Yeah but the only misconception he got is that value = sum Which is not the case. Edit: To be fair, the numberphile video explained it horribly wrong if I remember correctly. They made an updated video called "why - 1/12 is a gold nugged" that one's much better in explaining.
@Mika Hamari Could you somehow explain it to me? I am a high school student and my basic logic skills say that it is impossible to reach a negative result with positive additions. (Also english isn't my native language, so excuse some grammar or vocabulary mistakes).
yes they had a contradiction . the series doesn't converges .but they assumed it does converges and they used the properties of convergent series to find -1/12 .which is impossible since we are summing a positive integers . and the correct answer is that the sum approches infinity when n goes larger and larger .but what is more interesting is some how -1/12 is related to the series and it has applications in string theory and quantum mechanics even though it came from wrong assumption
Mika Hamari You can disprove convergence of all of those with all basic tests like D’alambert, Cauchy, Integral test and Leibniz for the +/- series, which are tools people learn on the 1st year of technical college. Really scary how few people talked about how flawed the numberphile video was
@@lupsik1 I think the big thing is that the majority of people are divided into two categories: People that have seen this all before in math classes but forgot some of the specifics and caveats, and people who haven't and trust professional mathematicians more than their own intuition. The latter group are the ones that would have been confused and bugging all the other math channels to explain it or something, which is what caused any of this. In reality, the numberphile video isn't "debunked", just properly contextualized and constrained. The issue with people bothering other math channels about the confusion is really the full extent of any damage that could have been done, at least that anybody should care about. If you're taking stuff from a youtube video and using it as the sole justification for anything you do on any math exam or really anything ever, then you have a bigger problem.
Another way to put this is this: the sum of all positive integers equals -1/12, for very specific definitions of the words "sum", "positive", "integers", and "equals".
I could swear, when I took number theory, one of the first homework problems was proving that the sum of two natural numbers is another natural number.
@@praharmitra 1. Every partial sum is, by recursion, the sum of two natural numbers, and hence must be a natural number. 2. The set of all partial sums is countably infinite.
@@l.w.paradis2108 I don't understand what your point is. Rational numbers are countably infinite. The infinite sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... is a sequence of rational numbers and each element of this sequence is a rational number. Yet, the limit of this sequence is pi which is not a rational number. Same goes for the sequence 1, 1+1/2^2, 1+1/2^2+1/3^2, 1+1/2^2+1/3^2+1/4^2,... where every element is a rational number but the limit is not.
@@sottallu It asserts such "solutions" exist but makes to claim as to which "solutions" those are. It's merely a warning not to be fooled by simplicity.
In a matter of fact, Mathologer told us to quit being real and start seeing imaginary! It's Numberphile who tried to project the power of complex and imaginary to the simplicity of real, hereby resulting in nonsense.
@@JusticeBackstrom No one is saying that the sum of the natural numbers is -1/12. That's just clickbait. It is obviously a divergent series with no real properties. But the Ramanujan Summation is used to apply a mathematically useful summation to a divergent infinite sum. It does find its way into things like String Theory.
The interesting thing about it is that physicists often really don't understand the deep subtleties of the maths they apply, abuse the maths in a way that makes every mathematician cringe, and get out a result, which is exactly in-line with how nature behaves (just think of normalization in QED).
Yeah, I actually couldn't watch it. I'm ten minutes in and all he's done is slag off the numberphile video and it's been boring for a solid five minutes. I'm out.
Having rewatched this for nostalgia:) it really reminds me of early math education in primary school, where you just get told stuff with no justification and even though most of the methods you learn there are common sensical, the point of math is to connect common sense with rigorous logic. And pretending something makes sense out of the blue is a really hard thing to unlearn and i think that sets a bunch of kids up to hate maths. Which is really a sad thing.
@@misanthrophexI have a BA in creative writing/English and now as a tutor, I also teach marh I can say with confidence that if primary school math involved more "philosophizing," the number of kids who "just don't like" it would drop significantly
@@pugsnhogz I would strongly argue it would be the opposite. The mere seconds (if that) of attention span these kids have precludes virtually any form of philosophizing as it relates to much of anything, especially math. Putting that aside, they probably wouldn't get it anyway. These are, for the most part, people who, when presented with math word problems, freak out. I've never understood why anyone would have an issue with word problems, but then again, I've never had an issue with math. I had to study for Calculus, etc. but very little in math classes prior to that.
The reason many teachers don't explain the equation is because they themselves do not know the explanation of the equation. They just pull out the book and tell the kids to memorize the equations and methods, and this is a very boring way to learn math.
@- RedBlazerFlame - The Zombie is like an Extension of the normal world: Your mathematical rules don't work here, human! 😈 Or you could say: This is the value you expect. The human is "converted" into a zombie, which actually makes sense
One of my favorite mathemathians is Cantor. He was German. Too bad he died a broken man because he was bullied because of his theory about cardinality.
Thanks. I never understood Numberphile's assumption that an infinite series can have a fixed value like 1/2. It seemed arbitrary to assign a value but the presenter acted like it was self evident.
Bro it was so poorly explained it seemed like they were just randomly throwing in series that would conveniently result in the desired -1/2. Laziness and math do not go hand in hand. Ever. Even on TH-cam... I was fortunate to immediately go into the numberphile comment section and see someone recommend this video.
Arguably it is assumable for some cases, because it is *true* for some cases - convergent series, as another reply states. But something does have to be a convergent series for things only true about convergent series to be true about it, so you have to at least have an intuition for whether a series will converge if you don't know for sure - and while my own test isn't 100% accurate, it DEFINITELY rules out series whose terms *increase rather than decrease*. My point being I agree that here was not the place to act like that was a given.
You did this in grammar school when you divided 1 by 3 and got 0.3333 . . . and so on to infinity. This means 3/10 + 3/100 + 3/1000 + 3/10,000 + . . . + 3/10^n + 3/10^(n +1) . . . for all *_N_*
Same here, never made sense to me why all of the POSITIVE, INTEGERS sum to a NEGATIVE, FRACTION. Always seemed completely backwards, and +infinity makes far more sense
@@rygerety8384 (1-1+1-1...)=1 or 0 now 2(1-1+1-1...)=2 or 0 so it is undefined.It could be 0 or another number because it is an infinite structure of conditions.You can say an infinite number is not a number.We calculate base on renormalized numbers. Infinity is not real in real life maybe,because if the world is real so it must be a limited structure of numbers,an well defined number that represents for physics laws. Zeno had said,time or motion is not real and you can't prove he wrong,no mathematics or physics solution can prove the cause and effect work in such a infinite manner.
Reminds me of the first time i learned about the dirac delta function in physics. I was basically told "there's some complicated math that proves this is correct but it works and that's all we really care about."
Well in the case of the Dirac delta, they are at least not giving wrong arguments why it works, do they? Btw: the foundations of distribution theory are really nice imo, worth checking out.
I have a lot of respect for Eddie Woo who also did the -1/12 proof. I knew there was something wrong with his strategy, and now I know exactly what it is. Thank you.
I just find it a bit dishonest (or very sloppy) they do not specify when the "super sum" (which is called I think Cesaro Summation), which assigns values to some infinite sums that are not necessarily convergent in the usual sense. The term "summation" needs also a big asterisk, since it's not the conventional sum you learn in primary school. In fact it's a swindle... the "Eilenberg-Mazur swindle", hehe
@@utkarshsaini5650, not even Ramanujan, it was Euler who first proved it, in the 1700s. This math has been around for years and there are multiple branches of physics-based around it, so if this video was accurate, which it's not, it would be one of the largest revelations for complex physics in the past 100 years
mathologer is great. as he points out, the shift in S2 is the culprit. if you did 3S2 where the last line got shifted back to the left, you get S2=-1/4, an S=1/12; also if you shift the 2nd line in 2S2 to the right twice instead of once, you get 2S2=-2S2-1, which also makes S2=-1/4
To be fair, he never said that this result was true, at last with the standard definition of a sum. He just redemonstrate the result to make people think about the mathematical logic, never saying if it's true or not
@@kristoferkoessel4354 Halte would be correct too, but it is more formal, which doesn't make much sense in this context. And Halt also means stop. In English there is a similar relationship of words. If somebody tells you to put something on hold you will probably stop doing something. Or if you are supposed to hold a door open for someone you also stop the door from moving. So Halte makes sense and the person you are talking to will understand you, so it is not a real issue. That rule also does not only apply to Halte. The e is often dropped from the verb, if you are telling somebody to do something, I can't even think of a word right now where it usually isn't dropped
So clearly what you wrote is all non-sense, but damn was it funny to read anyway. My favourite ones: "All scientists think light speed is c in the vacuum, they all wrong." Gee, I wonder what the light speed in vacuum is then... and what letter should we use to represent that value? "Iss is fake, AC systems cannot work in vacuum space" No, Iss is fake because there is no sound in space, so their alarm clocks wouldn't function properly. Get your facts straight. "If heat can radiate into space, [...], the whole universe will be at the same temperature, thermal equilibrium." *long stare* ... sure ... it's called heat death...
39:20 Also; even Ramanujan, for all the formal education he lacked, didn’t call the identity: ”Sum”, in his personal notes. He used the notation: ”c”, for: ”Constant”.
@@samueldeandrade8535 I agree. It *_IS_* a kind of a small thing. But a lot of people just want to misunderstand others, and will take any excuse to do so, however minor. That was a careful and smart move, to disarm such people.
The best complex logics/math film I have ever seen. By “complex” I mean “consisting of many, sometimes, non-trivial elements”. If I confess I am awarded Best University Lecturer for many years, it is only to pay tribute to the quality of this film - to keep things so ordered and clear is SIMPLY AMAZING! I do appreciate the apologies for not explaining why complex numbers needed to be introduced (but no fully explained) when analytical functions were being talked about. It gives a lot of security to a lay listener that all vital things were introduced even if no all were fully developed. Yes, the content still can be completely wrong (I am not an expert to judge) but it is certainly “CONSISTENT and COMPLETE” - in contrast to the film it was commenting. The detailed and well paced debate with the statements of Numberphile content were excellent. Well, it was really impressive. I do not subscribe to any channels and social media but believe me, I will be watching you regularly!!! Well done (you know it 😊).
Absolutely agree with you, I am a professional physicist so I can judge this video with some degree of expertise. It is absolutely brilliant. I was wondering how he would justify analytic continuation.... he succeeds even for a high school level educated person in my view. I am still dazed by the level of pedagogical expertise.
I have a slight suspicion who You are, and If I am correct - we might have passed eachother a few times on Madalinskiego. My late father spoke very highly of You. Odd, getting teary eyed under math video, of all things.. With the current level of growing mistrust of science, I am eternally grateful for those smarter than me being on guard for falsehoods. I understand the desire to simplify complex subjects but this is unacceptable, not because it's a mistake -as these happen to best of us, but because it seems to be almost consciously feeding into the "stupid scientists, power to the simple minds, they are hiding truths from you" type of the political climate and I viscerally hate anything that creates artificial divides between people, some of whom perhaps could be lured into the dark side of learning and reason still. Thank You, Mathologer.
Yes but he still assumes induction is valid forever and it isn't . The universe will stop you at a large number. You can't count forever. It is impossible. Physics will stop you from adding "one" to some large number and that will be the biggest number possible. You can't escape the universe.
Z -> Q loses single representation, Q -> R loses countability of the set, R -> C loses the order of numbers, C -> H loses commutativity of multiplication, H -> O loses associativity of multiplication. EDIT: s/looses/loses/g
26:14 - "now let's play a game." Me: sweet I love games *Shows a graph* Me: is this some kind of German game that I'm not structured/organized enough to understand?
I remember explaining how 1+2+3+... diverges in the comment section and people responded that I'm wrong since I'm not a university professor. So thank you very much for this video! Math is about truth, not educational authority.
"I remember explaining how 1+2+3+... diverges in the comment section " It does diverge. Everybody agrees that it diverges. The question of what it "equals" is conceptually separate and requires agreeing beforehand on what the word "equal" means. It's not at all true that the only possible meaning of "equal" for an infinite series is that of the limit of the partial sums. That is a choice, one which makes sense in many circumstances, but sometimes you may want a different one.
Vacuum Diagrams yes but then one has to make it very clear what equal means in a certain context, especially when the large amount of viewers might not be math students
"yes but then one has to make it very clear what equal means in a certain context" Indeed, but this applies to _convergent_ sums just in the same way. When I say that 2 + 2 = 4, I mean something quite different than when I say that 1 + 1/2 + 1/4 + 1/8 + ... = 2. The former is the result of a single addition, while the latter is a statement about convergence and limits. It's a nonstandard use of the equal sign, just like the use in 1 + 2 + 3 + 4 + ... = -1/12 is nonstandard.
This was like one of the first things they covered in undergrad, the series that alternates positive and negative 1 they told us to think about as a digital switch, it's either on (1) or it's off (0) and it can always be made to be in one of those states by adding an extra term but it can never behave like an analogue switch and be in a state that is some measure of two values it takes. Really helped me to understand why its sum cannot be assigned a value. This video made more clear outside of thay intuition.
Mathematicians reuse the same symbols with different meanings all of the time. It is much easier to say, here is this idea I am working with, and here is a nice symbol for it, than to come up with a brand new symbol for everything. Numberphile's problem was not putting a disclaimer up saying "Here is the standard meaning for this notation, and here is another idea that uses the same notation, but isn't the same thing." They should have made the distinction clear, instead of not mentioning it.
