool new things
@@Leviathan-gp2kv The guy's a college professor. If you get the answers wrong on his exam, you're going to fail the course. Period. You can be wrong while learning, but when it comes time to apply what you learned, it better be spot-on. Encouraging learning and encouraging mediocrity are two very different things.
@@KamuiAlmighty I don't necessarily mean in class. BTW is this college level math!!!??? But I mean even scientists makes mistakes without the mistakes you never come to a breakthrough. No one has is exactly right the first time
I have a theory as to why you cant solve it like you are trying to solve it. And heres WHY: 1 minus series-g does not equal 1/2, it equals to infinity. 2 times series-alfa does not equal 1/4, it equals to infinity. Series minus series-alfa does not equal -1/12, it equals to infinity. Don't ask me why, u, I dont know yet.
He accidentally damaged 1/12th of a whiteboard that belonged to his colleague. He apologised to his colleague and offered to replace the damaged part. His colleague said they will deliver one whiteboard to him on day one, then two whiteboards the next day, three whiteboards the day after and so on. Here we are on day two.
Long story short: adding, subtracting, dividing, and multiplying both sides of an equation is illegal when dealing with infinity, because: infinity + 1 is still equal to infinity.
@@Sixsince-dd2eu well, you can. Seems like people have a really hard time understanding that it doesn't work with classical definition of a sum of series. You can't even take a sum of infinite series in the first place, because, you know, it's infinite. You need to define first what sum of a series is and then talk about what is possible or not. In classical definition, that result would not work, but you can still assign values to non-convergent series, but you may lose some properties (like linearity). Think of - 1/12 as "generalized sum".
@@clawsie5543 Thats when you're assuming that the numbers don't end, which then will be equals to, infinity. But here we are not doing that we are just adding all the numbers up. In fact, thats the point of this theory. This is to prove whether there is an end to numbers or not Thats my view on this correct me if Im wrong
@@cnoz2378 I don't quite understand what you are saying. The point is that you can't just sum infinitely many numbers, because you can't apply normal algebraic rules. Algebraic rules can be applied only for finite number of numbers. In order to sum an infinite sequence, you first need to define how to do it. If you define it as the limit of partial sums (which is the "classical" definition), then you get that sum diverges towards infinity (what you would expect). But the result -1/12 doesn't mean that the limit of partial sums is -1/12, it is diferent way of summing than classical definition suggest, it is more general way.
I know why Eddie cant solve it like Eddie is trying to solve it. And heres WHY: Infinity-1 does NOT equal infinity. Hilbert himself said: some infinities are bigger than other infinities.
@@WhiteStripesStripiestFan I know why u cant solve it like you are trying to solve it. And heres WHY: 1 minus series-g does not equal 1/2, it equals to infinity. 2 times series-alfa does not equal 1/4, it equals to infinity. Series minus series-alfa does not equal -1/12, it equals to infinity. Don't ask me why, u, I dont know yet.
As an English and Maths teacher, I have a big passion for teaching the history of Maths. I always try to humanise mathematics by telling my students the stories of the people behind these theories and results. I feel it makes things more connected for students. Ramanujan's story is a story worth telling. The film "The man who knew infinity" is also a great film for exploring two modes of getting to mathematical results, inspiration or proof.
This has absolutely nothing to do with Ramanujan summation. Woo simply demonstrated that permuting the summands of a divergent series can change the behavior of the series.
I love the same thing (History of Maths)! I am a still a student and until I came upon the stories or the history of the people before us who discovered the things that we are studying today and how they approached them or how they failed and later people throught the time had other points of view of these problems and solved made me love mathematics even more than I used to. I really do believe that to understand and be able to solve a math problem ( famous hard problems especially) a bit of history of how the people before us tried to tackle these and their ideas is really useful.
@@Onoesmahpie For all your arrogance, you're dead wrong. Changing the order of the terms of a divergent series does not make it convergent using ordinary addition, because it's commutative. The super summation must be used to assign a result to a divergent series. What's shown in the video is a classic Ramanujan super summation. Btw, the proof shown in the video is just a trick (used by Ramanujan himself and still excellent to tickle the mind!). A more robust proof is much more complex (Riemann zeta function and analytic continuation get involved). A great deeper explanation is provided by Mathologer: ruclips.net/video/YuIIjLr6vUA/видео.html
That's really funny to say because we can assign -1 to 1+2+4+16 +... +2^n+... making it "when you add 1 to 1111...111 you get 0" that is indeed some kind of "memory overflow"
@@higorss Of course it's not a "memory overflow" (compturers don't have infinite memory), I'm saying it's "resembles" it. First, (my bad, I forgot to precise,) but 1111...111 is considered base 2, then 1111...111 doesn't really "exist", if you want mathematical rigorousness ; what we are dealing with is an application, let's call it S, that associate a value to a sequence : to finite sequences it associates the "true" sum, to a converging sequence it associates its sum (therefore S(u_n)=u_0+...+u_n when u_n is finite or converging). We want S to act "like" a sum, therfore we want S to be linear and also we want S(0,...)=S(...). This way we can give to some other sequences (the non converging ones) 1 unique possible value that we call "sum". So when I write 1111...111, in fact I'm speaking about S(1,...,1,...) that is perfectly well defined mathematicacly and has to be equal to -1 (in order for S to have the proprety of a sum, I can give a proof if you want). Then I notice that 1+S(1,...,1,...)=1-1=0. So when you write it as 1+1111...111=0 it "resembles" a memory overflow. In conclusion, base 2, modulo 2^n+1, the finite number 1111...111 (with n ones) + 1 = 0. That proprety is called "memory overflow" in computer sciences. My comment is saying that this proprety also appears with infinite sums. So when @Sion mentionned it, it was nice intuition.
That proofs that infinite divided by 2 not equal infinite. When he subtract Sa from S he lost half of the length of Numbers and yet he still compare them by location and asum that it is bigger by four times. Tricky
I’m out of college for a decade now and I still find his lectures fascinating. I wish I had access to them when I was in school. Thanks and keep uploading these captivating math videos.
And if that is the case cool, that does make for a very good class, BUT it also makes for a horrible youtube video, since we don't get to see that context, meaning he very well can be the start of false information spreading
Gummiel context is everything. Have a look at the rest of his RUclips clips and you’ll see that he is always teaching and never spoon feeding. It’d be like taking bits of what Mustafa wrote and claiming that he said “the whole point of this class was to ... spoon feed them correct ... answers.” He even ends by telling students to go and research this controversial proof - sounds like he’s trying to peek interest, inspire, challenge and encourage students to “think about solutions”, maybe even realise that even very smart people sometimes make fundamental errors.
Indeed you are right that context is everything here, and as I said that makes for an awesome class, but as a youtube video, it is BAD. You can't expect ppl to watch every single video of his, so they need to be able to stand alone as well, and at that this video does a terrible job
Gummiel surely you can expect people to watch the entire video and hear him say this is a controversial proof and you should go online and research it. We can’t assume that everyone is an idiot and can’t be bothered watching all of it.
@@mayal5206 this isnt really breaking math tho. It seems like it is, but most of the evidence points that negative numbers and positive numbers are on both sides of the origin. We thing of the numberline too much as one dimensional, which is actually breaking math. Infinite series and multidimensional math go hand in hand. This is a totally correct way of handling an infinite series, just not in the algebraic sense.
@@Predated2 - In what "sense" then? And what does "algebraic sense" mean? The number line is, by definition, a straight line and a line is, by definition, one-dimensional. Where did you get this notion that there are positive and negative numbers on both sides of the origin?
@@mayal5206 when a circle becomes bigger nd bigger up to an infinite radius, the point at which it's any tangent touches makes the origin of a number line an looks like a number line but actually it's a circle that's why it is 2 dimensional not the one.
"Have I made you think yet?" And there it is, the missing ingredient in many classrooms. I love that he doesn't push a lesson down their throats. He guides them through the thought process that made other people discover what we know today, in the hopes that these kids will do the same even of it's in other areas in life.
Exactly I'm not sure why Eddie Woo keeps saying the result is `controversial', it is just a nod to the fact that absolute convergence is a necessary requirement to re-arrange terms. He makes it seem to his students that math is all a big confusing mess where anything can be possible. He does a bad job at explaining this, and in fact seems to be conflating the Ramanujan summation technique with this basic flaw in reasoning.
@@Onoesmahpie I'm pretty sure the intention is to make the students think.. Sure, he can explain why this proof is wrong but the purpose is to make the students curious.. Why does a seemingly logical proof result in a completely nonsensical answer? By saying that it's "controversial", which must be exactly how it seems to high schoolers that aren't proficient in convergent and divergent series, he is sparking their interest in trying to learn more about it.
@@siriusthegrim I think Eddie is awesome, just pointing out that what he said was technically not true and could have been worded better. I do applaud him for trying to go above and beyond and spark curiosity in students by challenging their intuition and showing how we cannot assume generalizations of specific results hold, and how it is difficult to recognize one's fallacious reasoning sometimes.
@Shravan subtracting a positive number from a negative number always leads to large negative number. for example,consider -1-(3)=-4.it is possible to get big negative.
@@daaaaaaanny, you're twisting the meaning of the word bigger. Try looking at the numbers going higher or lower, the numbers you are suggesting causes it to go lower still, in the technical sense, though the number is "bigger" the fact that it has a "-" next to it causes it to become "smaller" not "bigger". Also adding decimals screw over the entirety of using the words "bigger" and "smaller" due to decimals becoming "bigger" in size yet also becoming "smaller", but not in the same way as negative numbers, and a negative decimal is getting "bigger", while the number itself is getting "bigger". My point is you're wrong due to the English language because there should be a word to describe a number getting "bigger" or "smaller", "higher" or "lower".
en.wikipedia.org/wiki/Riemann_series_theorem Did you even read anything about it? It only talks about rearranging conditionally convergent series to get other convergent series or divergent series. Nothing specifically about playing with divergent series.
Andrew Sp Also, the rearrangement Theorem does not say you cannot play with divergent series like this. It merely means infinite summation is not closed under any subset of the reals, nor is it commutative or associative. As long as Eddie hasn’t assumed either, this is all perfectly valid
Andrew Sp you do realize wikipedia is not an accurate source, any one can do anything to it, that's why colleges dont allow it as a credible source for papers. which reduces the credibility of this mathematician for suggesting his students go to wikipedia for anything, unless aussie wiki is better or something
If somebody is interested: Look up the discussion done by "Mathologer". In a nutshell: Infinite series need special care or you end up with nonsensical results. Infinite series only have a value, if the partial sums converge to a specific value. If they don't then they either have no value or the value is infinte. If this converge criterion is not taken into account, then you get nonsenical results. As said: Mathologer has discussed that issue. Look it up.
I knew you would have ended up seeing that non sense video as RUclips algo partially favors mathologer very much... Please see the video ramanujan summation from channel "singing banana" . you will get a new perspective that mathologer fooled us... Please see it for true knowledge sake.
@Stanislav Hronek I think the standard is to use an analytic continuation of the Riemann Zeta function at z = -1. Regardless of how this result is derived (and whether it is flawed or not), nature surprisingly seems to agree to this. In particular, the Casimir force (experimentally measurable) between two infinite plates in vacuum can be derived with this result.
Teachers like him are priceless. They can really spark your imagination and send you on a lifelong journey of learning. If every teacher and school was like this the world be a much better place I think.
8:49 time mark _ "The question is, what does it all mean?" It means that when manipulating series of numbers, it is very important to understand exactly what you are manipulating. Seemingly innocent truncating and shifting numbers over have dramatic and unintended consequences, when evaluating multiple series. For example, take an equation C = first series of repeating ones. Take the same equation C = second series of repeating ones BUT SHIFT IT OVER ONE PLACE. Now subtract the two equations, C - C = first series - second series. What do you get? 0 = 1. That is the flaw in this video's logic, you can't shift places of values in the series when evaluating multiple equations. For me, the bigger mystery is WHY does C = first series NOT EQUAL C = second series? Don't both equations (independently) resolve/converge to the same value? The only answer that I come with is that you have to be very careful when evaluating sets of infinities. As for this video, its biggest flaw isn't faulty logic. Arguably, the logic is impeccable. The flaw is a failure to identify the mathematical frame of reference. The playground being discussed is NOT REAL WORLD. Instead, it is something called "Analytic Continuation". It is a logical extrapolation that makes sense within its respective frame of reference. Eddie Loo should make this point clear in his video presentation. (I think maybe that omission is because he assumes that viewers understand this.)
@@joshuamoyer4141 The standard definition of the summation of a convergent series as the limit of the partial sums isn't a sum in the traditional sense either. A sum in the traditional sense is only defined for a summation of a finite number of integers. You can't start with the definition of addition and derive how the sum over an infinite set of integers should be defined. That definition will be an arbitrary choice, there are many possible choices. Defining it as the limit of the partial sums is one possible definition that has become the standard definition.
There was a video (Why 0! is 1) posted to Reddit/r/Math within the past week. It was top that day. A lot of people probably found this channel through that link and are just going through the videos (Like I am)
He accidentally damaged 1/12th of a whiteboard that belonged to his colleague. He apologised to his colleague and offered to replace the damaged part. His colleague said they will deliver one whiteboard to him on day one, then two whiteboards the next day and so on. Here we are on day two.
This proof was done by a famous Indian Mathematician Srinivasa Ramanujan (1887-1920). Its applications are in General Physics & Bosonian String Theory. Although it makes me think, I keep wondering can we do such operations with an infinite series? We can do them to finite series but in a never-ending series, it becomes difficult to comprehend. The value of S should be n(n+1)/2 where n is no. of terms & cannot be -1/12 definitely but this feels logical though...
@flat_earth_forever thank you "flat earth forever" you missed the point of the proofs (the first series you mentioned the actually does converge to 1/2 btw)
@flat_earth_forever it depends on how you approach it, much like many things in math. By some methods it doesnt converge, by others it does. I think it's much more interesting (and still mathematically coherent) to say it equals 1/2
If behaviour of an operation changes it must be because your operation changes, you just didn't notice. i.e. infinity is a limit for addition, any attempts to add infinity changes the concept of adding to the point where you're not doing the same thing as you were doing before.
You can do arithmetic operations to infinite series, but only when the series converges. If the series diverges (as the sum of the natural numbers does) doing any operation on it is not valid. What he's showing in this video his hogwash, and you can use the same "tricks" to make the sum equal to whatever you want. For example here's a video that "proves" that the sum is equal to -1/8: ruclips.net/video/6FTwMUL69u0/видео.html Why would this proof be any more valid than that one? (Answer is: Neither is a valid proof, because most operations done on divergent series are invalid.)
@@DjVortex-wThe operations are valid, provided you don't muck about with shifting terms or changing the number of terms. In this video calculating 2Sa by shifting and adding is "illegal", as is equating S-Sa to 4S since they have a different number of terms.
The fact that a series diverges does not mean that the sum cannot be evaluated to be a finite number. The limit of f(x) = x^x as x -> 0 = f(x) = 1, yet this by no means implies that 0^0 = 1. The same is true for summation.
The point of this is to get rid of arbitrary rules and see where that takes us. Why can't we assign a value to a divergent series? Just because it doesn't make sense doesn't mean it's impossible. 1-1+1-... can be written as inf Σ((-1)^n) n=0 Well that looks like a geometric series, and we know how to calculate infinite sums with that, so let's try it. 1/(1-(-1))=1/(1+1)=1/2 Well that sure looks familiar. And if you're not happy with pure algebra, throw in some calculus and take the limit as the base goes to -1, and you get the same answer. For all intents and purposes, the answer is 1/2. And the final series in this video does indeed equal -1/12. It shows up in physics, which just further confirms it.
I have a link in which expert mathematician Terrence Tao why the argument of convergence vs divergence is not convincing enough to present an objection to the result of the series as equal to 1/2.
Omg, this is what I hate the most about RUclips Comments, Its either People are believing This Proof or People are not believing This Proof or People dont know what to say, it's a Massive Mess
This is the cool part about mathematics... scientists are always trying to ask the right questions to find the right answer and expand knowledge, while mathematicians are simply chasing whatever curiosity their nerdiness comes up with, and then recording their results. A lot of important rules in mathematics have come to be, simply because some nerd was like "I wonder what happens if I ____?".
i think the most confusing part of this video is why is there a smaller white board attached onto a larger whiteboard? and why do i suspect it has something to do with the school ordering new boards to use up a budget so that they get the same budget next year, and ending up buying smaller boards than they already have.
Maybe some student didn’t like this teacher and smeared shit on the big whiteboard and instead of cleaning it up because education never gets funded properly they just put another whiteboard over the smeared shit.
This was first proved by "S Ramanujan- The Man Who Knew Infinity" He was a great Mathematician and today his theories are the basis for the study of Black Holes🙏
Ramanujan is the absolute goat of maths and for some reason we never learned about him in school. Im 31 now. I first heard about him at 29. It shouldnt be like that. If the guy was from a western country, I’m sure he would be as famous (or more) than Einstein to us.
Credits to S Ramanujun for bringing up this amazing theorem… He was a great Indian Mathematician who sent this exact theorem in a letter to a British Mathematician, G H Hardy on January 16, 1913 asking for this theorem to be published under his name.
No he didn't wanted this to publish. This proof was just found in his books. This proof is wrong btw and it's been debunked many times. Look up about Reimann Zeta function.
Heres something I thought: S=1+2+3+4+5+... S-S=1+2+3+4+5+...(-1-2-3-4-5-...) But hey, dont take that -1 from the 1, take it from the 2, and so on, we get: S-S=1+(2-1)+(3-2)+(4-3)+... S-S=1+1+1+1+1=infinity 0=infinity ????? Infinite sums are scary shit to work with
@@magicUFO Not correct, It may look like infinity series but it's not. The Sum of all the 1s come out of subtraction will be equal to last number and they both will cancel each other answer will be zero. S-S will remain zero. And if you think that last number to be a very big unimaginable number then dont worry the sum of all 1s will be equal to exact that number.
That one student that kept laughing more than the other students would have totally been me in this class. This is so entertaining to me (educationally). The fact that 1.8 Million people decided to go back to class for math of all things is amazing haha.
This is actually called the "Ramanujan Paradox" and was proved by a very well known mathematician in India and I have bean thinking for a long time how to find a loop hole and this is very tiring but whenever I watch a video on this it gets me excited
@@PubicGore the proof wasn't right but it's still named ramanujan paradox so you can't change that but yeah i didn't that the proof was wrong at that time
When you calculate the sum of Grandi's Series as 1/2, you're conflating the sum of a series with its Cesàro sum (Cesàro mean). The sum of an infinite series is the limit of its first n partial sums as n approaches infinity. The Cesàro sum/mean is the limit of the average of the first n partial sums as n approaches infinity. Those are related but clearly not the same.
"I must have made a mistake, right, right??" Absolutely. Two very big mistakes. Firstly you let a non converging sequence have a value. Then you manipulated several non absolutely converging sequences.
Jacopo Barberis A physician (Hendrick Casimir) had to make a calculation where this S sequence showed up and he decided to try with -1/12, and the answer he got was proven to be right few years later. So I'm not trying to say this makes sense but maybe you should understand that there is things you can't simply understand (Sorry for my bad english btw!) Bye
The result might be useful in physics, but that doesn't mean this proof is correct. You can get to the right answer with calculation mistakes. Then again, i don't think that's the right answer, if i'm not mistaken you have to define "convergence" in a really specific way to get to "1+2+...= -1/12", so that "=" symbol doesn't exactly mean what we think it means.
plz don't appeal to authority. That kind of thing is in contrast to the mathematical spirit and removes the core idea of analysis open to all. Simply arbitrarily calling someone stupid and saying they can never understand is kinda low, especially because there is no context on that claim about Casimir's result. I checked the literature and it does appear that he was considering the series in context of the zeta function linked to the observed vacuum energy, which is tied a meromorphic function with interesting pole placement that allowed him to solve one of his integrals (i.e. make the Lesbegue convergent). He was dealing purely with the analytic continuation of the zeta function, not this sum. For which it is unarguably defined at -1 to equal -1/12. But, no, it was not the sum he was working with.
You can't just act as if naming something is wrong if it doesn't lead to a contradiction. And manipulating series is fine so long as you don't rearrange them, which he didn't. He didn't come to a contradiction, he only came to something unintuitive and the only reason you think it's a contradiction is because you assume the false premise that it is equal to infinity.
You can't simply add, subtract, and noodle around with sums that are clearly divergent. It's not controversial, it's against the rules of math. The result -1/12 comes from far more complicated mathematics that involve analytical continuation of the Reimann Zeta function.