Obviously it's not always a great honour to be corrected in science. Some of the most renowned scientists of all time, including Newton, Kelvin, Edison were all challenged after having reached fame; their ideas about the universe and the contents of papers they had published were corrected, but they refused to accept and acknowledge these discoveries, many of which were ignored for a century before finally resurfacing providing solutions in other sciences. A great deal of this was the fact that basically all people are stubborn and will give in to power and fortune. You can think of it as great scientists being corrupted, or there being little to no difference in science emotionally from other endeavours. If you can acknowledge that you were indeed mistaken in your assumptions, then standing corrected may be a personal honour. But that actually has very little to do with being wrong. Most researchers for instance do not care about being right or wrong at all: providing an argument in the publishing of a discovery is just a formality. Being recognised for posing the right question and having the idea that sparked the study is a much greater honour. And when then someone comes afterwards and points out a mistake in a study you were the mind behind, you are quite simply flattered. Feeling honoured for being dissed in science is the worst pseudo spiritual zen bullshit myth I have to live with. It's just a mindset overrepresented by Hollywood movies.
Math isn't a rational subject: It's a system "we" created based off axioms which are accepted as true. (When a Contradiction occurs in Math- we either correct for the contradiction or avoid doing what caused error) Eugene Wigner wrote a really famous paper called: "The unreasonable effectiveness of mathematics in the natural sciences." *If there is an infinite amount of numbers between 1 & 2 (How do you get to Two?) *If it's Zero degrees outside and the weather man says it's going be twice as cold tomorrow as it is today. (What's the temperature going to be tomorrow? [ 2 x 0 = ? ] ~Not Zero you need to switch the formula. 1+1=3 When a Man and a Women enter a Dark-room- Nine-months later you have Three people... 'Math is litterally the Definition of *close enough;* The Great Pyramid of Giza is the most accurately aligned structure on earth- and it's still off 3/6 a degree True-North. (Rolls eyes) Don't get me wrong- Math is extremely important: Without Math we'd suck at 4th dimensional physics. But there's really only one number and that number is: *EVERYTHING*
Math is an observational tool, and while yes, we agreed to 1 = one object, 2 = two objects and so on to be the case, it doesn't change the fact that there was two objects in the first place. For your points: 1. Eugene Wigner, while being a wonderful physicist bringing light and joy to people arround the globe by some of his greater projects (sarcasm, obvs), absolutely did that. And he also has several others - "Maths being shit in economics", "Maths being shit in everything" and so on (obvious hyperboly is obvious). Reading through those articles (thank you for bringing it up in the first place, was an interesting read) - I came to a conclusion, that either: A - he is not aware, why does physics need some of the cooler stuff and how mathematics and physics are connected or B - he was just a hater for the sakes of it (especially when it comes to economics one, since Eugene seems to be fairly low knowledgable in the field). 2. By defining the step of your infinity in the first place. The one you mentioned is an uncountable (1;2) infinity 3. Extendanding an example to the concept - is a logical failure on your behalf (or wherever you took the quote from). One guy saying, that it will be twice as cold tommorow, when it is 0 today - isn't really the best example of human brain functioning in the first place 4. That is not really how babies work. If you want to be tehnical - throw in all of the variables (the baby doesn't appear out of nowhere, it has energy consumption throughout the whole process). Otherwise, I will extend your example on two rocks being left alone in the dark room for 9 months - and after that a third rock would magically appear 5. Great Pyramid of Giza - is "close enough" in your statement, not the other way around 6. You wouldn't be able to write your comment in the first place without math. Or watch the video for that matter. Or use TH-cam. Assuming you'd have Internet to open TH-cam. And an internet connection in the first place - to your PC, of course, if it'd exist 7. Hey look, I used numbers to make my comment easy to read. When were you born tho? Answer me in everythings please ^^ And also, if 0 degrees outside - you are a flat earther!
The average human has 9.x fingers and 9.y toes. Averages never claim to represent a single one of the values that went into calculating them. Another good example are population BMIs (body mass indexes) being applied to individuals. It's almost always wrong.
As someone who holds a PhD in analytic number theory, I appreciate the exposition here. The ideas are clearly presented and give a relatively complete explanation of the phenomenon occurring with -1/12. The explanation of analytic continuation was particularly nice, as this is a concept that's definitely tricky to pin down if you want to get into the technicalities around it. Glad to see some quality mathematics communication concerning the infamous Numberphile video.
Given your credentials, maybe you can answer this question from a non-mathematician. For the sequence 1/2+1/4+1/8... I had thought that, assuming the sequence is infinite, the sum would be an asymptote and not 1 because given infinite denominators you will simply get smaller and smaller fractions. What am I missing?
@@louzander This is just a matter of understanding vocabulary. When we speak about infinite sums, what we really mean is the limit (in the sense of calculus) of partial sums (that is, sums of finitely many terms). To say "the infinite sum equals x" is really to make a statement about limits. That is, the statement "the infinite sum equals x" is literally DEFINED TO MEAN that the sequence of partial sums (1/2, 1/2+1/4, 1/2+1/4+1/8, etc.) gets closer and closer to x. To use your language, "the sum being an asymptote" is the DEFINITION of equality in this scenario. If we're being more precise, we should say that "the infinite sum converges to x" rather than that it "equals" x. This is, of course, just a matter of semantics, and once one understands limits, an infinite sum "equalling" a number can be interpreted in a rigorous, precise, and unambiguous way. Hope that helps!
Excellent video. Unlike some, I don't think you were being harsh. When millions have viewed flawed information, a clear refutation can be seen as a public service.
Thanks a lot for your effort. I saw that numberphile video years ago when I began my studies and it confused me a lot because we've all been told you cannot do anything with divergent series. This video finally cleared things up for me.
John Deacon - that is a nicely-worded response, but it is, after all, written from the point of view of a physicist. I understand the points he makes, and he's quite right about the usefulness of analytic continuation - but that isn't the point. The point is that the audience of the video may have been given the impression that such things can be stated without context, as being strictly true. To me, it is clear that summing the natural numbers cannot possibly result in -1/12, UNLESS you state clearly that your context is one of analytic continuation. This is a subtlety unlikely to be understood by a general audience, and the complaint was that this was not made clear. I think this was a fair complaint. I differ from you about the style of Mathologer's video too - I don't think it was unpleasant. But of course, that is subjective and therefore not open to debate.
I keep coming back to this video every so often, and each time I am utterly amazed at how intuitive Burkard makes these complex topics. I appreciate that he is so careful with his terminology, and of course his graphics are awesome. It was so cool to have Burkard run down exactly the problems in the Numberphile calculation and how to "fix" them...when he did the transition from the Numberphile S-S_2 to zeta-eta I was blown away; in an instant, he transformed a simple, familiar, but false expression into a deep, rigorous, and true statement, highlighting the "simplicity" and "familiarity" behind things as complicated as power series in the complex plane. Literally one of the best math videos ever made.
Me too and i actually like the video and seen until the end and i just completed high school and some shit calculus and algebra from computer science.. Many time i wish i choosed math or phisics instead of cs
I stopped at the 10 minute mark too. Cause it felt he was done explaining the wrongness. "What else is there? An extra 30 minutes! What the hell... I don't remember signing up for this."
Wonderful stuff! The second half was way above my mathematical pay-grade, but I still understand much more than I did before. Great work! I had been duped by the -1/12 stuff.
Dupe isn’t the right word; this isn’t even necessarily a real rebuttal of the -1/12 sum. The result is controversial and this is a good argument against the result (which is counterintuitive which in itself isn’t meaningful). The whole thing, the controversy and the result, are more indicative of the clumsiness, errors and even perhaps uknowability of logic, math and the implicative language of trying to state it. The terms are very slippery and we get strange results in our minds when we try to manage it all. The argument made here is one, a robust and hardy one but it is no more ‘correct’ than other views.
you haven't been duped. -1/12 is a meaningful value assigned to an infinite series. this "sum" is not an actual sum in the traditional sense, but it was derived using real methods. in the context of a youtube video teaching about infinite series, numberphile was correct. in the context of a mathematics course that requires rigor and proper definitions, it was incomplete. we know that -1/12 works because it can be used in real world applications of physics.
@@LeNoLi. This last comment is what really interests me. What does "-1/12 works" or its utility in real world physics tell us about mathematical truth? I have in mind the use of infinitesimals, in Newtonian calculus - i.e., before the introduction of a "limit". These "ghosts of departed quantities" (as George Berkeley memorably called them) "worked" in physics, despite being, at core, inconsistent. This suggests to me that having real world applications in physics really doesn't necessarily tell us much.
The irony. You are being duped by thinking that we were duped. Terrence Tao just should that the -1/12 is valid and their is another numberfile vid on it.
I think you misunderstand. By "duped" I mean that I misunderstood something about the proof. I in no way intended to suggest that it is not "valid", in its own terms, but simply that I misunderstood the terms of the proof.@@sloaiza81
In (slight) defense of Numberphile, they did follow up with a much more informative discussion with Prof Edward Frenkel. Some aknowledgement of the flaws in that video that Mathologer is complaining about; the first thing we hear is Frenkel saying with some dismay "Oh... it's /you/ who made that video." He chuckles and shakes his head. Then what follows is some explanation of assignment rather than summing. They are very explicit: "[-1/12] is certainly not the result of summation of these numbers [1+2+3....]. It is something else, but what is it?" th-cam.com/video/0Oazb7IWzbA/w-d-xo.html
There is also the 'extra footage' video on Numberphile 2 which goes into greater depth of the math on the original- th-cam.com/video/E-d9mgo8FGk/w-d-xo.html
In this video (Frenkel's @ 10:19), Brady asks "My understanding of Math is it's very rigid and rigorous and it's never arbitrary, how can you throw away the dirt and keep the gold?". This question is the reason why I hated the 1+2+3...= -1/12 from the very first moment. Because that kind of destroys my view of Math (as the only concrete, unambiguous and objectively true tool we have). Mathologer if you're going to make a discussion video about this subject, PLEASE address this question.
Panting = breathing quickly. unpanting = not breathing quickly. So, "he unpants" could be interpreted as "he calms down and no longer pants". www.dictionary.com/browse/panting
Thank you for explaining analytic continuation in an actually good way. I've seen so many math TH-camrs talk about it and every time it boils down to "the most natural extension of a specific function," which, I imagine, would leave many questions in the audience's head. I can see myself understand this when I didn't already know what analytic continuation or any kind of analysis deals with. Really shows why derivatives shape a function which is not traditionally defined. Great job!
The way I always explained the "nonsensical" result of -1/12 coming from the Zeta function was this: The original zeta function is defined as the given sum, for only Re(z)>1. The analytically continued Zeta Function takes those same values for Re(z)>1, but is _not_ defined by the sum over its whole domain. I don't know if we know the closed form of the extended Zeta, but that form would relate -1 to -1/12 - and have nothing to do with the 1+2+3... Sum.
I think Mathologer deserves no criticism for this video. I like the Numberphile guys, but in that video, they presented a very misleading argument for the "sum" of these divergent series. The first rule in any, ANY argument regarding series is: "you can make some algebraic manipulations with series ONLY IF they converge". Notice the "IF". This is very important, because, with divergent series, you'll end up with nonsensical results applying algebraic manipulation. Let us check a stupid example. Let us suppose that I don't know if the following two series are convergent or divergent. S1 = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6... S2 = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6... Now, let us take, S1-S2, which, computating term by term, we get: S1 - S2 = (1/2 + 1/2) + (1/4+1/4) + (1/6+1/6) + ... = 1 + 1/2 + 1/3 + ... = S1 So, S1-S2 = S1, and thus, clearly, S2 = 0. Right?. WRONG. S2, as Leibniz discovered, converges to ln(2). The argument is invalid because S1 is a divergent series. So, my algebraic manipulation is invalid. The Numberphile guys should have made that very clear in the video, saying "these algebraic rules are only valid if the series are convergent. But, we'll be playful, and let's see what strange shennanigans happen if we ignore the convergence criteria". With that disclaimer, everything would be completely fine, but they failed to do so, so they deserve criticism in that regard.
@@SparelWood I think the Numberphile guys were trying to be informative regarding these "strange sums" which appear in advanced mathematics. But, of course, without any disclaimer, these identities are just nonsense. For example, we all know that "S1 = 1+1+1+1... = infinity". In fact, that is the main definition we use to explain people what infinity is!. But, let us again ignore any rules regarding convergence. S1 = 1+1+1+1+1+1+... S2 = 1-1+1-1+1-1+1-... S1 + S2 = 2+2+2+2+2+... = 2*S1 S1 = S2 So, given that we "know" that S2 = (1/2), then, S1 = (1/2). And thus, "infinity = (1/2)". So, even it is true that some process in physics in which the partial sum of a value can be considered "averaged" occurs in reality, but that is NOT an argument for justifying this kind of nonsense.
Thank you, Mathologer. This video finally explains what's going on in a rigorous and well-defined manner. I appreciate how you start from the standard definition of series summation, explain how that can be extended to "supersum" (Cesàro summation) and then go on to show the connection to the eta function. With that you actually write down the analytic continuation of the zeta function, which is really nice. For the first time I can see how to arrive at the result of -1/12 without handwaving or breaking the rules. Thank you for your high-quality videos.