+Sky Fox I remember being taught otherwise. I remember being told, that you can reorder a divergent series to make it converge. I remember being told a number of times by a number of people. While it seemed a bit hard to believe at the time, this sort of thing shows you can reorder divergent to make it converge. Do you have a math book that says otherwise?
@@tchgs11zdok15 No it's not, the sequence 1 -1 1 -1 diverges to per definition the sum over this sequence diverges as well so you can't assign a value to it
@@doomse150 watch mathologer video on this, he explains why most of it is wrong by defention but ultimately could be the answer with the right definition
It takes him from 7:10 to 7:54 to show the answer. It took me much less time to figure out that s-3s=4s since s alfa is 1/4 and s alfa is 3 times more than s, 1/4/-3=-1/12
Yeah sure, but showing the answer to a question at breakneck speeds doesn't make you a good teacher. If you want to explain something and have it understood, you must make your audience think before you give the answer, the way you thought about the answer while he was still explaining. The fact that he doesn't give away the answers at once makes him such a good teacher.
HE IS WRONG AND I SHALL PROVE IT The problem with is that he (mis)used algebra. At 6:14 he cancels out all odd numbers leaving him with infinite gaps which add up to 0+0+...=0*infinity0 Using limits of limits we can assign finite values to both 1-1+1-1+... and 1-2+3-4+... but not 1+2+3+... Let y=1+x+x^2+...=1/(1-x) when |x|1/2 so 1-1+1-1+...=1/2 As x->-1 w->1/4 so 1-2+3-4+...=1/4 As x->1 w->infinity so 1+2+3+...=infinity-1/12 Q.E.D.
The way I’m trying to make sense of it is like this: For the 1-1+1-1+1-1… series which equals 1/2: if I were to place an apple in a bowl the total number of apples would equal 1, and then if I were to remove the apple then the total number of apples would equal 0. If I did this for eternity, then one could observe the bowl at any point of time during that eternity and discover it being in one of two states. Thus, the probability of the bowl being in one of the two states at any point in time is 0.50 or 1/2. So, if we try to measure all of eternity in one instance, it would be most appropriate to define the bowl’s value of apples as some value that is respective of the probability 50% and the two possible states (1 and 0) such as 1/2. [Such phrasing is used because it’s probably not appropriate to define the answer as a probability; I don’t actually understand any of this, I’m only trying to explain how it could make sense for reasons that are beyond a common understanding of mathematics and physics]. This is basically similar to the quantum mechanical concept of superposition (which I don’t understand) where a particle can be spinning in two directions at once until it is observed, and thus the amplitude (complicated form of probability from my ignorant understanding) is used as a measurement. For the 1+2+3+4+5+6+7+8… series which equals (-1/12): If I had an infinitely large bowl and placed one, then 2, then 3, and so on and so forth apples into the bowl each second, and did so for eternity, then you could pause at any point during that eternity and measure the amount of apples in the bowl and get a hard value like 16 or 10,000. However, if we once again try to measure that eternity in one instance, then it would be most appropriate to somehow measure the probability of it being one specific number of apples, and then produce a value that incorporates that probability. But because each element of the series is different, the probability of it being one number is - from a premature and ignorant understanding of probabilities - 1 in an infinity, which seems impossible and also not negative. However, a quick google search tells me that negative probabilities can be used for “unobservable probabilities” which sounds kinda relevant. So, for some mathematical and confusing reason that pays respect to an elements probability of being a specific value when observed, -1/12 is accurate. Again, I don’t understand this and this isn’t an explanation, just a way of representing how these values might make sense for reasons we aren’t presently capable of understanding as random people who got this in our recommended I wrote this more as an intellectual exercise so I might’ve just said a lot of stupid shit
That's not how it works. The series S = 1 - 1 + 1 - 1 +... never converges. Neither does S = 1 + 2 + 3 +... The first one is undefined, the second is infinity. The S=-1/12 is the result of an indian mathematician experimenting with zeta functions and complex calculus that was misinterpreted for 1 + 2 + 3 + ... = -1/12 Also, don't apply physics to math. While math is the language of physics, it doesn't follow its laws
@@bigshrekhorner While S = 1 - 1 + 1 - 1 +.... doesn't converge, the average of all of it's states IS 1/2. That's obviously where the 1/2 comes from, it's just not actually equal to 1/2.
People here don't know that this result of Ramanujan has helped finding 26 dimensions of string theory and holds everytime. It is in the books and is still helping scientist... Many tried to prove it wrong cause its absurd...but it just works!!
He says, that S is EQUAL to -1/12, but it isn't. In order to make this true you have to think of the equal sign as anything BUT "is equal". More like "is associated with"
The way its derived here can be replicated but you can set the sum to any value you want, where as using a rigourous formulation, -1/12 would be the only value you could come up woth.
Great video, personally I disagree with the proof and here’s why. After an infinite number of steps in S, you will arrive at a number we cannot notate, same with 4S. In the division part of the proof you treat both series as if they follow the rules of countable numbers, which they can’t because they are infinity. Infinity and infinity x 4 are the same thing. The proof come out to -1/12 because you stop the series before it is completed. But if you wait until it’s done (ie infinite steps) they will be equal, and s will equal infinity.
The result is correct. -1/12 is proportional number. Zero, infinity and indefinable is also a number type. Whole proportion is already known as infinity. The result is 1/12 proportion of minus infinity and its total -12/12. But there are another thing reality is relative. When relativity is considered then infinity must be considered as relative fact it means it is not infinite. There are mathematical facts being used in math. Other than numbers there are facts as zero, infinity and indefinable. Numbers are apart by reality of zero because each number or thing or particle in universe is separated with emptiness. Numbers are also on the base of zero. There is one single zero and many numbers. Modern math has different zero types such as imaginary numbers but in classical math zero must be known as unified base and mentioned not as number. Zero is a base for numbers. Why I’m going in to this subject because infinity is not infinite. Infinity is ending by indefinable and it has proportional structure. Numbers and zero are together bur after last number there is zero then infinity getting started. This part up to zero is relative. Then relativity is infinity, which has dual structure. It is infinite but it is limited such as circle. Circle looks infinite cycle but it is limited object. The circle is base geometrical shape and its relativity is not a surprise. These are explanations of relativity, being two different things at the same time. Relative reality has layer and each layer has its own individual relativity. Such as numbers and zeros are relativity and numbers are relative to each other so on. On the infinity part, infinity has Its limit because after limited infinity indefinable must be placed. Indefinable is the end of infinity. From this structure we can understand -1/12 has a meaning. if we distinguish the layers and understand the layers in their own structure… By the way infinity is relative by its limited structure and the continuity of relativity leads us to the understanding of indefinable must be singular and it singular structure must be the source of relativity which makes it relative but singular at the same time. Actually it has another explanation but this is enough for -1/12
I have a problem with S-alpha. You "doubled" it by adding it to itself. BUT when you moved it over, you ensured that you are NOT doubling it. In order to pair all the numbers the way you do, there will always be one number left over at the end. Oh, and that missing number is either +infinity or -infinity! So you proved that the sum alternates between positive and negative infinity.
Of course, no one said that is actual, real proof. It 'looks' really great, just like a video I saw saying π = 4 and the proof definitely 'looks' real but it deceives us. S_a in this case would alternate between 1 and -1, and S would approach infinity.
this paradox is known as ramanujan paradox . Here ramanujan is an indian mathematician whose full name is Srinivasa Ramanujan. Proud to had such legend in India.
I'm 77 years old If I had had this man as a teacher in high school, or even in college, my life today would be totally different. This is an unlikely scenario since I graduated from college 15 years before he was born.
@Boz Scaggs Yes, Mathologer is one of the few popular math youtubers who actually seems to both understand and be capable of explaining the material in high level math topics.
Well, in S_alpha you always leave the last number out, which is an infinitely big number. In general adding/subtracting/multiplying on both sides of an equation breacks down with infinity because it follows different rules (such as 1+infinity infinity, 2*infinity = infinity, etcetera)
Exactly, when you look at the set of numbers that addition applies to, infinity is not in it. And arguably neither is zero since the behaviour of adding zero to something is fundamentally different from, for example, adding five.
The problem of this proof is that it contains infinite numbers. See: If we transfer the abstract math problem of the video to concrete numbers, we are able to see that the proof is just wrong. Sg = 1 - 1 + 1 That series is equal to 1. Now we do the exact thing that the teacher did: 1 - Sg = 1 - ( 1 - 1 + 1) 1 - Sg = 1 - 1 + 1 - 1 The new series on the right site is equal to 0. So we can not call it Sg because it is a complete different series. I call it Sn. So the new equation looks like this: 1 - Sg = Sn And that makes sense because Sg equals to 1 and Sn equals to 0. 1 - 1 = 0. The mistake in that proof is just the use of infinite series. Nobody ever knows what infinite is equal to. Sg could be 1 or 0. It depends where the series stops. If it stops at a 1+1 the series equals to 1. If it stops at a 1-1 the series equals to 0. But there is no real stop in the series, so there is no way to tell what Sg is equal to. Im sorry for my english. I am from Germany.
@@evandavid9087 B(1-1+1-1+...)=1-x+x²/2!-x³/3+...=e^-x, where B(S) denotes the Borel transform of a series Then if the Borel integral converges, the Borel sum of the series is equal to the value of the integral. Int[0, inf] (e^-x)(e^-x) dx=Int[0, inf] (e^-2x) dx=½
Its The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12!! Hey i have a suggestion for you..please give the credit to all the inventors from where you choose the topics and discuss the methods!! Also give some links from where everyone one will get the detail knowledge about the topics!! Bdy you are awsome man🙌🙌😇😇
Well, you are writing this on youtube, give credits to them. For english give credit to England, and because you're alive, give credit to you mom and pop too in this comment.
It is legitimate but what is not is the assumption that follows, that the Sα somehow gets a value, while the series diverges just like a (co)sine integral and has no limit and hence no sum whatsoever.
I would have loved to see the face of the students if the professor at the end went out and said: "and everything I just said untill now is wrong, goodbye"
The problem is that infinity isn't supposed to be used in the same manner as numbers or variables that carry limited amount of measure, infinity is more of a concept, and infinity with a series is even less suitable to be used in the same line as numbers. Even if you just rearrange some of the numbers in the S alpha series it would even make result of the series completely different; for example S alpha series itself is approaching negative infinity, but in 2S alpha he rearranged the series which technically it's no longer 2S alpha, instead of negative infinity plus negative infinity, now it becomes a series that alternates between 1 and 0
Andy Fu What he did is still completely valid a manipulation if you let the number N be the upper bound of summation and then evaluate the limit to infinity after the manipulations are done. It’ll leave you with the result 1 - S = S, which yields S = 1/2. Then, by taking the Cauchy product of the series with itself, you obtain B = 1 - 2 + 3 - 4 + ..., which equals 1/4 by the Cauchy product since B = S^2 and S = 1/2.
Your argument is basically, "if you rearrange the series you get something different" but the Riemann rearrangement theorem says that rearranging any conditionally convergent series changes it's value. So you have shown anything about the series S itself
The rules being used (algebra and also general division/multiplication) are for finite systems. A series is infinite, and thus the rules spit out weird numbers. Subtracting two definitive numbers is not the same as adding series. Weirdly however, -1/12 works. It is used in physics and actually makes sense when replacing infinites which pop out of a lot of physics equations. My head hurts.
@lessur dgreat When using the String Partition Function to determine the energy of a string based on all of its possible ground states, you end up adding (1+2+3+4+...) because a string can oscillate at the first harmonic, second harmonic, etc. The answer is -1/12, and in the context of the rest of the formula, it gets multiplied by 2 for a total of 24 dimensions perpendicular to the two dimensions of the string itself; so string theorists say there are 24+2 dimensions (they like to use that notation to make it clear that even though there are 26 total, 2 of them are not directions the string can oscillate in because the string exists in those dimensions).
Ramanujan included this proof in a letter he wrote to Oxford. I can't remember off the top of my head if it was his originally or if he rediscovered it without former association, exposure, or professional training. He was absolutely one of the greatest minds of mathematics, either way.
yeah why so many deny this in comment section. They reject the truth because their iq is not high enough to understand this. Even they act so smart and say that Ramanujan was wrong
Sir i loved the way and the things you teach. Can you tell me how you came up with this awesome questions and interesting explanation? So i can tell me school and other head of institutions to refer to that material instead of old curriculum which was taught.
@JHIGYASU STUDENT wow that's so proud feeling to hear that.. Can u tell me the resources where I can find interesting contributions of indian mathematicians.
I see all the comments like « you’re wrong, you can’t assign a value to a serie » accusing him of telling lies.... DUDES, He told you that it was a controversial result and that people always argue about it. So no need to try to convince us that he is wrong. Let us make our own thought of this. Also, his aim isn’t to prove anything but to make you THINK. And... clearly if you wanna talk about that kind of mathematics stuff, the comment section of RUclips isn’t the best place to argue....
so basically if someone wanted to give you $1 today, $2 tomorrow, and so on, you shouldn't accept the deal because they would be trying to steal from you.
That would only be the case if he lives infinitely long and never runs out of money. I would risk that 1/12 dollar on the bet that he wont.
@@MrPacoHamers Well no, because the value of an infinite stream of cash flows isn't adjusted based on the lifespan of the recipient, as the cash flows can be sold to another party at any point in time.
That's why the theory of a the price of a stock being equal to the present value of all its future cash flows is rooted in infinite time, not a human lifespan: you can SELL the stock at any time for immediate cash.
happy to be stolen，try hard on me
bruh
No. This sum only works in the case where the sum is infinite, meaning it never stops and therefore can never be evaluated.
The best part about this videos is my father thinks i'm studying
😂😂😂😂😂😂😂
You are.
You'd do better studying than watching this nonsense.
Lmao
maya L this is not nonsense this is math!!.
If you answer that in a calculus II test, your mark will converge to zero.
Haha!!...
Or even less
@@magicUFO well done
or maybe -1/12
Or be the derivated
"It could be wrong, that's fine. Mathematicians are wrong all the time, that's why they're discovering cool new stuff." And this is what makes a great teacher! None of that "you have to be right or you're failing and it's your fault" nonsense we often see (bad) teachers and educational systems do, which only discourages people from wanting to learn more and learn from their mistakes.
People are wrong all the time because they are just wrong, though, like in this video. Setting the sum of a divergent series to a discrete number is flat out wrong in mathematics. Being wrong doesn't always mean you're on the path to discovering something new.
@@eXJonSnow that's not the point though, it's about trying and proofing you're either right or wrong. That's how you learn and improve yourself. That's why it's allright to be wrong and how you discover cool new things
@@Leviathan-gp2kv The guy's a college professor. If you get the answers wrong on his exam, you're going to fail the course. Period. You can be wrong while learning, but when it comes time to apply what you learned, it better be spot-on.
Encouraging learning and encouraging mediocrity are two very different things.
@@KamuiAlmighty I don't necessarily mean in class. BTW is this college level math!!!??? But I mean even scientists makes mistakes without the mistakes you never come to a breakthrough. No one has is exactly right the first time
I have a theory as to why you cant solve it like you are trying to solve it. And heres WHY: 1 minus series-g does not equal 1/2, it equals to infinity. 2 times series-alfa does not equal 1/4, it equals to infinity. Series minus series-alfa does not equal -1/12, it equals to infinity.
Don't ask me why, u, I dont know yet.
Anyone gonna talk about the fact that he is writing on a whiteboard that is on a whiteboard ?
nah, he does it with style so its fine
i had a board like that in my class, the whiteboard was on a huge blackboard n it made no sense to me XD
He accidentally damaged 1/12th of a whiteboard that belonged to his colleague. He apologised to his colleague and offered to replace the damaged part. His colleague said they will deliver one whiteboard to him on day one, then two whiteboards the next day, three whiteboards the day after and so on. Here we are on day two.
#AustraliaMoment
Yeah really 😂😂
Long story short: adding, subtracting, dividing, and multiplying both sides of an equation is illegal when dealing with infinity, because: infinity + 1 is still equal to infinity.
Bryan Shortall also you can’t assign a value to a non-converging series (1-1+1-1+1-1+1-1+... does not approach any specific value)
@@Sixsince-dd2eu well, you can. Seems like people have a really hard time understanding that it doesn't work with classical definition of a sum of series. You can't even take a sum of infinite series in the first place, because, you know, it's infinite. You need to define first what sum of a series is and then talk about what is possible or not. In classical definition, that result would not work, but you can still assign values to non-convergent series, but you may lose some properties (like linearity). Think of - 1/12 as "generalized sum".
@@clawsie5543 Thats when you're assuming that the numbers don't end, which then will be equals to, infinity. But here we are not doing that we are just adding all the numbers up. In fact, thats the point of this theory. This is to prove whether there is an end to numbers or not
Thats my view on this correct me if Im wrong
@@cnoz2378 I don't quite understand what you are saying. The point is that you can't just sum infinitely many numbers, because you can't apply normal algebraic rules. Algebraic rules can be applied only for finite number of numbers. In order to sum an infinite sequence, you first need to define how to do it. If you define it as the limit of partial sums (which is the "classical" definition), then you get that sum diverges towards infinity (what you would expect). But the result -1/12 doesn't mean that the limit of partial sums is -1/12, it is diferent way of summing than classical definition suggest, it is more general way.
infinity < (infinity +1) ;P
The US government is trying this with the national debt.
Damn, nice thought, rofl
hahah....hope they will success
government debt isn't real. its a number on a bit of paper with no relevance in the slightest.
@@dabeveryday9991 he was joking.................................................
@@idunnowhattonamemyself9935 are you in his brain? How would you know that?
Math: Am I joke to you?
Mathematicians: Yes. An infinite sum of jokes, even.
Ramanujan
Odd
An infinite sum of jokes is 1/12 of not a joke.
I know why Eddie cant solve it like Eddie is trying to solve it. And heres WHY: Infinity-1 does NOT equal infinity.
Hilbert himself said: some infinities are bigger than other infinities.
@@WhiteStripesStripiestFan I know why u cant solve it like you are trying to solve it. And heres WHY: 1 minus series-g does not equal 1/2, it equals to infinity. 2 times series-alfa does not equal 1/4, it equals to infinity. Series minus series-alfa does not equal -1/12, it equals to infinity.
Don't ask me why, u, I dont know yet.
"have i made you think yet?"
no but you made me think why is there a whiteboard on top of another whiteboard 🤔
Ian Thai so he did make you think
Ian Thai lmao dumbass
Its for double exp
@@tzielsky lol xD
Holy shit
-1/12 is nature's buffer overflow
haha that can be true :D
Lol computer science
Verri that*
reality.sys is corrupt. Reformat universe?
No No man the system of numbers was created by human beings. No accuse to god.
You may be cool...
But you’ll never be “writing -1/12 without looking” cool 7:49
The way he drew a dot on the board when saying "after all" and striking a pose was truly epic.
@@dhruvupreti5120 you can’t
@@dhruvupreti5120 show off
True finesse
its 2:15 in the morning and im not sure if i dream or does he realy blow my mind
As an English and Maths teacher, I have a big passion for teaching the history of Maths. I always try to humanise mathematics by telling my students the stories of the people behind these theories and results. I feel it makes things more connected for students. Ramanujan's story is a story worth telling. The film "The man who knew infinity" is also a great film for exploring two modes of getting to mathematical results, inspiration or proof.
This has absolutely nothing to do with Ramanujan summation. Woo simply demonstrated that permuting the summands of a divergent series can change the behavior of the series.
I love the same thing (History of Maths)! I am a still a student and until I came upon the stories or the history of the people before us who discovered the things that we are studying today and how they approached them or how they failed and later people throught the time had other points of view of these problems and solved made me love mathematics even more than I used to. I really do believe that to understand and be able to solve a math problem ( famous hard problems especially) a bit of history of how the people before us tried to tackle these and their ideas is really useful.
@@Onoesmahpie wtf he literally invented the series. Moreover, his mathematics inventions are what used to study the behaviour of black holes nowadays
@@Onoesmahpie For all your arrogance, you're dead wrong. Changing the order of the terms of a divergent series does not make it convergent using ordinary addition, because it's commutative. The super summation must be used to assign a result to a divergent series. What's shown in the video is a classic Ramanujan super summation.
Btw, the proof shown in the video is just a trick (used by Ramanujan himself and still excellent to tickle the mind!). A more robust proof is much more complex (Riemann zeta function and analytic continuation get involved). A great deeper explanation is provided by Mathologer: ruclips.net/video/YuIIjLr6vUA/видео.html
Our history teacher despises mathematics
Everytime the word "plus" and "infinity" comes into the same sentence, mathematics turn weird. I love it.