To all commenters. I'm sorry that this comment is so long and ask you to be patient. The debate in the comment section whether Mathologer is rude/too late/ignoring other Numberphile videos on the subject is making me smile, so I'll put my two cents, too :) Numberphile made a video about a subject which is completely counter-intuitive. So it went viral, to the point that my father, who is 50+ years old electrical engineer, completely unconcerned with mathematics other than that helps to do his job in reality and barely speaking English, and even some medical doctor I went to (knowing that I studied physics), both claimed to me that the sum of all positive numbers is -1/12 ... That doctor even stated that nowadays mathematics is incomprehensible :) That's exactly the point which drives people like Mathologer out of their minds - claiming such counter-intuitive statements without proper disclaimers (I'm not even saying proper context, like Zeta function and analytical continuation). One guy in comments says (I'm paraphrasing) "All natural numbers can be written as a sum of 1s. So, 1+2+3+4+...=1+(1+1)+(1+1+1)+...=1+1+1+1+1... You say that 1+2+3+4=-1/12 and 1+1+1+1=-1/2. So now -1/12=-1/2 ??? " I guess that some people, uninvolved in mathematics, thought to themselves after seeing that video "And these people get paid for that ?" Numberphile should have added only one minute, saying that: "equals sign in these equations should be understood as "is assigned to", not "is equal to" " and "these calculations are not intended as a proof, they merely show what answer is to be expected from more rigorous methods". That's it. Everyone (almost) would be happy. Instead, all we heard was "astounding", "amazing" and "correct". Someone says (I'm paraphrasing) "How dares Mathologer cite Numberphile out of context? Numberphile did two other videos on the subject, which (more or less) address the issues with the first video. Mathologer ignores that. " Mathologer is perfectly aware of this. He even links one of them ("Why -1/12 is a gold nugget") in his description. The reason is simple: view count. The first two Numberphile videos on that subject, which completely miss to point out the crucial distinction between "is equal to" and "is assigned to" have been viewed 7.7 M times combined as of 2018 July. The one which discuses the subject properly ("Why -1/12 is a gold nugget") has been viewed only 1.6 M times. The difference is those confused people inundating comment sections. Another person says (I'm para...) " The goal of Numberphile channel is to make mathematics interesting to wider audience. Don't expect rigour there. Anyone who is wiling to get deeper understanding should follow the links and research themselves." Well, this youtuber forgot that he is commenting in ... TH-cam :) Content providers in TH-cam, especially those who want to appeal to "wider audience", should keep in mind "least action principle" - most people these days will spend the least effort to get information. Those who will research seriously, I assume, are those who already find mathematics interesting + small minority newly engaged. Most people, I guess, come there just to see "what interesting video did Numberphile upload today ?" I even suspect that many people rejected the video as nonsense, not wanting to have anything to do with divergent sums anymore, barring further research. All in all, I don't think that Mathologer is rude or incorect, I think he is right on the money (except that cameraman. He should have kept his jokes off-record.)
1. You fail to actually address the rudeness. There is a clear tone of condescension throughout the video, not just from the cameraman. Who is factually correct is irrelevant to whether Mathologer was rude, which he was, by standard observation of tonality and wording. Your comment rather comes off as ‘I think he's right, therefore he wasn't rude’, which is a nonsense argument. 2. Your argument is essentially that this video is to address misconceptions of people who viewed the Numberphile video and misunderstood it. Meanwhile, this video actually directly tells Numberphile they are wrong, repeatedly. For what? Not being able to control what their viewers say and do? No. You don't get to blame Numberphile for that. Your suggestions for what they should have said may have affected things… but you fail to provide a reason why they would know those suggestions would be necessary BEFORE THE VIDEO WAS MADE AND PUBLISHED. Funny; those suggestions are followed in the other videos that both you and Mathologer handwave away… almost like it doesn't matter what Numberphile does or doesn't do, they're just wrong because of what people watching them do. Either your understanding of this video's purpose is incorrect, or both your and Mathologer's understanding of responsibility is crude.
Badly Drawn Turtle Hm, on a second thought I guess I gave Mathologer a pass to being condescending, because he is right. Ok, I can somewhat concede this point. However, that first Numberphile video was just doomed to be interpreted incorectly. I believe this was because he was asking physicists to explain it. Physicists are less concerned with nuances in mathematics, and more concerned with applications, which in this case was knowing what number can be assigned to this sum. When Numberphile came to mathematician, namely Edward Frenkel, who has seen the video, Edward immediately understood that the solution was not explaining rigour, details, zeta function and all that, but an abstract meaning of that hapless equals sign. In fact, an advanced physics textbook is shown in an original video, and there is an arrow instead of equals sign. They did not explain that crucial detail which would have made a lot of people happier.
I’m learning data structures and algorithms and came to this video after a teacher told me to check out numberphile’s -1/12 video. So glad I pulled that thread and landed here where you made it all make “sense”. I would have laid awake in bed for far too long trying to wrap my head the bogus numberphile solution.
Confused 1+2+3+…=-1/12 comments originating from that infamous 2014 Numberphile video keep flooding the comment sections of my and other math TH-camrs videos. And so I think it’s time to have another serious go at setting the record straight. In this video I’ll do just that by having a really close look at the bizarre calculation at the center of the Numberphile video and then stating clearly what is wrong with it, how to fix it, and how to reconnect it to the genuine math that the Numberphile professors had in mind originally. Lots of nice maths to look forward to: non-standard summation methods for divergent series, the eta function a very well-behaved sister of the zeta function, the gist of analytic continuation in simple words, some more of Euler’s mathemagical tricks, etc. This is my second attempt at doing this topic justice. This video is partly in response to feedback that I got on my first video. What a lot of you were interested in were more details about the analytic continuation business and the strange Numberphile/Ramanujan calculations. Responding to these requests, in this video I am taking a very different approach from the first video and really go all out and don't hold back in any respect. The result is a video that is a crazy 41.44 (almost 42 :) minutes long.
Thanks for that. I'm not realy mathematicly educated, but i enjoy watching your videos and thank you for clearing that myth out which i myself believed
Thanks for your video. I regularly watch both numberphile and your videos and love them both. Not being a mathematician but being in science I really appreciate them. Likewise I know that in science arrogance spurs easily and often egos simple don't match even where facts have the reason. I was a bit surprised by the aggressive nature of your video, I just hope you pointed out their mistake directly to numberphile guys before doing this video. I reckon that may have been the case and they didn't took it well and that led to the tone of this video.
MY AUNT: But, the way that I calculated it, you owe me money for my purchasing all of this. *Everyone stares at us.* ME: Please excuse my dear Aunt Sally.
you may me 10000000000000000000000000000000000000000000000000000000000000000 dolllars. i tell u to keep giving me money and i will pay u back. soon enough i keep getting money from u infinitely and i say it can be represented by 1 + 2 + 3..... and he is like yea whtver give me back my money. and i say nope, i owe u -1/12 of a dollar, which means u owe me 1/12 of a dollar GG (ps: ty for all the money hehe
There was a video made by Numberphile called, "Why -1/12 is a gold nugget", where the professor, Edward Frankel, made it clear on what the identity "1+2+3+...=-1/12" really meant.
Would be fair to mention that video as well. Otherwise the term 'misled' could be partially true for your video. It's clear math videos like to be 0 or 1 :) Great video, my issue is just a small footnote.
Yes, that's also the case with Numberphile of course, but their videos are shorter so they cut (too many) corners. I just like the 'gold nugget' metaphor and wanted your opinion. Maybe you have another (better) metaphor. But like I said before, it's only a footnote in an otherwise very well made video, the effort really shows!
Isn't that the video that compared the infinity-ness of the series as a bunch of dirt that can be swept away, leaving a gold nugget behind. I found that almost as troubling as the first. It was like an explanation why it's true without explaining how it's true.
I graduated college a while ago, haven’t done any math in a while, but I really love this video. You are so concise and clear with your explanations. You make me miss math class lol
Thank you for discussing this. I have been in endless discussions trying to point out to others exactly what you have stated. It is easy to get caught up in all sorts of paradoxes when applying rules for finite math to infinite series. One must be careful when applying algebraic rules and arithmetic in these cases. GH
The thing about maths is that mathematians always care about and give the general case whereas physicists in physics always cares about and give the special case And yes Richard Feynman said something like this
Video is pretty good, if long, but I was not a fan of Grumpy Background Voice, who didn't seem to be making any actual contribution to the content, just kind of dissing half-heartedly.
Oh my god this video is amazing thank you very much for making this. Here are my answers to your challenges and some question I have at the end of this comment. On 22:22: Series: 1+0-1+0+1+0-1+0+1+... Partial sums: 1, 1, 0, 0, 1, 1, 0, 0, 1, ... Partial averages of partial sums: 1, 1, 2/3, 1/2, 3/5, 2/3, 4/7, 1/2, 5/9, ... -> 1/2 Therefore the supersum of the series is 1/2. So I think you made a minor mistake taking the wrong example as this does not prove your point. Here is an example which does prove your point: Series: 1-1+0+1-1+0+1-1+0+1-1+0+... Partial sums: 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, ... Partial averages of partial sums: 1, 1/2, 1/3, 1/2, 2/5, 1/3, 3/7, 3/8, 1/3, 2/5, 4/11, 1/3, ... -> 1/3 Therefore the supersum of the series is 1/3. Therefore supersumming is not invariant under adding infinitely many zeroes. On 23:10: Funnily enough, every extension from N to Z to Q to R to C is mostly invented in order to add structure. The structures added are additive inverse, multiplicative inverse, completion and roots respectively. Some things you might consider a loss could be the following: You lose well-orderedness, completion, countability (but regain completion) and uniqueness of roots and logarithms respectively. On 23:25: If 1+2+3+4+... supersums to some S, then: 0=S-2S+S= 1+2+3+4+... ...-2-4-6-... ......+1+2+... =1+0+0+...=1. This is obviously a contradiction. From this we can conclude that it is impossible to define some ubersum with the three desired properties such that the series 1+2+3+4+... falls in the domain of the ubersum. From this we can conclude that the series has no supersum, because supersums have the three desired properties. On 38:40: Do I understand correctly that this means that if Re(z)>0 then zeta(z)=0 if, and only if, eta(z)=0? And because Re(z)>0 implies eta(z)=\sum_{n=1}^\infty((-1)^(n+1)/n^z), finding zeroes for the Riemann-zeta function just corresponds to finding z with Re(z)=1/2 such that this series is 0? (Assuming the Riemann hypothesis.) Because that is simply amazing! Edit: I really want to thank you for this video, because I was always very curious how it is possible that the argument given in the numberphile video just happens to give the same result as analytic continuation. I always refused to believe this is a coincidence. So thanks so much for showing why this is actually not a coincidence!
I'm really glad that I stumbled onto this. A while back I started to look at a video on the "-1/12" claim. Very quickly it started to sound like hocus-pocus to me, so I bailed out. It has been a very long time since I studied mathematics, and I didn't want to try to figure out what they were doing. Your presentation, on the other hand, is crystal clear and it makes perfect sense to me with my traditional education in mathematics, including infinite series.
In an earlier Numberphile video, Dr James Grime described S_1 as PSEUDO-convergent, which I think is the most accurate description, since it doesn't *really* converge to 1/2.
Here's the relevant video: th-cam.com/video/PCu_BNNI5x4/w-d-xo.html And here are a couple of other videos he's made on his own channel about infinite sums: th-cam.com/video/7fGoins7q3s/w-d-xo.html th-cam.com/video/dwYPOi-Hfg8/w-d-xo.html
Glad to hear :) As you can probably tell from my final remark I was quite relieved when we finally finished filming this one. 40 minutes is a long time when you try to get every word just right :)
The Numberphile video in question seems to violate the principle, "Make it as simple as you can, but no simpler." Simplicity is a noble goal, and I laud those who try to make complex ideas understandable to a wider audience, but simplicity has boundaries beyond which it becomes simplistic or simply wrong.
People taking this video as offensive have little respect for mathematics. In the mathematical community proofs must be truths not follower fights in terms of what channel i like better. The way is presented may get some angry but the proof seems to be correctly developed
the problem is not his proof, but something no serious scientist would do: quoting parts of someone else work without considering the other half of their work. Numberphile themselves added two more videos to their introductory video which went viral. In these videos (esp. "why -1/12 is a gold nugget") they explain in more detail how -1/12 actually differs from a convergent sum and why it is still meaningful. What Mythologer does here is quoting and attacking (yes attacking. The headline of this video and the way it is presented is sensationalist and honestly a bit disappointing, since it is in general good content) part of someones work, ignoring other parts completely. Especially if the part of work you quote is a video made to introduce non math-PhD people on the internet to interesting and "mindblowing" concepts in mathmatics.
But it's not the right answer. The correct answer is that the infinite series 1+2+3+4+... is divergent. It does not converge to -1/12. This is what Mathologer has pointed out. If an infinite series diverges it diverges. Stating "it diverges" is stating the correct answer.
I am not basing my conclusion on intiuition but rather on regular summation. If we derive an expression for the partial sums of 1+2+3+4+... (i.e. Sn=n(n+1)/2 ) we find that the partial sums get increasingly larger as n->infinity thus the series is divergent with respect to regular summation and is a valid and correct answer. If we use zeta function regularization (i.e Reimann Zeta function) / Reimann summation we can assign values to otherwise divergent summations. Applying such techniques we can indeed correctly answer 1+2+3+4+... + = -1/12. Such results have value and meaning in Physics and I stand corrected in my assertion that it is the wrong answer. n the contect of regular summation however we find ever increasing partial sums and we conclude the series s divergent which in this latter context is correct although not particularly useful if you're a Physicist. :) Nonetheless 1+2+3+4+5+6+... is divergent is correct with respect to it's regular sum which is proven when we look at the limit of the expression for partial sums S = n(n+1)/2 as n-> infinity which is clearly divergent therefor 1+2+3+4+5+6+... is divergent. Q.E.D.
22:09: The supersum 1+0-1+0+1+0-1+.... is still 1/2. However if you insert zeros like this: 1-1+0+1-1+0+1-1+0+... then the supersum indeed will change to 1/3
ohthis Shiny but the sum depends on how you add the terms. If you add 1 and then -1 etc you get a different result (if any really) which will be different if you add them in groups
Finally. FINALLY. I'm no expert of course, but it was not very hard to realize that Numberphile's "proof" makes no sense, and finally someone talks about it.
The difference is really about math versus analytical science. The series has value in much the same was as the "rule of 72" has value in compound interest, in that it is useful even if it doesn't make sense on the face of it (although the rule of 72 is far easier to understand as a shorthand estimation).