This comment made me think about infinity, PLUS it`s ridiculous.
Truer words couldn’t have been spoken
"Infinity" is kind of taking limit
i love it too
1+1≠∞
"We subtracted something and it got bigger."
That's clearly a memory overflow error in the reality matrix !
Bruh
That's really funny to say because we can assign -1 to 1+2+4+16 +... +2^n+... making it "when you add 1 to 1111...111 you get 0" that is indeed some kind of "memory overflow"
@@baptiste2b31 there isn't a memory overflow. When you say that it exists you're saying that there's no number greater than 1111...111
@@higorss Of course it's not a "memory overflow" (compturers don't have infinite memory), I'm saying it's "resembles" it.
First, (my bad, I forgot to precise,) but 1111...111 is considered base 2, then 1111...111 doesn't really "exist", if you want mathematical rigorousness ; what we are dealing with is an application, let's call it S, that associate a value to a sequence :
to finite sequences it associates the "true" sum, to a converging sequence it associates its sum (therefore S(u_n)=u_0+...+u_n when u_n is finite or converging). We want S to act "like" a sum, therfore we want S to be linear and also we want S(0,...)=S(...).
This way we can give to some other sequences (the non converging ones) 1 unique possible value that we call "sum".
So when I write 1111...111, in fact I'm speaking about S(1,...,1,...) that is perfectly well defined mathematicacly and has to be equal to -1 (in order for S to have the proprety of a sum, I can give a proof if you want).
Then I notice that 1+S(1,...,1,...)=1-1=0. So when you write it as 1+1111...111=0 it "resembles" a memory overflow.
In conclusion, base 2, modulo 2^n+1, the finite number 1111...111 (with n ones) + 1 = 0. That proprety is called "memory overflow" in computer sciences. My comment is saying that this proprety also appears with infinite sums. So when @Sion mentionned it, it was nice intuition.
That proofs that infinite divided by 2 not equal infinite.
When he subtract Sa from S he lost half of the length of Numbers and yet he still compare them by location and asum that it is bigger by four times. Tricky
Teacher: the test won't be hard.
The test:
This video is why I don't like methmatix
@@stevethea5250 methmatix
@@stevethea5250 lol
@@stevethea5250 Were you on Methmatix when making that comment?
@@turtle_demigod wish i wuzza methmatix head
I’m out of college for a decade now and I still find his lectures fascinating. I wish I had access to them when I was in school. Thanks and keep uploading these captivating math videos.
You can't add and subtract infinite series. If you could, you could prove 1 = 2.
David Messer just out of curiosity, would you mind showing me?
Faisal S = 1+1+1+1+1+1+1+1+1+... then S+1=1+1+1+1+1+1+1+1... which means S+1=S, which means 1=0
Priest Plaxis
That’s simpler. :)
Priest Plaxis thanks
Sahil Naik Yes, but by the same logic 1+2+3+4+5+6+7+8+... equals infinity, which it in fact, does.
The whole point of this class was to get the students to THINK .....not spoon feed them correct/incorrect answers . Good teacher
And if that is the case cool, that does make for a very good class, BUT it also makes for a horrible youtube video, since we don't get to see that context, meaning he very well can be the start of false information spreading
Gummiel context is everything. Have a look at the rest of his RUclips clips and you’ll see that he is always teaching and never spoon feeding. It’d be like taking bits of what Mustafa wrote and claiming that he said “the whole point of this class was to ... spoon feed them correct ... answers.”
He even ends by telling students to go and research this controversial proof - sounds like he’s trying to peek interest, inspire, challenge and encourage students to “think about solutions”, maybe even realise that even very smart people sometimes make fundamental errors.
Indeed you are right that context is everything here, and as I said that makes for an awesome class, but as a youtube video, it is BAD. You can't expect ppl to watch every single video of his, so they need to be able to stand alone as well, and at that this video does a terrible job
Gummiel surely you can expect people to watch the entire video and hear him say this is a controversial proof and you should go online and research it. We can’t assume that everyone is an idiot and can’t be bothered watching all of it.
@@GummieI he mentioned a lot of times that this is a contradictory topic
I just love how infinity always tries to break maths
People mishandling infinite series break math.
Maya is right. If you treat Infinity as a number, math might appear broken. But it's not broken.
@@mayal5206 this isnt really breaking math tho. It seems like it is, but most of the evidence points that negative numbers and positive numbers are on both sides of the origin. We thing of the numberline too much as one dimensional, which is actually breaking math. Infinite series and multidimensional math go hand in hand. This is a totally correct way of handling an infinite series, just not in the algebraic sense.
@@Predated2 - In what "sense" then? And what does "algebraic sense" mean? The number line is, by definition, a straight line and a line is, by definition, one-dimensional. Where did you get this notion that there are positive and negative numbers on both sides of the origin?
@@mayal5206 when a circle becomes bigger nd bigger up to an infinite radius, the point at which it's any tangent touches makes the origin of a number line an looks like a number line but actually it's a circle that's why it is 2 dimensional not the one.
"Have I made you think yet?"
And there it is, the missing ingredient in many classrooms. I love that he doesn't push a lesson down their throats. He guides them through the thought process that made other people discover what we know today, in the hopes that these kids will do the same even of it's in other areas in life.
I really like the guy who's always mind blown at some point in every video
Alternative title: Why you can't rearrange a divergent series
LOL. Perfect!
This is.a.better answer than -1/12: ruclips.net/video/yjjvAyLgntM/видео.html
Exactly I'm not sure why Eddie Woo keeps saying the result is `controversial', it is just a nod to the fact that absolute convergence is a necessary requirement to re-arrange terms. He makes it seem to his students that math is all a big confusing mess where anything can be possible. He does a bad job at explaining this, and in fact seems to be conflating the Ramanujan summation technique with this basic flaw in reasoning.
@@Onoesmahpie I'm pretty sure the intention is to make the students think.. Sure, he can explain why this proof is wrong but the purpose is to make the students curious.. Why does a seemingly logical proof result in a completely nonsensical answer? By saying that it's "controversial", which must be exactly how it seems to high schoolers that aren't proficient in convergent and divergent series, he is sparking their interest in trying to learn more about it.
@@siriusthegrim I think Eddie is awesome, just pointing out that what he said was technically not true and could have been worded better. I do applaud him for trying to go above and beyond and spark curiosity in students by challenging their intuition and showing how we cannot assume generalizations of specific results hold, and how it is difficult to recognize one's fallacious reasoning sometimes.
"We subtracted something and it got bigger."
I want you as my math teacher sometime...
Well technically subtracting a negative makes it bigger
I want him as my Mohel.
@@daaaaaaanny but he subtracted a positive...
@Shravan subtracting a positive number from a negative number always leads to large negative number. for example,consider
-1-(3)=-4.it is possible to get big negative.
@@daaaaaaanny, you're twisting the meaning of the word bigger. Try looking at the numbers going higher or lower, the numbers you are suggesting causes it to go lower still, in the technical sense, though the number is "bigger" the fact that it has a "-" next to it causes it to become "smaller" not "bigger". Also adding decimals screw over the entirety of using the words "bigger" and "smaller" due to decimals becoming "bigger" in size yet also becoming "smaller", but not in the same way as negative numbers, and a negative decimal is getting "bigger", while the number itself is getting "bigger". My point is you're wrong due to the English language because there should be a word to describe a number getting "bigger" or "smaller", "higher" or "lower".
*S. Ramanujan* - "The man who knew infinity"
Somewhere in heaven he is smiling at us. 🙏🙏
Infinity was conceptualized because of zero
He probably wrote this to annoy us even after his death lol😂
@@bosongod2830 will it's definitely useful. So it doesn't matter if it's annoying when it's useful
And the best dad joke award goes to: 0:07
Owen Wilson kkkkkkkkkkkk
Lol Owen Wilson with Trump's hair 🤣
lmao :D
This was given by S Ramanujan
That's India 🇮🇳
Exactly!!
Bobs and vagene
@@mithat9398 XDDDDD
Ramanujan did use this version one yeah, but more impressively he came up with Ramanujan summation which is why the result is really -1/12
There's a thing called Riemann's Rearrangement Theorem which states that you can't play like that with divergent infinite series
Andrew Sp In the general case, no, but in this particular case it can be done carefully.
en.wikipedia.org/wiki/Riemann_series_theorem
Did you even read anything about it? It only talks about rearranging conditionally convergent series to get other convergent series or divergent series. Nothing specifically about playing with divergent series.
Andrew Sp Also, the rearrangement Theorem does not say you cannot play with divergent series like this. It merely means infinite summation is not closed under any subset of the reals, nor is it commutative or associative. As long as Eddie hasn’t assumed either, this is all perfectly valid
Andrew Sp you do realize wikipedia is not an accurate source, any one can do anything to it, that's why colleges dont allow it as a credible source for papers. which reduces the credibility of this mathematician for suggesting his students go to wikipedia for anything, unless aussie wiki is better or something
Peter Cottantail ok you guys are right now stop it with this.
If somebody is interested: Look up the discussion done by "Mathologer".
In a nutshell: Infinite series need special care or you end up with nonsensical results.
Infinite series only have a value, if the partial sums converge to a specific value. If they don't then they either have no value or the value is infinte.
If this converge criterion is not taken into account, then you get nonsenical results.
As said: Mathologer has discussed that issue. Look it up.
I knew you would have ended up seeing that non sense video as RUclips algo partially favors mathologer very much...
Please see the video ramanujan summation from channel "singing banana" . you will get a new perspective that mathologer fooled us... Please see it for true knowledge sake.
@@modhere2448 well said.
'Approximates to' is not the same as 'equals'. Hence S != 1/4 only approximates to it at infinity.
This is ramanujan summation, which differs from summation. So the sum of all counting numbers is not -1/12.
Bumder
Idk much about this but would the sum of all counting numbers =0 because got their positive and negative counterparts which add up to equal 0
@@merth.6423 counting numbers mean positive numbers
This could also be a Cesaro summation.
@Stanislav Hronek I think the standard is to use an analytic continuation of the Riemann Zeta function at z = -1. Regardless of how this result is derived (and whether it is flawed or not), nature surprisingly seems to agree to this. In particular, the Casimir force (experimentally measurable) between two infinite plates in vacuum can be derived with this result.
“We subtracted something and it got bigger. “ I want you as my accountant! 🤣
Teachers like him are priceless. They can really spark your imagination and send you on a lifelong journey of learning. If every teacher and school was like this the world be a much better place I think.
8:49 time mark _ "The question is, what does it all mean?"
It means that when manipulating series of numbers, it is very important to understand exactly what you are manipulating. Seemingly innocent truncating and shifting numbers over have dramatic and unintended consequences, when evaluating multiple series.
For example, take an equation C = first series of repeating ones. Take the same equation C = second series of repeating ones BUT SHIFT IT OVER ONE PLACE. Now subtract the two equations, C - C = first series - second series. What do you get? 0 = 1. That is the flaw in this video's logic, you can't shift places of values in the series when evaluating multiple equations.
For me, the bigger mystery is WHY does C = first series NOT EQUAL C = second series? Don't both equations (independently) resolve/converge to the same value? The only answer that I come with is that you have to be very careful when evaluating sets of infinities.
As for this video, its biggest flaw isn't faulty logic. Arguably, the logic is impeccable. The flaw is a failure to identify the mathematical frame of reference. The playground being discussed is NOT REAL WORLD. Instead, it is something called "Analytic Continuation". It is a logical extrapolation that makes sense within its respective frame of reference. Eddie Loo should make this point clear in his video presentation. (I think maybe that omission is because he assumes that viewers understand this.)
6:04 automatic english subtitles be like "no no, I love you!" "Ohhh I see what you mean"
Hehe
That awkward moment when you set non-converging series equal to discrete numbers
Yes, "Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series"
Keep reading the Wikipedia article:
“Although the Ramanujan summation of a divergent series is not a sum in the traditional sense”
@@joshuamoyer4141 The standard definition of the summation of a convergent series as the limit of the partial sums isn't a sum in the traditional sense either. A sum in the traditional sense is only defined for a summation of a finite number of integers. You can't start with the definition of addition and derive how the sum over an infinite set of integers should be defined. That definition will be an arbitrary choice, there are many possible choices. Defining it as the limit of the partial sums is one possible definition that has become the standard definition.
Correct answer deserves more upvotes...
I hate to be that Grammar Nazi asshole, but *discrete. "Discreet" means secret, "discrete" is the word you're looking for.
This was first explained by srinivasa Ramanujan sir
Proud to be an indian🇮🇳
the part i love most about your vids is that it's literally free information. no asking for subs, no hit the like, just raw math and i'm here for it.
I'd love to view this video's analytics. Most comments here are from the past week and I wonder why.
Bob McMullan I got here from his 0 to the power of 0 video
Try socialblade. [C O M]/youtube/user/misterwootube/ :)
within the period of a single second I saw his subscribers increase by 20
Michael Rector smart
There was a video (Why 0! is 1) posted to Reddit/r/Math within the past week. It was top that day. A lot of people probably found this channel through that link and are just going through the videos (Like I am)
this result is used in string theory and used in some calculation in quantum physics . this ramanujan infinite series
Math teacher : Series alpha.
Back bencher : "Are you serious?"
Math teacher : "Yes I am serious (laughs)" LOL!
*EVERYONE* : Video 📹
*ME 🤔* : White Board on White Board ⬜
Oh god you’re right
He accidentally damaged 1/12th of a whiteboard that belonged to his colleague. He apologised to his colleague and offered to replace the damaged part. His colleague said they will deliver one whiteboard to him on day one, then two whiteboards the next day and so on. Here we are on day two.
@@MrAlRats I pray you are kidding 😉
one is likely a projector and the other a whiteboard
It's not a projector screen, there's one rolled up above
"A glitch in the matrix" is all that needs to be said.
Im actually more impressed by this guy's talent to write on the board with an incredible speed or without looking and the board while writing
This proof was done by a famous Indian Mathematician Srinivasa Ramanujan (1887-1920). Its applications are in General Physics & Bosonian String Theory. Although it makes me think, I keep wondering can we do such operations with an infinite series? We can do them to finite series but in a never-ending series, it becomes difficult to comprehend. The value of S should be n(n+1)/2 where n is no. of terms & cannot be -1/12 definitely but this feels logical though...
@flat_earth_forever thank you "flat earth forever" you missed the point of the proofs (the first series you mentioned the actually does converge to 1/2 btw)
@flat_earth_forever it depends on how you approach it, much like many things in math. By some methods it doesnt converge, by others it does. I think it's much more interesting (and still mathematically coherent) to say it equals 1/2
If behaviour of an operation changes it must be because your operation changes, you just didn't notice.
i.e. infinity is a limit for addition, any attempts to add infinity changes the concept of adding to the point where you're not doing the same thing as you were doing before.
You can do arithmetic operations to infinite series, but only when the series converges. If the series diverges (as the sum of the natural numbers does) doing any operation on it is not valid. What he's showing in this video his hogwash, and you can use the same "tricks" to make the sum equal to whatever you want.
For example here's a video that "proves" that the sum is equal to -1/8: ruclips.net/video/6FTwMUL69u0/видео.html
Why would this proof be any more valid than that one? (Answer is: Neither is a valid proof, because most operations done on divergent series are invalid.)
@@DjVortex-wThe operations are valid, provided you don't muck about with shifting terms or changing the number of terms.
In this video calculating 2Sa by shifting and adding is "illegal", as is equating S-Sa to 4S since they have a different number of terms.
This is Ramanujan Paradox.who was a great Mathematician.proud of u sir.your thought was amazing that make the people to think different.
I think we can't write
1-1+1-1+1-1+.....to infinity= 1/2
because its non-convergent series.
Yes it is a divergent series but we arnt taking normal sums, we cant. Hence we are taking a special summation of the series.
The fact that a series diverges does not mean that the sum cannot be evaluated to be a finite number. The limit of f(x) = x^x as x -> 0 = f(x) = 1, yet this by no means implies that 0^0 = 1. The same is true for summation.
The point of this is to get rid of arbitrary rules and see where that takes us. Why can't we assign a value to a divergent series? Just because it doesn't make sense doesn't mean it's impossible.
1-1+1-... can be written as
inf
Σ((-1)^n)
n=0
Well that looks like a geometric series, and we know how to calculate infinite sums with that, so let's try it.
1/(1-(-1))=1/(1+1)=1/2
Well that sure looks familiar. And if you're not happy with pure algebra, throw in some calculus and take the limit as the base goes to -1, and you get the same answer. For all intents and purposes, the answer is 1/2. And the final series in this video does indeed equal -1/12. It shows up in physics, which just further confirms it.
I have a link in which expert mathematician Terrence Tao why the argument of convergence vs divergence is not convincing enough to present an objection to the result of the series as equal to 1/2.
terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/#comment-487580
2+2 is four minous one that is 3 quick mathhhhh
learn grammar
I think you might mean minus
just sauce raw sauce
babes, man's not hot
Grammatical grammars
Mom: What are you watching ?
Me : If u add too many positive numbers u will get negative,..!
And not just a negative, but a negative fraction.
The video was supposed to filter out the retards from passing his course. You fail.
Eh... As every programmer will tell you. But if you get a fraction, you dun bad.
Omg, this is what I hate the most about
RUclips Comments, Its either People
are believing This Proof or People are
not believing This Proof or People dont
know what to say, it's a Massive Mess
This is the cool part about mathematics... scientists are always trying to ask the right questions to find the right answer and expand knowledge, while mathematicians are simply chasing whatever curiosity their nerdiness comes up with, and then recording their results. A lot of important rules in mathematics have come to be, simply because some nerd was like "I wonder what happens if I ____?".
i think the most confusing part of this video is why is there a smaller white board attached onto a larger whiteboard? and why do i suspect it has something to do with the school ordering new boards to use up a budget so that they get the same budget next year, and ending up buying smaller boards than they already have.
Maybe some student didn’t like this teacher and smeared shit on the big whiteboard and instead of cleaning it up because education never gets funded properly they just put another whiteboard over the smeared shit.
If the school ordered a board they didn't need, why would they put it in a classroom instead of a storage room or the dumpster?
all of those series are (i don't know how to say this in english) "no convergentes" so it's imposible to get a number
It's not technically possible
But this is what would happen if it *was*
En inglés sería así: "Divergent series". Hope it helped ya.
I think it's called a diverging series or a divergent series
Actually the first one(the one he didnt show) is convergent, its telescoping
This was first proved by "S Ramanujan- The Man Who Knew Infinity"
He was a great Mathematician and today his theories are the basis for the study of Black Holes🙏
ruclips.net/video/uhNGCn_3hmc/видео.html
If any one interested in Ramanujan
Ramanujan is the absolute goat of maths and for some reason we never learned about him in school. Im 31 now. I first heard about him at 29. It shouldnt be like that. If the guy was from a western country, I’m sure he would be as famous (or more) than Einstein to us.
@@Whiskypapa GOAT
@@ryanchowdhary965 Then why is he called ram anujan (what do you mean rams are sheep?)
@@TheRenegade... greatest of all time (GOAT)
Credits to S Ramanujun for bringing up this amazing theorem… He was a great Indian Mathematician who sent this exact theorem in a letter to a British Mathematician, G H Hardy on January 16, 1913 asking for this theorem to be published under his name.
No he didn't wanted this to publish. This proof was just found in his books. This proof is wrong btw and it's been debunked many times.
Look up about Reimann Zeta function.
Heres something I thought:
S=1+2+3+4+5+...
S-S=1+2+3+4+5+...(-1-2-3-4-5-...)
But hey, dont take that -1 from the 1, take it from the 2, and so on, we get:
S-S=1+(2-1)+(3-2)+(4-3)+...
S-S=1+1+1+1+1=infinity
0=infinity
?????
Infinite sums are scary shit to work with
If 1+1+1+...=∞ then 1+2+3+4+...=∞
∞-∞≠ because ∞ is not a number
@@magicUFO correct
Well, you'll have to subtract the 1 like all other numbers before going on to step 2.
Cheers.
You can’t add or subtract non-converging series’.
@@magicUFO Not correct, It may look like infinity series but it's not. The Sum of all the 1s come out of subtraction will be equal to last number and they both will cancel each other answer will be zero. S-S will remain zero. And if you think that last number to be a very big unimaginable number then dont worry the sum of all 1s will be equal to exact that number.