Dude, it's not numberphile's proof, it's a proof that has been for many many years by so many mathematicians, like Grandi and Ramanujan and others, Numberphile did not create or invent anything, and what they delivered is correct to what was presented in the past, now weather it's correct or wrong is another story, this video right here is not the best mathematician in the world and certainly not better than Ramanujan and Grandi and others, so therefore I wouldn't take his words for granted.
You do calculations with convergent series to derive a valid formula, but that derivation would be invalid for divergent series. Then, after the derivation with convergent series, you set s = -1 which makes the initial series divergent and hence invalidates your derived formula. Therefore the rest is just pure nonsense.
a) If 1+2+3+...= -1/12, then b) 2+4+6+... = 2*(-1/12) = -1/6, and therefore c) 1+3+5+... = -1/12 - -1/6 = 1/12 d) Taking c-a gives us (1-1)+(3-2)+(5-3)+(7-4)+...= 1/12 -(-1/12) e) Simplifying, we get 0+1+2+3+...= 1/6 f) Therefore -1/12 = 1/6 And if you want to, you can keep going and generate all kinds of other values for 1+2+3+... I posted this (or something like it) on the Numberphile video.
Yes, that's the sort of contradictory calculation I was after :) Maybe try to come up with something that does not require zapping infinitely many interspersed 0s, so restrict yourself to the three basic properties that I highlighted in the video :)
OK, how's this: a) 1+2+3+... = -1/12 b) 1-1+1-1+... = 1/2 c) a+b = -1/12 + 1/2 = 5/12 d) (1+1)+(2-1)+(3+1)+(4-1)+...= 5/12 e) Simplifying, we get, 2+1+4+3+6+5+...= 5/12, which is the same as 1+2+3+4+5+6+... f) Therefore -1/12 = 5/12. Or is re-ordering a problem also?
Perhaps even better: a) 1+2+3+4+...= -1/12 b) Insert just one zero: 0+1+2+3+...= -1/12 c) Adding a+b gives 1+3+5+7+... = -1/6 d) Insert just one zero again: 0+1+3+5+... = -1/6 e) Add a+d giving 1+3+6+9+12+...= -1/4 f) Subtract that initial 1: 3+6+9+12+... = -5/4 g) Divide by 3: 1+2+3+4+...= -5/12 h) Therefore -1/12 = -5/12. Ha! Nothing infinite done. Added two zeros, one at a time, and subtracted one once.
Adding or subtracting sequences was permitted, but adding or subtracting an individual number like one wasn't a permitted operation was it? I'm more interested in the fact that a-b implies an infinite sequence of ones sums to zero. Shifting that new sequence again and subtracting from itself easily proves that 1=0, implying that the shift operation isn't allowed for these 'supersums'.
Thanks for this great video! I think there's also another way to reason about this: Given the infinite series S = 1 − 1 + 1 − 1 + 1… the conclusion was made (by summing it with a shifted copy of itself) that S + S = 1. However a silent assumption is made here that S is an actual number in the first place. It was assumed that S ∈ ℝ (or ℂ if you prefer) from which it follows that the expression S + S is a well-defined mathematical expression that has a meaning, from which one can conclude that S = ½ using the usual manipulations. However if S ∉ ℝ then what is S? Then the expression S + S lacks any definition of what it means and makes as much sense as the expressions "yesterday + the moon" or "the square root of yellow". Thus to complete the proof, one would have to show that symbolic manipulation on S have a meaningful definition and there exists a sequence of valid manipulations on it that lead to S = ½. That could for example be done by showing that S ∈ ℝ, but that is unfortunately not feasible. That is the missing part of the proof. And of course it's invalid to conclude that S ∈ ℝ because ½ ∈ ℝ ∧ S = ½, because that would be begging the question (a circular argument).
But you should know that "while dealing with real numbers, addition and substraction on them results in real answer" but it is not always true for multiplication and division so as far as the series given in this video fall under this law we can consider them to be equal to s and (s€R)
You should invite the professor over at Numberphile to a discussion of the topic. You could live stream a hangout, or something. It could be interesting.
Not likely to happen. When he was called out in the comments/on twitter, he got incredibly defensive and wrote a whole blog post on how “actually this is totally allowed and you’re all wrong”
Awesome video! My summary: 1+2+3+... = -1/12 is wrong when we use the standard summation for the infinite series from 1st year calculus, since 1+2+3+... series diverges. However, if we use a very different Ramanujan summation method, then 1+2+3+... = -1/12 is true. The problem with the Numberphile video is that they used the standard summation method incorrectly to prove the -1/12 result. This might give millions of people a false idea that 1+2+3+... = -1/12 is true for standard summation. I think making mistakes is ok. Only those who do nothing don't make them. So good job Numberphile, keep the ball rolling! :)
It should also be said that there are standard meanings for symbols like + and =, and if somebody intends to change their meanings he is socially obliged to explain them in advance. It is much better to retain the standard meanings and use different symbols for operations or things that have nonstandard properties. For example, a right-pointing arrow can represent a transformation that does not preserve equality, as in the string-theory text exhibited on the Numberphile "ASTOUNDING" video.
Well said! And good point about the arrow, it would be more suitable here, since 1+2+3+... is not equal to -1/12 in the usual sense. But we can say that we can use so-and-so method to associate the -1/12 number with the 1+2+3+... series.
I disagree about using another method to make a mathematical claim. That’s like saying, I scrambled the letters of the English language into my crypto code, made some sentences and then descrambled it and came up with the perfectly reasonable answer to your question (which is exactly wrong!!). Mathematics is a rigorous scientific method. You obey the rules. It’s that simple.
Nice to see someone do this. I randomly stumbled across someone still in university (I think an engineering program) bringing up these sums, I think as fun puzzles. I quickly put up proofs of their divergence, I might be a chemist but I was taught well enough to test a series for convergence before running off with it in my math classes (that and the sum of all natural numbers is obviously divergent). I was vaguely aware of non-standard summations such as cesaro sums and brought up that those series can be assigned summation values, but struggled to explain the nuance of the difference between being able to assign a value and the sum being that value. If only I could go back in time and have actually studied mathematics instead of science.
I really love your videos and I deeply appreciate and admire your intellect and talent in mathematics. You always explain concepts so well. Phenomenal work.
When the numberphile guys said "so this series alternates between 1 and 0, so the sum must be 0.5" I was like, "what, no, it doesn't work like that", but since I only have 'high school' maths and they're professors, I went along with it. I am feeling relieved and validated now that youtube has recommend me this. I'll be honest, started to struggle to follow around the zeta/eta part, but at least thanks to the first half of this vid I can rest assured the 0.5 thing was indeed nonsense
One can do all kinds of stuff with the Grandi's series, for example I can make it equal to 1 by writing 1-1+1-1+... = 1 - (1-1) - (1-1) - .. = 1 + 0 + 0 + .. = 1 and I can take it even further and make it equal to any number X by writing 1-1+1-1+... = (1-1) + (1-1) + .. = 0 + 0 + ... = (X-X) + (X-X) + ... = X - (X-X) - (X-X) = X - 0 - 0 -.. = X This series is actually the most profound counter example for unjustified arithmetic operations with infinite series. It's one of the first things a math major learns in the theory of infinite series. It's incredible how dishonest that Numberphile video was in that regard.
Wow, did not know about the sequence 1-1+1-1… not having a sum. Though it makes sense when u consider that one cannot evaluate oscillating functions, e.g. sinx or cosx, as they go to infinity.
only near the end of this video when i'd gotten long lost in the proof did my brain suddenly pull out the much shorter summary of all of it from my college calculus class: "the method that gives -1/12 is for predicting an output at the very end of a function, but not every function you find a limit for calculates a cumulative sum of things" (edit: or in this case one very specific sum of things versus others)
The beauty of math is that "alternate facts" are not possible. When someone misunderstands, their misunderstanding can always be analyzed and exposed unambiguously, and forced into regeneracy.
You mean 'alternative facts' which means 'replacement or substitute facts' 'Alternate facts' is a command to contantly swap 2 facts, which is a totally different meaning.
@@SteveNicholasLondon No, that's the verb which is pronounced /ˈɔl.tɚ.neɪt/ (or some analogous pronunciation). But the spelling "alternate" also represents an adjective, pronounced /ˈɔl.tɚ.nət/ (or some analogous pronunciation), which in some English vernaculars is synonymous with "alternative". So Sara did, in fact, mean "alternate facts", with the intended meaning the same as what you would write/say as "alternative facts".
@@Veeno00 Only in America. Why not use alternative and not alternate if alternative is what is meant? It's this American degradation of the English language that typically leads to American pilots shooting their own planes out of the sky and landing in rivers. Why not use the correct, unambiguous words?
@@SteveNicholasLondon Oh, you're one of those people. 🙄 In retrospect, I should have guessed that you weren't merely ignorant but just intentionally wrong. The modern English word "deer" comes from the Old English word "dēor", which meant generally "wild animal", not specifically one particular species. The Old English word which carries the meaning of the modern English "deer" is "heorot". If you want to stick to your principle, stop calling heorotas "deer". Why not use "heorot" instead of "deer" if "heorot" is what is meant? Why not use the correct, unambiguous words? Language change is not degradation. If it were, then English itself should already be rolled back to Proto-Germanic, or perhaps even Proto-Indo-European, or maybe even further back? The way you phrase it, one would think that British English is somehow a pristine language devoid of ambiguity, which is most certainly not the case - for example, in every single variety of English, the -ing forms of verbs can often ambiguously represent either an action noun or an adjective ("flying aeroplanes" - is this what pilots do, or is this a collection of aeroplanes that are currently flying?). In any case, the ambiguities already present in English are certainly worse than the potential ambiguity of the modified meaning of "alternate", which can easily be disambiguated from context. To answer your question, the reason to use the word "alternate" is because "alternate" IS what is meant - the word is being used appropriately according to its meaning. By pretending that the only valid meaning of this (or any other) word is prescribed by your vernacular of English, all you accomplish is that you make it seem like you just don't know English well enough.
Unlike the previous times I watched this video, I managed to understand convergent vs divergent series, and basically understood everything up to the Zeta function. Now I'm watching just trying to understand xD
This is the math equivalent of a diss track.
Math Battle
😂😂
loooooooooool
Universal Kombat dont you mean -1/12 more important things
Yeah but the only misconception he got is that value = sum
Which is not the case.
Edit: To be fair, the numberphile video explained it horribly wrong if I remember correctly. They made an updated video called "why - 1/12 is a gold nugged" that one's much better in explaining.
@Multorum Unum 😐
Never I thought I would see the day that a maths channel gets exposed by another maths channel
@Mika Hamari Could you somehow explain it to me? I am a high school student and my basic logic skills say that it is impossible to reach a negative result with positive additions. (Also english isn't my native language, so excuse some grammar or vocabulary mistakes).
@Mika Hamari So, is there a fault on the calculations?
yes they had a contradiction . the series doesn't converges .but they assumed it does converges and they used the properties of convergent series to find -1/12 .which is impossible since we are summing a positive integers . and the correct answer is that the sum approches infinity when n goes larger and larger .but what is more interesting is some how -1/12 is related to the series and it has applications in string theory and quantum mechanics even though it came from wrong assumption
Mika Hamari
You can disprove convergence of all of those with all basic tests like D’alambert, Cauchy, Integral test and Leibniz for the +/- series, which are tools people learn on the 1st year of technical college.
Really scary how few people talked about how flawed the numberphile video was
@@lupsik1 I think the big thing is that the majority of people are divided into two categories: People that have seen this all before in math classes but forgot some of the specifics and caveats, and people who haven't and trust professional mathematicians more than their own intuition. The latter group are the ones that would have been confused and bugging all the other math channels to explain it or something, which is what caused any of this.
In reality, the numberphile video isn't "debunked", just properly contextualized and constrained. The issue with people bothering other math channels about the confusion is really the full extent of any damage that could have been done, at least that anybody should care about. If you're taking stuff from a youtube video and using it as the sole justification for anything you do on any math exam or really anything ever, then you have a bigger problem.
Another way to put this is this: the sum of all positive integers equals -1/12, for very specific definitions of the words "sum", "positive", "integers", and "equals".
Mainly sum and equals but yeah
Or just use lim x-> 0 x+1 bc 0+1 = 1 the series is divergent.
Also, 1/12
No, only sum.
Very much agree with this one, context is everything
I could swear, when I took number theory, one of the first homework problems was proving that the sum of two natural numbers is another natural number.
how did that go?
Two, yes. Finite, yes. Infinite? No.
@@praharmitra if 1+2 is natural, then the result, 3+4 must also be natural. It'll always be natural even when you do it infinite times.
@@praharmitra 1. Every partial sum is, by recursion, the sum of two natural numbers, and hence must be a natural number.
2. The set of all partial sums is countably infinite.
@@l.w.paradis2108 I don't understand what your point is. Rational numbers are countably infinite. The infinite sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... is a sequence of rational numbers and each element of this sequence is a rational number. Yet, the limit of this sequence is pi which is not a rational number. Same goes for the sequence 1, 1+1/2^2, 1+1/2^2+1/3^2, 1+1/2^2+1/3^2+1/4^2,... where every element is a rational number but the limit is not.
Things are heating up in the Math community of TH-cam.
Things about to get lukewarm up in this piece
Waiting for Numberphile's response.
keemstar and scarce will be all over this in no time.
I'm waiting for the disstrack
The maths drama is the best drama. These guys don't mess around.
Watch out for the diss equations - they can be savage.
"For every difficult problem there is a solution that is simple, easily understood, and wrong." H L Mencken
This sounds relevant only when you don't know who the author of the quote is.
Minakami Yuki What’s wrong with Mencken?
The original solution is also simple and easily understood by mathematicians of this era. Does that mean that even the original solution is wrong?
@@sottallu It asserts such "solutions" exist but makes to claim as to which "solutions" those are. It's merely a warning not to be fooled by simplicity.
Kinda like the opposite of Occam's razor
Numberphile is like the fun uncle. Whereas Mathologer is the Dad who smacks you on the head and says "get real son"
i^2
@@MrOllitheOne = -1
shit just became real
hey i, get real!
i : (grabs friend)
In a matter of fact, Mathologer told us to quit being real and start seeing imaginary! It's Numberphile who tried to project the power of complex and imaginary to the simplicity of real, hereby resulting in nonsense.
The -1/12 thing always seemed more like a party trick than a genuine maths solution.
But it is still a real solution and an important one.
@@JohnSmith-gu6hf no.
@@JusticeBackstrom Numberphiles did another video on this recently that is worth the view.
@@JohnSmith-gu6hf I've seen like 5 of their videos on this. It's still a party trick because thet's not how math works.
@@JusticeBackstrom No one is saying that the sum of the natural numbers is -1/12. That's just clickbait. It is obviously a divergent series with no real properties. But the Ramanujan Summation is used to apply a mathematically useful summation to a divergent infinite sum. It does find its way into things like String Theory.
Forget Logan Paul and Shane Dawson, numberphile vs mathologer is the real youtube drama of 2018
There is no drama just mistakes
Don't forget #shitholegate lmao
Or rather, don't forget to forget it
Numberphile just made the mistakes of picking Physics professors instead of real mathematicians to present some of their videos.
The interesting thing about it is that physicists often really don't understand the deep subtleties of the maths they apply, abuse the maths in a way that makes every mathematician cringe, and get out a result, which is exactly in-line with how nature behaves (just think of normalization in QED).
DavidSmyth666 so this is what future arguments look like
Numberphile on Schrödingers cat:
The cat is half dead, meaning it's probably in a coma.
poor cat
thats right what it is, he calculated an expected value, not a sum :-)
Lmao
No they meant the cat is alive and dead. It was in a state of quantum uncertainty. Unless observed the cat is alive and dead not half dead.
@@Alex-hj2jd Yes, we know. It's a joke.
"And this is where Numberphile takes a bow... BUT"
- 35 minutes left.
...and then the real fun stuff starts!
@@amogorkon ...and then the imaginary fun stuff starts!
@@αγρ-κ6λ lol
nice pfp
Yeah, I actually couldn't watch it. I'm ten minutes in and all he's done is slag off the numberphile video and it's been boring for a solid five minutes. I'm out.
Having rewatched this for nostalgia:) it really reminds me of early math education in primary school, where you just get told stuff with no justification and even though most of the methods you learn there are common sensical, the point of math is to connect common sense with rigorous logic. And pretending something makes sense out of the blue is a really hard thing to unlearn and i think that sets a bunch of kids up to hate maths. Which is really a sad thing.
Not much philosophizing in primary school math though... Some people just don't like math, some people just don't like poetry. Some like both.
@@misanthrophexI have a BA in creative writing/English and now as a tutor, I also teach marh
I can say with confidence that if primary school math involved more "philosophizing," the number of kids who "just don't like" it would drop significantly
@@misanthrophex Arguing for uncaused causes.
@@pugsnhogz I would strongly argue it would be the opposite. The mere seconds (if that) of attention span these kids have precludes virtually any form of philosophizing as it relates to much of anything, especially math. Putting that aside, they probably wouldn't get it anyway. These are, for the most part, people who, when presented with math word problems, freak out. I've never understood why anyone would have an issue with word problems, but then again, I've never had an issue with math. I had to study for Calculus, etc. but very little in math classes prior to that.
The reason many teachers don't explain the equation is because they themselves do not know the explanation of the equation. They just pull out the book and tell the kids to memorize the equations and methods, and this is a very boring way to learn math.
*start of video*
"This is a serious video so I'm wearing black"
*later*
Zombie + Human = 2 Zombies
You forgot; "Und now we discuss Supersum" and switches into Black Superman shirt.
One does not simply change t-shirt 4 times in a video and gets away with it...
oh wait. He did.
@- RedBlazerFlame - The Zombie is like an Extension of the normal world: Your mathematical rules don't work here, human! 😈
Or you could say: This is the value you expect. The human is "converted" into a zombie, which actually makes sense
@- RedBlazerFlame - Other types don't have the exact same properties as numbers.
Obviously the. Total of positive numbers is not equal to a negative number. There is at least one step wrong . It should be found.
Your German accent automatically raises your math credibility by 3 points.
:)
If it was Asian, it would be +100
@@AbhijitZimare1 i don' belive you
One of my favorite mathemathians is Cantor. He was German. Too bad he died a broken man because he was bullied because of his theory about cardinality.
I thought it was Indian
You killed my party trick
It'll be fine. You can still be an illusionist.
Your part trick is still alive see from 41:15
Matt Parker's card trick, my friend :)
What kind of parties have you been going to?
Do 1=2 proof
Thanks. I never understood Numberphile's assumption that an infinite series can have a fixed value like 1/2. It seemed arbitrary to assign a value but the presenter acted like it was self evident.
Bro it was so poorly explained it seemed like they were just randomly throwing in series that would conveniently result in the desired -1/2. Laziness and math do not go hand in hand. Ever. Even on TH-cam... I was fortunate to immediately go into the numberphile comment section and see someone recommend this video.
The sum of an infinite series of numbers can be a fixed value if it is convergent (e.g. 1/2 + 1/4 + 1/8 + 1/16 + ... = 1) as the video explains
@@candylover6419 search for "sum of convergent series"
Arguably it is assumable for some cases, because it is *true* for some cases - convergent series, as another reply states. But something does have to be a convergent series for things only true about convergent series to be true about it, so you have to at least have an intuition for whether a series will converge if you don't know for sure - and while my own test isn't 100% accurate, it DEFINITELY rules out series whose terms *increase rather than decrease*. My point being I agree that here was not the place to act like that was a given.
You did this in grammar school when you divided 1 by 3 and got 0.3333 . . . and so on to infinity. This means 3/10 + 3/100 + 3/1000 + 3/10,000 + . . . + 3/10^n +
3/10^(n +1) . . . for all *_N_*
I can't believe I am just now finding this video. The -1/12 thing has been confounding me for years. Well explained, thank you.
Same here, never made sense to me why all of the POSITIVE, INTEGERS sum to a NEGATIVE, FRACTION. Always seemed completely backwards, and +infinity makes far more sense
Same here!
@@rygerety8384 (1-1+1-1...)=1 or 0 now 2(1-1+1-1...)=2 or 0 so it is undefined.It could be 0 or another number because it is an infinite structure of conditions.You can say an infinite number is not a number.We calculate base on renormalized numbers.
Infinity is not real in real life maybe,because if the world is real so it must be a limited structure of numbers,an well defined number that represents for physics laws.
Zeno had said,time or motion is not real and you can't prove he wrong,no mathematics or physics solution can prove the cause and effect work in such a infinite manner.
same here
That Numberphile video was nothing short of vicious. I literally hate them for doing that.
Reminds me of the first time i learned about the dirac delta function in physics. I was basically told "there's some complicated math that proves this is correct but it works and that's all we really care about."
Well in the case of the Dirac delta, they are at least not giving wrong arguments why it works, do they?
Btw: the foundations of distribution theory are really nice imo, worth checking out.
Not satisfying at all
I remember loving Laplace Transformation until I found the Dirac Delta function felt like a brick wall.
Physicists being physicists
@@keineangabe8993 And at least they don’t try to change the definitions; e.g., try to pass off Ramanujan-summation as standard summation 😅.
Y'all so focused on James vs Tati vs Jeffrey while this right here is some high quality tea
Thats a quality evaluation, Fonn the Human
Quali-tea
@@alexwang982 Shh.... you are not welcome here. You are not # e^(pi•i) after all.
Omg sisterrrrrr
Jason -e^(pi•i)
I have a lot of respect for Eddie Woo who also did the -1/12 proof. I knew there was something wrong with his strategy, and now I know exactly what it is. Thank you.
I just find it a bit dishonest (or very sloppy) they do not specify when the "super sum" (which is called I think Cesaro Summation), which assigns values to some infinite sums that are not necessarily convergent in the usual sense. The term "summation" needs also a big asterisk, since it's not the conventional sum you learn in primary school. In fact it's a swindle... the "Eilenberg-Mazur swindle", hehe
I don't think you did.
@@utkarshsaini5650, not even Ramanujan, it was Euler who first proved it, in the 1700s. This math has been around for years and there are multiple branches of physics-based around it, so if this video was accurate, which it's not, it would be one of the largest revelations for complex physics in the past 100 years
mathologer is great. as he points out, the shift in S2 is the culprit. if you did 3S2 where the last line got shifted back to the left, you get S2=-1/4, an S=1/12; also if you shift the 2nd line in 2S2 to the right twice instead of once, you get 2S2=-2S2-1, which also makes S2=-1/4
To be fair, he never said that this result was true, at last with the standard definition of a sum. He just redemonstrate the result to make people think about the mathematical logic, never saying if it's true or not
Everybody gangsta till there’s math TH-camr drama.
Numberphile (Brits): It’s -1/12th
Mathologer (Germans): Halt mein Bier
*-1/12th
It's '' halt mein Bier''*
MattixHQ Sorry guys 😂 you get the point...
MattixHQ wait but halt=stop right? Halte=hold? Or am I just retarded please tell me...
@@kristoferkoessel4354 Halte would be correct too, but it is more formal, which doesn't make much sense in this context. And Halt also means stop. In English there is a similar relationship of words. If somebody tells you to put something on hold you will probably stop doing something. Or if you are supposed to hold a door open for someone you also stop the door from moving. So Halte makes sense and the person you are talking to will understand you, so it is not a real issue.
That rule also does not only apply to Halte. The e is often dropped from the verb, if you are telling somebody to do something, I can't even think of a word right now where it usually isn't dropped
I was watching 8 mile ending rap battles and this came up
Not disappointed this is a very mathematical diss track
Did you end up finishing 8 miles, or was that too much of a diss-track-tion? Alright, I'll go hide...
@@XavierDesroches
This is math war, very brutal war
Screw nitwit 8 mile crap... this is real rhyme and reason not just random rhyming words by a dumb rapper looking for a pissing contest.
Crab Synth whoosh
man, we really need new video for this "Does -1/12 Protect Us From Infinity? - Numberphile"
"Kids in primary school should be able to follow it!"
He should meet my coworkers...
what is your line of work tho?
@@A_Box Physicist, sadly
;'(
Jesse Kucharek he should meet me.
Emphasis on "should."
@@jessers1712 Remember to blink twice.
Yooooo Mathologer throwing the shade at Numberphile... This calls for a math off!!!
I think they would prefer a maths off.
Geez that would be a close call, depending who from Numberphile would fight Mathologer.
I've heard "math duels" were the main income source of mathematicians from few centuries ago.
*sharpens division symbols*
Now if we can get the Vatican in on this fight we'll have the scene set for some epic Math Drama!
Numberphile: 1+2+3...=-1/12
Mathologer: Impressive, every word in that sentence was wrong.
Ohhhhhh yesssss, Star Wars references ^_^
Mr Banana808 What is wrong with you
Mr Banana808 Are you an actual banana?
Francesco Santi
his pharmaceutical clock has dilated.
So clearly what you wrote is all non-sense, but damn was it funny to read anyway. My favourite ones:
"All scientists think light speed is c in the vacuum, they all wrong."
Gee, I wonder what the light speed in vacuum is then... and what letter should we use to represent that value?
"Iss is fake, AC systems cannot work in vacuum space"
No, Iss is fake because there is no sound in space, so their alarm clocks wouldn't function properly. Get your facts straight.
"If heat can radiate into space, [...], the whole universe will be at the same temperature, thermal equilibrium."
*long stare* ... sure ... it's called heat death...
39:20 Also; even Ramanujan, for all the formal education he lacked, didn’t call the identity: ”Sum”, in his personal notes. He used the notation: ”c”, for: ”Constant”.
Kinda po-tei-to, po-tah-to. But, yeah, was a careful move.
@@samueldeandrade8535 I agree. It *_IS_* a kind of a small thing. But a lot of people just want to misunderstand others, and will take any excuse to do so, however minor. That was a careful and smart move, to disarm such people.
Oh god, mathematical hell is gotta be like 10 times worse than regular hell.
-1/12 time worse :)
😂😂😂😂😂😂😂
All you do is math problems there...chilling
I would have said pi time worse.
it is the analytical extension of regular hell
1+2+3+...=-1/12 is a Parker sum.
Ben McDaniel And that is?
A funny joke: th-cam.com/video/aOT_bG-vWyg/w-d-xo.html
When something in math isn't quite right, you name it after Matt Parker: th-cam.com/video/aOT_bG-vWyg/w-d-xo.html
Ben McDaniel Ah, that guy. I recognize him. Thanks a lot.
I LOLed :D
Mathematical équivalent of a diss track
The best complex logics/math film I have ever seen. By “complex” I mean “consisting of many, sometimes, non-trivial elements”. If I confess I am awarded Best University Lecturer for many years, it is only to pay tribute to the quality of this film - to keep things so ordered and clear is SIMPLY AMAZING! I do appreciate the apologies for not explaining why complex numbers needed to be introduced (but no fully explained) when analytical functions were being talked about. It gives a lot of security to a lay listener that all vital things were introduced even if no all were fully developed. Yes, the content still can be completely wrong (I am not an expert to judge) but it is certainly “CONSISTENT and COMPLETE” - in contrast to the film it was commenting. The detailed and well paced debate with the statements of Numberphile content were excellent. Well, it was really impressive. I do not subscribe to any channels and social media but believe me, I will be watching you regularly!!! Well done (you know it 😊).
Absolutely agree with you, I am a professional physicist so I can judge this video with some degree of expertise. It is absolutely brilliant. I was wondering how he would justify analytic continuation.... he succeeds even for a high school level educated person in my view. I am still dazed by the level of pedagogical expertise.