Why does he have a whiteboard on a whiteboard????
THINK.
He's hardcore.
Mother of weirdness, what the heck.
That one student that kept laughing more than the other students would have totally been me in this class. This is so entertaining to me (educationally). The fact that 1.8 Million people decided to go back to class for math of all things is amazing haha.
This is actually called the "Ramanujan Paradox" and was proved by a very well known mathematician in India and I have bean thinking for a long time how to find a loop hole and this is very tiring but whenever I watch a video on this it gets me excited
He proved it? No, doing so he assumed that the common ratio of a geometric series did not satisfy |r|
@@PubicGore the proof wasn't right but it's still named ramanujan paradox so you can't change that but yeah i didn't that the proof was wrong at that time
@@krishiv2021 I never said it wasn't called the Ramanujan paradox.
When you calculate the sum of Grandi's Series as 1/2, you're conflating the sum of a series with its Cesàro sum (Cesàro mean).
The sum of an infinite series is the limit of its first n partial sums as n approaches infinity.
The Cesàro sum/mean is the limit of the average of the first n partial sums as n approaches infinity.
Those are related but clearly not the same.
"I must have made a mistake, right, right??"
Absolutely. Two very big mistakes. Firstly you let a non converging sequence have a value. Then you manipulated several non absolutely converging sequences.
Jacopo Barberis A physician (Hendrick Casimir) had to make a calculation where this S sequence showed up and he decided to try with -1/12, and the answer he got was proven to be right few years later. So I'm not trying to say this makes sense but maybe you should understand that there is things you can't simply understand (Sorry for my bad english btw!) Bye
The result might be useful in physics, but that doesn't mean this proof is correct. You can get to the right answer with calculation mistakes. Then again, i don't think that's the right answer, if i'm not mistaken you have to define "convergence" in a really specific way to get to "1+2+...= -1/12", so that "=" symbol doesn't exactly mean what we think it means.
Zygo Petalum Right in the spot
plz don't appeal to authority. That kind of thing is in contrast to the mathematical spirit and removes the core idea of analysis open to all. Simply arbitrarily calling someone stupid and saying they can never understand is kinda low, especially because there is no context on that claim about Casimir's result. I checked the literature and it does appear that he was considering the series in context of the zeta function linked to the observed vacuum energy, which is tied a meromorphic function with interesting pole placement that allowed him to solve one of his integrals (i.e. make the Lesbegue convergent). He was dealing purely with the analytic continuation of the zeta function, not this sum. For which it is unarguably defined at -1 to equal -1/12. But, no, it was not the sum he was working with.
You can't just act as if naming something is wrong if it doesn't lead to a contradiction. And manipulating series is fine so long as you don't rearrange them, which he didn't. He didn't come to a contradiction, he only came to something unintuitive and the only reason you think it's a contradiction is because you assume the false premise that it is equal to infinity.
4:15 "I've got news for you"
"Oh for God sake"
Woo: did I make you think yet?
Everybody: *SMILES*
You can't simply add, subtract, and noodle around with sums that are clearly divergent. It's not controversial, it's against the rules of math. The result -1/12 comes from far more complicated mathematics that involve analytical continuation of the Reimann Zeta function.
Yes but it is still based on the idea that 1-1+1-1+1-...=1/2.
+Sky Fox I remember being taught otherwise. I remember being told, that you can reorder a divergent series to make it converge. I remember being told a number of times by a number of people. While it seemed a bit hard to believe at the time, this sort of thing shows you can reorder divergent to make it converge. Do you have a math book that says otherwise?
analytical continuation has the same basis of the proof if this is wrong analytical continuation is wrong
Dude, the point is: you can reorder series that are NOT absolute convergent in ways to make them converge to any number you can think of.
There's no math gods. Nobody can tell you that you "can't" do anything, especially not so in mathematics.
All the lies began with Sg=1/2
No actually that is ok as a supersum
@@tchgs11zdok15 No it's not, the sequence 1 -1 1 -1 diverges to per definition the sum over this sequence diverges as well so you can't assign a value to it
@@doomse150 watch mathologer video on this, he explains why most of it is wrong by defention but ultimately could be the answer with the right definition
@@tchgs11zdok15 What do you mean by "the right definition" if you define stuff the way you want it obviously can work
@@doomse150 just watch it, it's like a freaking 40 minute video that addresses everything you can address about this
It makes me sad that he didn't even mention Ramanujan here, while introducing infinite sum series.
Yes I was waiting for it.
Me too
I clicked it for him
Agree.. Ramanujan Summation could have been mentioned
Gonna cry?
Mr Woo, I am a 61 yer old engineer and economist... I have worked in banks all my life... I admire your way of teaching... May God bless you...
At 3:27 when he says “this series”, he activates Siri on my phone 😂
😂😂😂💔
What a surprise
A perfect example of how interesting subject matter nullifies misbehaviour. Excellent work.
This has to be the best way i have seen this prof done
It takes him from 7:10 to 7:54 to show the answer. It took me much less time to figure out that s-3s=4s since s alfa is 1/4 and s alfa is 3 times more than s, 1/4/-3=-1/12
Yeah sure, but showing the answer to a question at breakneck speeds doesn't make you a good teacher. If you want to explain something and have it understood, you must make your audience think before you give the answer, the way you thought about the answer while he was still explaining. The fact that he doesn't give away the answers at once makes him such a good teacher.
Speed I can't believe ur level of immaturity is so high.
"proof"
HE IS WRONG AND I SHALL PROVE IT
The problem with is that he (mis)used algebra.
At 6:14 he cancels out all odd numbers leaving him with infinite gaps
which add up to 0+0+...=0*infinity0
Using limits of limits we can assign finite values to
both 1-1+1-1+... and 1-2+3-4+... but not 1+2+3+...
Let y=1+x+x^2+...=1/(1-x) when |x|1/2 so 1-1+1-1+...=1/2
As x->-1 w->1/4 so 1-2+3-4+...=1/4
As x->1 w->infinity so 1+2+3+...=infinity-1/12
Q.E.D.
The way I’m trying to make sense of it is like this:
For the 1-1+1-1+1-1… series which equals 1/2:
if I were to place an apple in a bowl the total number of apples would equal 1, and then if I were to remove the apple then the total number of apples would equal 0. If I did this for eternity, then one could observe the bowl at any point of time during that eternity and discover it being in one of two states. Thus, the probability of the bowl being in one of the two states at any point in time is 0.50 or 1/2. So, if we try to measure all of eternity in one instance, it would be most appropriate to define the bowl’s value of apples as some value that is respective of the probability 50% and the two possible states (1 and 0) such as 1/2. [Such phrasing is used because it’s probably not appropriate to define the answer as a probability; I don’t actually understand any of this, I’m only trying to explain how it could make sense for reasons that are beyond a common understanding of mathematics and physics]. This is basically similar to the quantum mechanical concept of superposition (which I don’t understand) where a particle can be spinning in two directions at once until it is observed, and thus the amplitude (complicated form of probability from my ignorant understanding) is used as a measurement.
For the 1+2+3+4+5+6+7+8… series which equals (-1/12):
If I had an infinitely large bowl and placed one, then 2, then 3, and so on and so forth apples into the bowl each second, and did so for eternity, then you could pause at any point during that eternity and measure the amount of apples in the bowl and get a hard value like 16 or 10,000. However, if we once again try to measure that eternity in one instance, then it would be most appropriate to somehow measure the probability of it being one specific number of apples, and then produce a value that incorporates that probability. But because each element of the series is different, the probability of it being one number is - from a premature and ignorant understanding of probabilities - 1 in an infinity, which seems impossible and also not negative. However, a quick google search tells me that negative probabilities can be used for “unobservable probabilities” which sounds kinda relevant. So, for some mathematical and confusing reason that pays respect to an elements probability of being a specific value when observed, -1/12 is accurate. Again, I don’t understand this and this isn’t an explanation, just a way of representing how these values might make sense for reasons we aren’t presently capable of understanding as random people who got this in our recommended
I wrote this more as an intellectual exercise so I might’ve just said a lot of stupid shit
That's not how it works. The series S = 1 - 1 + 1 - 1 +... never converges. Neither does S = 1 + 2 + 3 +... The first one is undefined, the second is infinity. The S=-1/12 is the result of an indian mathematician experimenting with zeta functions and complex calculus that was misinterpreted for 1 + 2 + 3 + ... = -1/12
Also, don't apply physics to math. While math is the language of physics, it doesn't follow its laws
@@bigshrekhorner While S = 1 - 1 + 1 - 1 +.... doesn't converge, the average of all of it's states IS 1/2. That's obviously where the 1/2 comes from, it's just not actually equal to 1/2.
People here don't know that this result of Ramanujan has helped finding 26 dimensions of string theory and holds everytime.
It is in the books and is still helping scientist...
Many tried to prove it wrong cause its absurd...but it just works!!
It is 'wrong' in how it is presented on the board. It is talking about two different things.
He says, that S is EQUAL to -1/12, but it isn't. In order to make this true you have to think of the equal sign as anything BUT "is equal". More like "is associated with"
You cant add up divergent series and shift the cols... you learn that in the first semester of any math course
The way its derived here can be replicated but you can set the sum to any value you want, where as using a rigourous formulation, -1/12 would be the only value you could come up woth.
Gone Crazy It had been proven, several ways, before Ramanujan.
Salute to some Indian guy for giving us this. ❤️ Ramanujan ❤️
For those who didn't know, this was formulated by an Indian mathematician Srinivasa Ramanujan
🇮🇳
& he is from bihar
@@AbhishekRaj-dt9fi bro he was from madras (tamil nadu)
@@anupamtiwari9020 mathematician jyade bihari hote hain isiliye lga
Great video, personally I disagree with the proof and here’s why.
After an infinite number of steps in S, you will arrive at a number we cannot notate, same with 4S. In the division part of the proof you treat both series as if they follow the rules of countable numbers, which they can’t because they are infinity. Infinity and infinity x 4 are the same thing. The proof come out to -1/12 because you stop the series before it is completed. But if you wait until it’s done (ie infinite steps) they will be equal, and s will equal infinity.
I remember seeing this on another channel; numberphile
ya, they got it totally wrong there and it went sorta viral too.. total fuckup
The result is correct. -1/12 is proportional number.
Zero, infinity and indefinable is also a number type.
Whole proportion is already known as infinity. The result is 1/12 proportion of minus infinity and its total -12/12.