I have a slight suspicion who You are, and If I am correct - we might have passed eachother a few times on Madalinskiego. My late father spoke very highly of You. Odd, getting teary eyed under math video, of all things..
With the current level of growing mistrust of science, I am eternally grateful for those smarter than me being on guard for falsehoods. I understand the desire to simplify complex subjects but this is unacceptable, not because it's a mistake -as these happen to best of us, but because it seems to be almost consciously feeding into the "stupid scientists, power to the simple minds, they are hiding truths from you" type of the political climate and I viscerally hate anything that creates artificial divides between people, some of whom perhaps could be lured into the dark side of learning and reason still.
Thank You, Mathologer.
Yes but he still assumes induction is valid forever and it isn't . The universe will stop you at a large number. You can't count forever. It is impossible. Physics will stop you from adding "one" to some large number and that will be the biggest number possible. You can't escape the universe.
Z -> Q loses single representation,
Q -> R loses countability of the set,
R -> C loses the order of numbers,
C -> H loses commutativity of multiplication,
H -> O loses associativity of multiplication.
EDIT: s/looses/loses/g
Cool :)
Must admit i had to look up octonions, but had enough knowledge to do the rest!
Why stopping there? We also have the Sedenions. ;)
O -> S looses alternativety of multiplication.
What do we loose going from Reals to Surreals? (Honest question. Those exist.)
Heyo, cool!
26:14 - "now let's play a game."
Me: sweet I love games
*Shows a graph*
Me: is this some kind of German game that I'm not structured/organized enough to understand?
To some people (like me) gragh (maths) is a game
@@irongolem5539 and you're losing
@@nolann2382 You are always losing a game of graphs
I remember explaining how 1+2+3+... diverges in the comment section and people responded that I'm wrong since I'm not a university professor. So thank you very much for this video! Math is about truth, not educational authority.
But... they are! en.wikipedia.org/wiki/Indiana_Pi_Bill (end sarcasm)
That was a sad day
"I remember explaining how 1+2+3+... diverges in the comment section "
It does diverge. Everybody agrees that it diverges. The question of what it "equals" is conceptually separate and requires agreeing beforehand on what the word "equal" means. It's not at all true that the only possible meaning of "equal" for an infinite series is that of the limit of the partial sums. That is a choice, one which makes sense in many circumstances, but sometimes you may want a different one.
Vacuum Diagrams yes but then one has to make it very clear what equal means in a certain context, especially when the large amount of viewers might not be math students
I'm pretty sure Appealing to Authority is a logical fallacy. So, I wonder why people use it...
"yes but then one has to make it very clear what equal means in a certain context"
Indeed, but this applies to _convergent_ sums just in the same way. When I say that 2 + 2 = 4, I mean something quite different than when I say that 1 + 1/2 + 1/4 + 1/8 + ... = 2. The former is the result of a single addition, while the latter is a statement about convergence and limits. It's a nonstandard use of the equal sign, just like the use in 1 + 2 + 3 + 4 + ... = -1/12 is nonstandard.
This was like one of the first things they covered in undergrad, the series that alternates positive and negative 1 they told us to think about as a digital switch, it's either on (1) or it's off (0) and it can always be made to be in one of those states by adding an extra term but it can never behave like an analogue switch and be in a state that is some measure of two values it takes. Really helped me to understand why its sum cannot be assigned a value. This video made more clear outside of thay intuition.
That is a very helpful analogy!
Mathematics equivalent of a diss video
haha yes! Mathologer is basically Eminem
Mathematicians reuse the same symbols with different meanings all of the time. It is much easier to say, here is this idea I am working with, and here is a nice symbol for it, than to come up with a brand new symbol for everything.
Numberphile's problem was not putting a disclaimer up saying "Here is the standard meaning for this notation, and here is another idea that uses the same notation, but isn't the same thing." They should have made the distinction clear, instead of not mentioning it.
Obviously it's not always a great honour to be corrected in science. Some of the most renowned scientists of all time, including Newton, Kelvin, Edison were all challenged after having reached fame; their ideas about the universe and the contents of papers they had published were corrected, but they refused to accept and acknowledge these discoveries, many of which were ignored for a century before finally resurfacing providing solutions in other sciences. A great deal of this was the fact that basically all people are stubborn and will give in to power and fortune. You can think of it as great scientists being corrupted, or there being little to no difference in science emotionally from other endeavours. If you can acknowledge that you were indeed mistaken in your assumptions, then standing corrected may be a personal honour. But that actually has very little to do with being wrong. Most researchers for instance do not care about being right or wrong at all: providing an argument in the publishing of a discovery is just a formality. Being recognised for posing the right question and having the idea that sparked the study is a much greater honour. And when then someone comes afterwards and points out a mistake in a study you were the mind behind, you are quite simply flattered. Feeling honoured for being dissed in science is the worst pseudo spiritual zen bullshit myth I have to live with. It's just a mindset overrepresented by Hollywood movies.
Math isn't a rational subject: It's a system "we" created based off axioms which are accepted as true.
(When a Contradiction occurs in Math- we either correct for the contradiction or avoid doing what caused error)
Eugene Wigner wrote a really famous paper called:
"The unreasonable effectiveness of mathematics in the natural sciences."
*If there is an infinite amount of numbers between 1 & 2 (How do you get to Two?)
*If it's Zero degrees outside and the weather man says it's going be twice as cold tomorrow as it is today.
(What's the temperature going to be tomorrow? [ 2 x 0 = ? ] ~Not Zero you need to switch the formula.
1+1=3
When a Man and a Women enter a Dark-room-
Nine-months later you have Three people...
'Math is litterally the Definition of *close enough;*
The Great Pyramid of Giza is the most accurately aligned structure on earth-
and it's still off 3/6 a degree True-North. (Rolls eyes)
Don't get me wrong- Math is extremely important:
Without Math we'd suck at 4th dimensional physics.
But there's really only one number and that number is: *EVERYTHING*
Math is an observational tool, and while yes, we agreed to 1 = one object, 2 = two objects and so on to be the case, it doesn't change the fact that there was two objects in the first place. For your points:
1. Eugene Wigner, while being a wonderful physicist bringing light and joy to people arround the globe by some of his greater projects (sarcasm, obvs), absolutely did that. And he also has several others - "Maths being shit in economics", "Maths being shit in everything" and so on (obvious hyperboly is obvious). Reading through those articles (thank you for bringing it up in the first place, was an interesting read) - I came to a conclusion, that either: A - he is not aware, why does physics need some of the cooler stuff and how mathematics and physics are connected or B - he was just a hater for the sakes of it (especially when it comes to economics one, since Eugene seems to be fairly low knowledgable in the field).
2. By defining the step of your infinity in the first place. The one you mentioned is an uncountable (1;2) infinity
3. Extendanding an example to the concept - is a logical failure on your behalf (or wherever you took the quote from). One guy saying, that it will be twice as cold tommorow, when it is 0 today - isn't really the best example of human brain functioning in the first place
4. That is not really how babies work. If you want to be tehnical - throw in all of the variables (the baby doesn't appear out of nowhere, it has energy consumption throughout the whole process). Otherwise, I will extend your example on two rocks being left alone in the dark room for 9 months - and after that a third rock would magically appear
5. Great Pyramid of Giza - is "close enough" in your statement, not the other way around
6. You wouldn't be able to write your comment in the first place without math. Or watch the video for that matter. Or use TH-cam. Assuming you'd have Internet to open TH-cam. And an internet connection in the first place - to your PC, of course, if it'd exist
7. Hey look, I used numbers to make my comment easy to read. When were you born tho? Answer me in everythings please ^^
And also, if 0 degrees outside - you are a flat earther!
“On my home planet, this symbol stands for
S U P E R S U M”
“This is not my planet, is it?”
The fallacy of the first series reminds me of the analysis of the human race that concludes the average human has one boob and one ball.
lol
Underrated comment, that's actually funny as hell, I was thinking of an analogy and this is a perfect one!
that's really just a bimodal distribution situation, not sure if it's quite applicable to the fallacy at work here. but it's funny as hell
PFFFFFTTTT dang!
The average human has 9.x fingers and 9.y toes. Averages never claim to represent a single one of the values that went into calculating them. Another good example are population BMIs (body mass indexes) being applied to individuals. It's almost always wrong.
As someone who holds a PhD in analytic number theory, I appreciate the exposition here. The ideas are clearly presented and give a relatively complete explanation of the phenomenon occurring with -1/12. The explanation of analytic continuation was particularly nice, as this is a concept that's definitely tricky to pin down if you want to get into the technicalities around it. Glad to see some quality mathematics communication concerning the infamous Numberphile video.
Can I ask you something?
@@Manaschoudhary3636 sure
Given your credentials, maybe you can answer this question from a non-mathematician.
For the sequence 1/2+1/4+1/8... I had thought that, assuming the sequence is infinite, the sum would be an asymptote and not 1 because given infinite denominators you will simply get smaller and smaller fractions.
What am I missing?
@@louzander This is just a matter of understanding vocabulary. When we speak about infinite sums, what we really mean is the limit (in the sense of calculus) of partial sums (that is, sums of finitely many terms). To say "the infinite sum equals x" is really to make a statement about limits. That is, the statement "the infinite sum equals x" is literally DEFINED TO MEAN that the sequence of partial sums (1/2, 1/2+1/4, 1/2+1/4+1/8, etc.) gets closer and closer to x.
To use your language, "the sum being an asymptote" is the DEFINITION of equality in this scenario. If we're being more precise, we should say that "the infinite sum converges to x" rather than that it "equals" x. This is, of course, just a matter of semantics, and once one understands limits, an infinite sum "equalling" a number can be interpreted in a rigorous, precise, and unambiguous way.
Hope that helps!
@@joshuastucky that was extremely helpful and very interesting! Thank you!
Excellent video. Unlike some, I don't think you were being harsh. When millions have viewed flawed information, a clear refutation can be seen as a public service.
That's the way I look at it :)
Agreed. Can't fix a problem if you won't admit there is one.
Thanks a lot for your effort. I saw that numberphile video years ago when I began my studies and it confused me a lot because we've all been told you cannot do anything with divergent series. This video finally cleared things up for me.
th-cam.com/video/0Oazb7IWzbA/w-d-xo.html
John Deacon - that is a nicely-worded response, but it is, after all, written from the point of view of a physicist. I understand the points he makes, and he's quite right about the usefulness of analytic continuation - but that isn't the point. The point is that the audience of the video may have been given the impression that such things can be stated without context, as being strictly true. To me, it is clear that summing the natural numbers cannot possibly result in -1/12, UNLESS you state clearly that your context is one of analytic continuation. This is a subtlety unlikely to be understood by a general audience, and the complaint was that this was not made clear. I think this was a fair complaint.
I differ from you about the style of Mathologer's video too - I don't think it was unpleasant. But of course, that is subjective and therefore not open to debate.
I keep coming back to this video every so often, and each time I am utterly amazed at how intuitive Burkard makes these complex topics. I appreciate that he is so careful with his terminology, and of course his graphics are awesome. It was so cool to have Burkard run down exactly the problems in the Numberphile calculation and how to "fix" them...when he did the transition from the Numberphile S-S_2 to zeta-eta I was blown away; in an instant, he transformed a simple, familiar, but false expression into a deep, rigorous, and true statement, highlighting the "simplicity" and "familiarity" behind things as complicated as power series in the complex plane. Literally one of the best math videos ever made.
"If you've made it this far you know..." I stopped knowing at the 10 minute mark
Well, I found it releatively easy to follow along.... then again... I have a math degree ;P
@@constantly-confused5736 I'm 14 and I understood it
Me too and i actually like the video and seen until the end and i just completed high school and some shit calculus and algebra from computer science.. Many time i wish i choosed math or phisics instead of cs
@@alexandrubragari1537 rip bro
I stopped at the 10 minute mark too. Cause it felt he was done explaining the wrongness.
"What else is there? An extra 30 minutes! What the hell... I don't remember signing up for this."
Wonderful stuff! The second half was way above my mathematical pay-grade, but I still understand much more than I did before. Great work! I had been duped by the -1/12 stuff.
Dupe isn’t the right word; this isn’t even necessarily a real rebuttal of the -1/12 sum. The result is controversial and this is a good argument against the result (which is counterintuitive which in itself isn’t meaningful). The whole thing, the controversy and the result, are more indicative of the clumsiness, errors and even perhaps uknowability of logic, math and the implicative language of trying to state it. The terms are very slippery and we get strange results in our minds when we try to manage it all. The argument made here is one, a robust and hardy one but it is no more ‘correct’ than other views.
you haven't been duped. -1/12 is a meaningful value assigned to an infinite series. this "sum" is not an actual sum in the traditional sense, but it was derived using real methods. in the context of a youtube video teaching about infinite series, numberphile was correct. in the context of a mathematics course that requires rigor and proper definitions, it was incomplete.
we know that -1/12 works because it can be used in real world applications of physics.
@@LeNoLi. This last comment is what really interests me. What does "-1/12 works" or its utility in real world physics tell us about mathematical truth? I have in mind the use of infinitesimals, in Newtonian calculus - i.e., before the introduction of a "limit". These "ghosts of departed quantities" (as George Berkeley memorably called them) "worked" in physics, despite being, at core, inconsistent. This suggests to me that having real world applications in physics really doesn't necessarily tell us much.
The irony. You are being duped by thinking that we were duped. Terrence Tao just should that the -1/12 is valid and their is another numberfile vid on it.
I think you misunderstand. By "duped" I mean that I misunderstood something about the proof. I in no way intended to suggest that it is not "valid", in its own terms, but simply that I misunderstood the terms of the proof.@@sloaiza81
In (slight) defense of Numberphile, they did follow up with a much more informative discussion with Prof Edward Frenkel. Some aknowledgement of the flaws in that video that Mathologer is complaining about; the first thing we hear is Frenkel saying with some dismay "Oh... it's /you/ who made that video." He chuckles and shakes his head. Then what follows is some explanation of assignment rather than summing. They are very explicit: "[-1/12] is certainly not the result of summation of these numbers [1+2+3....]. It is something else, but what is it?" th-cam.com/video/0Oazb7IWzbA/w-d-xo.html
Yes, I actually like that video with Edward Frenkel, he is a very good mathematician and really knows what he is talking about :)
Lesson learned: Don't ask a physicist to explain number theory.
Nillie I still think they were meming hard and were just joking in that video. ^^
There is also the 'extra footage' video on Numberphile 2 which goes into greater depth of the math on the original- th-cam.com/video/E-d9mgo8FGk/w-d-xo.html
In this video (Frenkel's @ 10:19), Brady asks "My understanding of Math is it's very rigid and rigorous and it's never arbitrary, how can you throw away the dirt and keep the gold?". This question is the reason why I hated the 1+2+3...= -1/12 from the very first moment. Because that kind of destroys my view of Math (as the only concrete, unambiguous and objectively true tool we have).
Mathologer if you're going to make a discussion video about this subject, PLEASE address this question.
**stares at the length of the video**
**stares at the fully loaded coffee machine**
**unpants**
**presses play**
unpants?
ok, you do you ;P
Panting = breathing quickly.
unpanting = not breathing quickly.
So, "he unpants" could be interpreted as "he calms down and no longer pants".
www.dictionary.com/browse/panting
Nah I just fap while I drink coffee and think about math. XD
This is hardcore math.
Do you actually think anybody on the internet is wearing pants?
Thank you for explaining analytic continuation in an actually good way. I've seen so many math TH-camrs talk about it and every time it boils down to "the most natural extension of a specific function," which, I imagine, would leave many questions in the audience's head.
I can see myself understand this when I didn't already know what analytic continuation or any kind of analysis deals with. Really shows why derivatives shape a function which is not traditionally defined.
Great job!
3blue1brown defines it pretty well. It's most natural because the derivative is constant and it preserves angles.
The exact definition of the analytic continuation is that the value and derivative of the function is the same as the data given at all point.
The way I always explained the "nonsensical" result of -1/12 coming from the Zeta function was this:
The original zeta function is defined as the given sum, for only Re(z)>1. The analytically continued Zeta Function takes those same values for Re(z)>1, but is _not_ defined by the sum over its whole domain. I don't know if we know the closed form of the extended Zeta, but that form would relate -1 to -1/12 - and have nothing to do with the 1+2+3... Sum.
I think Mathologer deserves no criticism for this video. I like the Numberphile guys, but in that video, they presented a very misleading argument for the "sum" of these divergent series. The first rule in any, ANY argument regarding series is: "you can make some algebraic manipulations with series ONLY IF they converge". Notice the "IF". This is very important, because, with divergent series, you'll end up with nonsensical results applying algebraic manipulation.
Let us check a stupid example. Let us suppose that I don't know if the following two series are convergent or divergent.
S1 = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6...
S2 = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6...
Now, let us take, S1-S2, which, computating term by term, we get:
S1 - S2 = (1/2 + 1/2) + (1/4+1/4) + (1/6+1/6) + ... = 1 + 1/2 + 1/3 + ... = S1
So, S1-S2 = S1, and thus, clearly, S2 = 0. Right?. WRONG. S2, as Leibniz discovered, converges to ln(2). The argument is invalid because S1 is a divergent series. So, my algebraic manipulation is invalid.
The Numberphile guys should have made that very clear in the video, saying "these algebraic rules are only valid if the series are convergent. But, we'll be playful, and let's see what strange shennanigans happen if we ignore the convergence criteria". With that disclaimer, everything would be completely fine, but they failed to do so, so they deserve criticism in that regard.
And they further state their math is valid because it "shows up in physics." Thats the part that irritated me.
@@SparelWood I think the Numberphile guys were trying to be informative regarding these "strange sums" which appear in advanced mathematics. But, of course, without any disclaimer, these identities are just nonsense.
For example, we all know that "S1 = 1+1+1+1... = infinity". In fact, that is the main definition we use to explain people what infinity is!. But, let us again ignore any rules regarding convergence.
S1 = 1+1+1+1+1+1+...
S2 = 1-1+1-1+1-1+1-...
S1 + S2 = 2+2+2+2+2+... = 2*S1
S1 = S2
So, given that we "know" that S2 = (1/2), then, S1 = (1/2). And thus, "infinity = (1/2)". So, even it is true that some process in physics in which the partial sum of a value can be considered "averaged" occurs in reality, but that is NOT an argument for justifying this kind of nonsense.
@@elasiduo108 ....... " because it shows up in physics" ...... LMAO.
what a beautiful comment, and great counterexample. well said!
This is the best counterexample ive seen
Thank you, Mathologer. This video finally explains what's going on in a rigorous and well-defined manner. I appreciate how you start from the standard definition of series summation, explain how that can be extended to "supersum" (Cesàro summation) and then go on to show the connection to the eta function. With that you actually write down the analytic continuation of the zeta function, which is really nice. For the first time I can see how to arrive at the result of -1/12 without handwaving or breaking the rules. Thank you for your high-quality videos.
"This is not mathematics. Don't use it. Otherwise, you will burn in mathematical hell."
xD
Blananas2 wow a new religion have been born is Math Religion.
Blananas2 wow a new religion have been born is Math Religion.
Mathematical Hell = Being doomed to make wrong predictions about the world
You are tortured with people using 3 for pi and x for sin(x)
OMG 314 LIKES
THANK YOU! The -1/12 meme has gone way too far.
It's not a meme, it's used widely in physics and maths
Nonsense comment. It's a perfectly valid evaluation of this series. Mathologer is an annoying pedantist.
41:20
@@madlad4206 where exactly?
To all commenters.
I'm sorry that this comment is so long and ask you to be patient.
The debate in the comment section whether Mathologer is rude/too late/ignoring other Numberphile videos on the subject is making me smile, so I'll put my two cents, too :)
Numberphile made a video about a subject which is completely counter-intuitive. So it went viral, to the point that my father, who is 50+ years old electrical engineer, completely unconcerned with mathematics other than that helps to do his job in reality and barely speaking English, and even some medical doctor I went to (knowing that I studied physics), both claimed to me that the sum of all positive numbers is -1/12 ... That doctor even stated that nowadays mathematics is incomprehensible :)
That's exactly the point which drives people like Mathologer out of their minds - claiming such counter-intuitive statements without proper disclaimers (I'm not even saying proper context, like Zeta function and analytical continuation). One guy in comments says (I'm paraphrasing) "All natural numbers can be written as a sum of 1s. So, 1+2+3+4+...=1+(1+1)+(1+1+1)+...=1+1+1+1+1... You say that 1+2+3+4=-1/12 and 1+1+1+1=-1/2. So now -1/12=-1/2 ??? " I guess that some people, uninvolved in mathematics, thought to themselves after seeing that video "And these people get paid for that ?"
Numberphile should have added only one minute, saying that:
"equals sign in these equations should be understood as "is assigned to", not "is equal to" " and
"these calculations are not intended as a proof, they merely show what answer is to be expected from more rigorous methods".
That's it. Everyone (almost) would be happy. Instead, all we heard was "astounding", "amazing" and "correct".
Someone says (I'm paraphrasing) "How dares Mathologer cite Numberphile out of context? Numberphile did two other videos on the subject, which (more or less) address the issues with the first video. Mathologer ignores that. " Mathologer is perfectly aware of this. He even links one of them ("Why -1/12 is a gold nugget") in his description. The reason is simple: view count. The first two Numberphile videos on that subject, which completely miss to point out the crucial distinction between "is equal to" and "is assigned to" have been viewed 7.7 M times combined as of 2018 July. The one which discuses the subject properly ("Why -1/12 is a gold nugget") has been viewed only 1.6 M times. The difference is those confused people inundating comment sections.
Another person says (I'm para...) " The goal of Numberphile channel is to make mathematics interesting to wider audience. Don't expect rigour there. Anyone who is wiling to get deeper understanding should follow the links and research themselves." Well, this youtuber forgot that he is commenting in ... TH-cam :) Content providers in TH-cam, especially those who want to appeal to "wider audience", should keep in mind "least action principle" - most people these days will spend the least effort to get information. Those who will research seriously, I assume, are those who already find mathematics interesting + small minority newly engaged. Most people, I guess, come there just to see "what interesting video did Numberphile upload today ?" I even suspect that many people rejected the video as nonsense, not wanting to have anything to do with divergent sums anymore, barring further research.
All in all, I don't think that Mathologer is rude or incorect, I think he is right on the money (except that cameraman. He should have kept his jokes off-record.)
lol
This is the most precise explanation I've read in this whole comment war. Well done.
seacaptain72
Thank you.
1. You fail to actually address the rudeness. There is a clear tone of condescension throughout the video, not just from the cameraman. Who is factually correct is irrelevant to whether Mathologer was rude, which he was, by standard observation of tonality and wording. Your comment rather comes off as ‘I think he's right, therefore he wasn't rude’, which is a nonsense argument.
2. Your argument is essentially that this video is to address misconceptions of people who viewed the Numberphile video and misunderstood it. Meanwhile, this video actually directly tells Numberphile they are wrong, repeatedly. For what? Not being able to control what their viewers say and do? No. You don't get to blame Numberphile for that. Your suggestions for what they should have said may have affected things… but you fail to provide a reason why they would know those suggestions would be necessary BEFORE THE VIDEO WAS MADE AND PUBLISHED. Funny; those suggestions are followed in the other videos that both you and Mathologer handwave away… almost like it doesn't matter what Numberphile does or doesn't do, they're just wrong because of what people watching them do. Either your understanding of this video's purpose is incorrect, or both your and Mathologer's understanding of responsibility is crude.
Badly Drawn Turtle
Hm, on a second thought I guess I gave Mathologer a pass to being condescending, because he is right. Ok, I can somewhat concede this point. However, that first Numberphile video was just doomed to be interpreted incorectly. I believe this was because he was asking physicists to explain it. Physicists are less concerned with nuances in mathematics, and more concerned with applications, which in this case was knowing what number can be assigned to this sum. When Numberphile came to mathematician, namely Edward Frenkel, who has seen the video, Edward immediately understood that the solution was not explaining rigour, details, zeta function and all that, but an abstract meaning of that hapless equals sign. In fact, an advanced physics textbook is shown in an original video, and there is an arrow instead of equals sign. They did not explain that crucial detail which would have made a lot of people happier.
TOP 10 ANIME FIGHTS OF ALL TIME
Anime?
The strongest attack in his arsenal:
Serious Series: Infinite Sum!!
respect
jajajajajaaj
Vegetto vs Buuhan: Mathematics Edition.
This guy was awesome in Raiders of the Lost Ark :)
I’m learning data structures and algorithms and came to this video after a teacher told me to check out numberphile’s -1/12 video. So glad I pulled that thread and landed here where you made it all make “sense”. I would have laid awake in bed for far too long trying to wrap my head the bogus numberphile solution.
I watched this like 2 years ago and it's been recommended to me again tonight at 0:30 am and by god I'll be watching it again
Confused 1+2+3+…=-1/12 comments originating from that infamous 2014 Numberphile video keep flooding the comment sections of my and other math TH-camrs videos. And so I think it’s time to have another serious go at setting the record straight. In this video I’ll do just that by having a really close look at the bizarre calculation at the center of the Numberphile video and then stating clearly what is wrong with it, how to fix it, and how to reconnect it to the genuine math that the Numberphile professors had in mind originally.
Lots of nice maths to look forward to: non-standard summation methods for divergent series, the eta function a very well-behaved sister of the zeta function, the gist of analytic continuation in simple words, some more of Euler’s mathemagical tricks, etc.
This is my second attempt at doing this topic justice. This video is partly in response to feedback that I got on my first video. What a lot of you were interested in were more details about the analytic continuation business and the strange Numberphile/Ramanujan calculations. Responding to these requests, in this video I am taking a very different approach from the first video and really go all out and don't hold back in any respect. The result is a video that is a crazy 41.44 (almost 42 :) minutes long.
Thanks for that. I'm not realy mathematicly educated, but i enjoy watching your videos and thank you for clearing that myth out which i myself believed
Mathologer what happened to the plain black shirt at start 😁
Sorry to be a dick but 41.44 minutes /= 41 minutes 44 seconds
Didn't you mean a "series go" :)
Thanks for your video. I regularly watch both numberphile and your videos and love them both. Not being a mathematician but being in science I really appreciate them. Likewise I know that in science arrogance spurs easily and often egos simple don't match even where facts have the reason. I was a bit surprised by the aggressive nature of your video, I just hope you pointed out their mistake directly to numberphile guys before doing this video. I reckon that may have been the case and they didn't took it well and that led to the tone of this video.
Teacher: “What’s 1+2+3... forever?”
Me: “Infinity”
Teacher: “Wrong. It’s -1/12”
Me: *_”DID I STUTTER.”_*
MY AUNT: But, the way that I calculated it, you owe me money for my purchasing all of this.
*Everyone stares at us.*
ME: Please excuse my dear Aunt Sally.
you may me 10000000000000000000000000000000000000000000000000000000000000000 dolllars. i tell u to keep giving me money and i will pay u back. soon enough i keep getting money from u infinitely and i say it can be represented by 1 + 2 + 3..... and he is like yea whtver give me back my money. and i say nope, i owe u -1/12 of a dollar, which means u owe me 1/12 of a dollar GG (ps: ty for all the money hehe
@@grantorino2325 :O
This video proves you wrong too.
@@roseCatcher_How so?