But there are another thing reality is relative. When relativity is considered then infinity must be considered as relative fact it means it is not infinite.
There are mathematical facts being used in math. Other than numbers there are facts as zero, infinity and indefinable.
Numbers are apart by reality of zero because each number or thing or particle in universe is separated with emptiness. Numbers are also on the base of zero. There is one single zero and many numbers. Modern math has different zero types such as imaginary numbers but in classical math zero must be known as unified base and mentioned not as number. Zero is a base for numbers.
Why I’m going in to this subject because infinity is not infinite. Infinity is ending by indefinable and it has proportional structure.
Numbers and zero are together bur after last number there is zero then infinity getting started. This part up to zero is relative. Then relativity is infinity, which has dual structure. It is infinite but it is limited such as circle. Circle looks infinite cycle but it is limited object. The circle is base geometrical shape and its relativity is not a surprise.
These are explanations of relativity, being two different things at the same time.
Relative reality has layer and each layer has its own individual relativity. Such as numbers and zeros are relativity and numbers are relative to each other so on. On the infinity part, infinity has Its limit because after limited infinity indefinable must be placed.
Indefinable is the end of infinity.
From this structure we can understand -1/12 has a meaning. if we distinguish the layers and understand the layers in their own structure…
By the way infinity is relative by its limited structure and the continuity of relativity leads us to the understanding of indefinable must be singular and it singular structure must be the source of relativity which makes it relative but singular at the same time. Actually it has another explanation but this is enough for -1/12
I have a problem with S-alpha. You "doubled" it by adding it to itself. BUT when you moved it over, you ensured that you are NOT doubling it. In order to pair all the numbers the way you do, there will always be one number left over at the end. Oh, and that missing number is either +infinity or -infinity!
So you proved that the sum alternates between positive and negative infinity.
haha same thought
Of course, no one said that is actual, real proof. It 'looks' really great, just like a video I saw saying π = 4 and the proof definitely 'looks' real but it deceives us. S_a in this case would alternate between 1 and -1, and S would approach infinity.
Oh yeah !!
There is no end, it's an infinite series.
Except these series are logically consistent and appear in the real world with these values when you perform quantum experiments
this paradox is known as ramanujan paradox . Here ramanujan is an indian mathematician whose full name is Srinivasa Ramanujan. Proud to had such legend in India.
I'm 77 years old If I had had this man as a teacher in high school, or even in college, my life today would be totally different. This is an unlikely scenario since I graduated from college 15 years before he was born.
Drink!
I'm gonna guess before watching the video: ±∞
Let's see why I was wrong :)
@Boz Scaggs Yes, Mathologer is one of the few popular math youtubers who actually seems to both understand and be capable of explaining the material in high level math topics.
Christian ma sei ovunque
You are not wrong.
Cosa ci fa lei qui?
I have devised a truly marvelous counter-proof for this, but alas, this comment box is too small to contain it.
try harder next time you might win a Nobel laureate with typing fingers of yours
gidmanone I don't think you get it... he is referring to Fermats last Theorem
Jacobi Carter this is a parker reply
Hello fermat
r/iamverysmart
This is called the “Ramanujan Paradox” which was given the great Indian Mathematician Ramanujan!
Well, in S_alpha you always leave the last number out, which is an infinitely big number. In general adding/subtracting/multiplying on both sides of an equation breacks down with infinity because it follows different rules (such as 1+infinity infinity, 2*infinity = infinity, etcetera)
Exactly, when you look at the set of numbers that addition applies to, infinity is not in it. And arguably neither is zero since the behaviour of adding zero to something is fundamentally different from, for example, adding five.
The real question is :
Why is there a witheboard stuck to another witheboard?
In the end I was like: WHAT DOES THIS ALL MEAN?! I'M TOO INVESTED IN THIS WHY DID IT STOP THERE?!😂😭
The Mathematician Ramanujan made this theory, he did these research with a chalk in a temple, said that the goddess helped him.
Mr. Eddie Woo, if you ever see this.. Thank you so much for your videos. Puts me in a good mood instantly. Love how beautiful numbers re
The problem of this proof is that it contains infinite numbers. See:
If we transfer the abstract math problem of the video to concrete numbers, we are able to see that the proof is just wrong.
Sg = 1 - 1 + 1
That series is equal to 1. Now we do the exact thing that the teacher did:
1 - Sg = 1 - ( 1 - 1 + 1)
1 - Sg = 1 - 1 + 1 - 1
The new series on the right site is equal to 0. So we can not call it Sg because it is a complete different series. I call it Sn. So the new equation looks like this:
1 - Sg = Sn
And that makes sense because Sg equals to 1 and Sn equals to 0.
1 - 1 = 0.
The mistake in that proof is just the use of infinite series. Nobody ever knows what infinite is equal to. Sg could be 1 or 0. It depends where the series stops. If it stops at a 1+1 the series equals to 1. If it stops at a 1-1 the series equals to 0. But there is no real stop in the series, so there is no way to tell what Sg is equal to.
Im sorry for my english. I am from Germany.
Grandi's series is well known to equal ½, and it can be shown in many more technical ways if that would help you.
Eoin Lanier I don’t know where you’re getting this from but that is not the case
@@evandavid9087 B(1-1+1-1+...)=1-x+x²/2!-x³/3+...=e^-x, where B(S) denotes the Borel transform of a series
Then if the Borel integral converges, the Borel sum of the series is equal to the value of the integral.
Int[0, inf] (e^-x)(e^-x) dx=Int[0, inf] (e^-2x) dx=½
Its The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12!! Hey i have a suggestion for you..please give the credit to all the inventors from where you choose the topics and discuss the methods!! Also give some links from where everyone one will get the detail knowledge about the topics!!
Bdy you are awsome man🙌🙌😇😇
Well, you are writing this on youtube, give credits to them. For english give credit to England, and because you're alive, give credit to you mom and pop too in this comment.
@@thisbevibhor if you don't know what citation in academia, go and learn
I can't wrap my head around how shifting the sequence right to double it is a legit mathematical process.
Completely agree, that but blew my mind. Imagine if we did that in matrices
It's not legitimate. That's the point of the exercise.
@@mychaelsmith6874 watch the video dude
@@ssgoko88 you cannot group infinite series because grouping implies that it's finite
It is legitimate but what is not is the assumption that follows, that the Sα somehow gets a value, while the series diverges just like a (co)sine integral and has no limit and hence no sum whatsoever.
I would have loved to see the face of the students if the professor at the end went out and said: "and everything I just said untill now is wrong, goodbye"
The problem is that infinity isn't supposed to be used in the same manner as numbers or variables that carry limited amount of measure, infinity is more of a concept, and infinity with a series is even less suitable to be used in the same line as numbers. Even if you just rearrange some of the numbers in the S alpha series it would even make result of the series completely different; for example S alpha series itself is approaching negative infinity, but in 2S alpha he rearranged the series which technically it's no longer 2S alpha, instead of negative infinity plus negative infinity, now it becomes a series that alternates between 1 and 0
Andy Fu What he did is still completely valid a manipulation if you let the number N be the upper bound of summation and then evaluate the limit to infinity after the manipulations are done. It’ll leave you with the result 1 - S = S, which yields S = 1/2. Then, by taking the Cauchy product of the series with itself, you obtain B = 1 - 2 + 3 - 4 + ..., which equals 1/4 by the Cauchy product since B = S^2 and S = 1/2.
Your argument is basically, "if you rearrange the series you get something different" but the Riemann rearrangement theorem says that rearranging any conditionally convergent series changes it's value. So you have shown anything about the series S itself
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Why is this video ending at exactly the moment when it gets interesting?
just waiting for jeff bezos to wake up one day to -1/12 of a dollar in his bank account.
Underrated comment
The really interesting question, which wasn’t answered is “what does it all mean?”
ruclips.net/video/BpfY8m2VLtc/видео.html
@@reazzae2979 Well i've got some bad news for you, physicist use this result in physics and it actually works =)
@@moutii how do you know?
The rules being used (algebra and also general division/multiplication) are for finite systems.
A series is infinite, and thus the rules spit out weird numbers.
Subtracting two definitive numbers is not the same as adding series.
Weirdly however, -1/12 works. It is used in physics and actually makes sense when replacing infinites which pop out of a lot of physics equations.
My head hurts.
String Theory
@lessur dgreat string theory proofs of dimensions and quantum theory
@lessur dgreat When using the String Partition Function to determine the energy of a string based on all of its possible ground states, you end up adding (1+2+3+4+...) because a string can oscillate at the first harmonic, second harmonic, etc. The answer is -1/12, and in the context of the rest of the formula, it gets multiplied by 2 for a total of 24 dimensions perpendicular to the two dimensions of the string itself; so string theorists say there are 24+2 dimensions (they like to use that notation to make it clear that even though there are 26 total, 2 of them are not directions the string can oscillate in because the string exists in those dimensions).
It was first evaluated by Srinivasa Ramanujan of INDIA. 🇮🇳🇮🇳🇮🇳
Ramanujan included this proof in a letter he wrote to Oxford. I can't remember off the top of my head if it was his originally or if he rediscovered it without former association, exposure, or professional training.
He was absolutely one of the greatest minds of mathematics, either way.
If infinite is not consider as a number then the results will be like this..
True
yeah why so many deny this in comment section. They reject the truth because their iq is not high enough to understand this. Even they act so smart and say that Ramanujan was wrong
Indian Mathematician Srinivasa Ramanujan actually stated this.
This work was by sir ramanujan and is used in the mathematics to explain blackholes.
not this math, the one that is used in black holes is mock theta function but yea he was the one who invented it.
Sir i loved the way and the things you teach. Can you tell me how you came up with this awesome questions and interesting explanation? So i can tell me school and other head of institutions to refer to that material instead of old curriculum which was taught.
@JHIGYASU STUDENT wow that's so proud feeling to hear that..
Can u tell me the resources where I can find interesting contributions of indian mathematicians.
I see all the comments like « you’re wrong, you can’t assign a value to a serie » accusing him of telling lies.... DUDES, He told you that it was a controversial result and that people always argue about it. So no need to try to convince us that he is wrong. Let us make our own thought of this. Also, his aim isn’t to prove anything but to make you THINK. And... clearly if you wanna talk about that kind of mathematics stuff, the comment section of RUclips isn’t the best place to argue....
I just discovered this guy, I wish more teachers where like this.