God bless finally, I was sick and tired of hearing people try to tell me that adding infinitely positive numbers equals a negative number.
There was a video made by Numberphile called, "Why -1/12 is a gold nugget", where the professor, Edward Frankel, made it clear on what the identity "1+2+3+...=-1/12" really meant.
Yes, a very nice video :)
Would be fair to mention that video as well. Otherwise the term 'misled' could be partially true for your video. It's clear math videos like to be 0 or 1 :) Great video, my issue is just a small footnote.
I link to it and lost of other relevant thing in the description. There is only so much you can say in a video :)
Yes, that's also the case with Numberphile of course, but their videos are shorter so they cut (too many) corners. I just like the 'gold nugget' metaphor and wanted your opinion. Maybe you have another (better) metaphor. But like I said before, it's only a footnote in an otherwise very well made video, the effort really shows!
Isn't that the video that compared the infinity-ness of the series as a bunch of dirt that can be swept away, leaving a gold nugget behind. I found that almost as troubling as the first. It was like an explanation why it's true without explaining how it's true.
I graduated college a while ago, haven’t done any math in a while, but I really love this video. You are so concise and clear with your explanations. You make me miss math class lol
Right on! I love math too. It is great, like music. So concise so clear. Nothing greater.
This comment summarizes my feelings also.
Thank you for discussing this. I have been in endless discussions trying to point out to others exactly what you have stated. It is easy to get caught up in all sorts of paradoxes when applying rules for finite math to infinite series. One must be careful when applying algebraic rules and arithmetic in these cases.
GH
The thing about maths is that mathematians always care about and give the general case
whereas physicists in physics always cares about and give the special case
And yes Richard Feynman said something like this
You are an excellent teacher and explain complex concepts in a readily understandable way. ❤️
Video is pretty good, if long, but I was not a fan of Grumpy Background Voice, who didn't seem to be making any actual contribution to the content, just kind of dissing half-heartedly.
couldn't agree more about the pot shots coming from the Henchman
your right, thats not smart, but I understand his point. It is like when Sheldon tries to trap his rage about schrödingers cat.
Couldn't agree more, he should've been dissing with all his heart.
This is the Mathologer's video, he doesn't have a problem with it, and the videographer actually does contribute.
He is contributing, representing you the ignorant public.
Oh my god this video is amazing thank you very much for making this.
Here are my answers to your challenges and some question I have at the end of this comment.
On 22:22:
Series:
1+0-1+0+1+0-1+0+1+...
Partial sums:
1, 1, 0, 0, 1, 1, 0, 0, 1, ...
Partial averages of partial sums:
1, 1, 2/3, 1/2, 3/5, 2/3, 4/7, 1/2, 5/9, ... -> 1/2
Therefore the supersum of the series is 1/2.
So I think you made a minor mistake taking the wrong example as this does not prove your point.
Here is an example which does prove your point:
Series:
1-1+0+1-1+0+1-1+0+1-1+0+...
Partial sums:
1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, ...
Partial averages of partial sums:
1, 1/2, 1/3, 1/2, 2/5, 1/3, 3/7, 3/8, 1/3, 2/5, 4/11, 1/3, ... -> 1/3
Therefore the supersum of the series is 1/3.
Therefore supersumming is not invariant under adding infinitely many zeroes.
On 23:10:
Funnily enough, every extension from N to Z to Q to R to C is mostly invented in order to add structure.
The structures added are additive inverse, multiplicative inverse, completion and roots respectively.
Some things you might consider a loss could be the following:
You lose well-orderedness, completion, countability (but regain completion) and uniqueness of roots and logarithms respectively.
On 23:25:
If 1+2+3+4+... supersums to some S, then:
0=S-2S+S=
1+2+3+4+...
...-2-4-6-...
......+1+2+...
=1+0+0+...=1.
This is obviously a contradiction.
From this we can conclude that it is impossible to define some ubersum with the three desired properties such that the series 1+2+3+4+... falls in the domain of the ubersum. From this we can conclude that the series has no supersum, because supersums have the three desired properties.
On 38:40:
Do I understand correctly that this means that if Re(z)>0 then zeta(z)=0 if, and only if, eta(z)=0? And because Re(z)>0 implies eta(z)=\sum_{n=1}^\infty((-1)^(n+1)/n^z), finding zeroes for the Riemann-zeta function just corresponds to finding z with Re(z)=1/2 such that this series is 0? (Assuming the Riemann hypothesis.) Because that is simply amazing!
Edit:
I really want to thank you for this video, because I was always very curious how it is possible that the argument given in the numberphile video just happens to give the same result as analytic continuation. I always refused to believe this is a coincidence. So thanks so much for showing why this is actually not a coincidence!
Very nice summary of most important points. Should be stickied
The partial averages are wrong. The second aveeage isn't 1, but 1/2
Manuel Ochoa (1+1)/2=1/2?
I'm really glad that I stumbled onto this. A while back I started to look at a video on the "-1/12" claim. Very quickly it started to sound like hocus-pocus to me, so I bailed out. It has been a very long time since I studied mathematics, and I didn't want to try to figure out what they were doing. Your presentation, on the other hand, is crystal clear and it makes perfect sense to me with my traditional education in mathematics, including infinite series.
In an earlier Numberphile video, Dr James Grime described S_1 as PSEUDO-convergent, which I think is the most accurate description, since it doesn't *really* converge to 1/2.
Gimme a link fam I wanna see Grime :)
Here's the relevant video: th-cam.com/video/PCu_BNNI5x4/w-d-xo.html
And here are a couple of other videos he's made on his own channel about infinite sums:
th-cam.com/video/7fGoins7q3s/w-d-xo.html
th-cam.com/video/dwYPOi-Hfg8/w-d-xo.html
Thanks!
Then the infinite sum doesn't *really* converge to -1/12... because it just doesn't converge at all. It goes to infinity.
willprogresivo I agree I'm just here for the maths drama ;)
Of course the - 1/12 meme will be the first video of the year
yeah. of F*CKIN' course.
oh no, not mohamed adibou.
is it a meme? where?
Guy in TH-cam. Facebook and among mathematicians
At least if one of us owe a numberphille fan an infinite amount of money they’s pay us 1/12 bucks back
Wow, I did not realise that video went for 40 minutes, it was all so intriguing I just kept watching without noticing how long it had been.
Glad to hear :) As you can probably tell from my final remark I was quite relieved when we finally finished filming this one. 40 minutes is a long time when you try to get every word just right :)
Mathologer, Thank you for your time, patience, and hard work for all your videos.
With the new numberphile videos, I think this topic needs an update. :D
which new vid
@@ArnavTHR the one about -1/12 protecting us from infinity.
2i/24, open your mind, open your mind. You live in a hologram. All who believe in infinite series are duped by reps. You know... Tiles. Reps-tiles.
More data after contact. Cant share. ReptileAI deletes.
Dang, already removed even the thing before that. Lets try it bitbybit.
"Do not use it, or you will burn in mathematical hell!"
StarlightVisual 200th like
it is used in string theory
Major Homer - The string theory is a bunch of nosense
NICK
.....so says someone who can't spell
He switched his t-shirt while he was talking, thats what I call a mathemagician (5:31)
Know where most magicians are born? Magichigan.
@@RationallySkeptical
*M A G I C H E L L E O B A M A*
M A G A (?)
He's wearing four different T-shirts in the video.
@@RationallySkeptical Unibomber?
Very very nice. I liked the approach of telling you how to make the arguments rigorous and giving enough detail to see how that might be possible.
The Numberphile video in question seems to violate the principle, "Make it as simple as you can, but no simpler." Simplicity is a noble goal, and I laud those who try to make complex ideas understandable to a wider audience, but simplicity has boundaries beyond which it becomes simplistic or simply wrong.
Alot of work and thought here into presenting teaching steps. What an enjoyable video. I learned so much here.
it just bugs me at a simple level because a divergent series does not converge to an answer. glad to see i'm not crazy.
People taking this video as offensive have little respect for mathematics. In the mathematical community proofs must be truths not follower fights in terms of what channel i like better.
The way is presented may get some angry but the proof seems to be correctly developed
Welcome to the snowflake generation. Where the truth doesn't matter anymore, only if you "hurt people's feelings" (TM)
the problem is not his proof, but something no serious scientist would do: quoting parts of someone else work without considering the other half of their work. Numberphile themselves added two more videos to their introductory video which went viral. In these videos (esp. "why -1/12 is a gold nugget") they explain in more detail how -1/12 actually differs from a convergent sum and why it is still meaningful. What Mythologer does here is quoting and attacking (yes attacking. The headline of this video and the way it is presented is sensationalist and honestly a bit disappointing, since it is in general good content) part of someones work, ignoring other parts completely. Especially if the part of work you quote is a video made to introduce non math-PhD people on the internet to interesting and "mindblowing" concepts in mathmatics.
But it's not the right answer. The correct answer is that the infinite series 1+2+3+4+... is divergent. It does not converge to -1/12. This is what Mathologer has pointed out. If an infinite series diverges it diverges. Stating "it diverges" is stating the correct answer.
In a mathematical context it's not an attack nor is it sensationalist.It's only an attack if one is defending a channel or brand.
I am not basing my conclusion on intiuition but rather on regular summation. If we derive an expression for the partial sums of 1+2+3+4+... (i.e. Sn=n(n+1)/2 ) we find that the partial sums get increasingly larger as n->infinity thus the series is divergent with respect to regular summation and is a valid and correct answer. If we use zeta function regularization (i.e Reimann Zeta function) / Reimann summation we can assign values to otherwise divergent summations. Applying such techniques we can indeed correctly answer 1+2+3+4+... + = -1/12. Such results have value and meaning in Physics and I stand corrected in my assertion that it is the wrong answer. n the contect of regular summation however we find ever increasing partial sums and we conclude the series s divergent which in this latter context is correct although not particularly useful if you're a Physicist. :) Nonetheless 1+2+3+4+5+6+... is divergent is correct with respect to it's regular sum which is proven when we look at the limit of the expression for partial sums S = n(n+1)/2 as n-> infinity which is clearly divergent therefor 1+2+3+4+5+6+... is divergent. Q.E.D.
So much attention to detail in a long video. Great work
22:09: The supersum 1+0-1+0+1+0-1+.... is still 1/2. However if you insert zeros like this: 1-1+0+1-1+0+1-1+0+... then the supersum indeed will change to 1/3
Well spotted :)
I just worked through that, got ½, and naturally assumed I'd got it horribly wrong as usual. Thanks for the clarification.
Um, Alex was being sarcastic. He was asserting that adding zeroes could change their wrong answer to a different wrong answer.
Can’t you also get 1 if you say 1+(-1+1)+(-1+1)+(-1+1)...?
ohthis Shiny but the sum depends on how you add the terms. If you add 1 and then -1 etc you get a different result (if any really) which will be different if you add them in groups
He's got the perfect condescending sidekick...I love this.
Are you German? I find it a little bit annoying.
@@Doeff8 Whilst condenscending yourself. I love it.
lol
@@Doeff8 Fine to be Writing this Comment & It is not about this Video.
Yea I really really don’t like his assistant.
Finally. FINALLY. I'm no expert of course, but it was not very hard to realize that Numberphile's "proof" makes no sense, and finally someone talks about it.
The difference is really about math versus analytical science. The series has value in much the same was as the "rule of 72" has value in compound interest, in that it is useful even if it doesn't make sense on the face of it (although the rule of 72 is far easier to understand as a shorthand estimation).
Thanks for giving me something to look up.
It had been done before, and beautifully, by 3blue1brown 2016-12-09 th-cam.com/video/sD0NjbwqlYw/w-d-xo.html
Dude, it's not numberphile's proof, it's a proof that has been for many many years by so many mathematicians, like Grandi and Ramanujan and others, Numberphile did not create or invent anything, and what they delivered is correct to what was presented in the past, now weather it's correct or wrong is another story, this video right here is not the best mathematician in the world and certainly not better than Ramanujan and Grandi and others, so therefore I wouldn't take his words for granted.
Nicolás Ortíz But I bet you thought it "makes no sense" for all the wrong reasons.
You do calculations with convergent series to derive a valid formula, but that derivation would be invalid for divergent series. Then, after the derivation with convergent series, you set s = -1 which makes the initial series divergent and hence invalidates your derived formula. Therefore the rest is just pure nonsense.
a) If 1+2+3+...= -1/12, then
b) 2+4+6+... = 2*(-1/12) = -1/6, and therefore
c) 1+3+5+... = -1/12 - -1/6 = 1/12
d) Taking c-a gives us (1-1)+(3-2)+(5-3)+(7-4)+...= 1/12 -(-1/12)
e) Simplifying, we get 0+1+2+3+...= 1/6
f) Therefore -1/12 = 1/6
And if you want to, you can keep going and generate all kinds of other values for 1+2+3+...
I posted this (or something like it) on the Numberphile video.
Yes, that's the sort of contradictory calculation I was after :) Maybe try to come up with something that does not require zapping infinitely many interspersed 0s, so restrict yourself to the three basic properties that I highlighted in the video :)
OK, how's this:
a) 1+2+3+... = -1/12
b) 1-1+1-1+... = 1/2
c) a+b = -1/12 + 1/2 = 5/12
d) (1+1)+(2-1)+(3+1)+(4-1)+...= 5/12
e) Simplifying, we get, 2+1+4+3+6+5+...= 5/12, which is the same as 1+2+3+4+5+6+...
f) Therefore -1/12 = 5/12.
Or is re-ordering a problem also?
Perhaps even better:
a) 1+2+3+4+...= -1/12
b) Insert just one zero: 0+1+2+3+...= -1/12
c) Adding a+b gives 1+3+5+7+... = -1/6
d) Insert just one zero again: 0+1+3+5+... = -1/6
e) Add a+d giving 1+3+6+9+12+...= -1/4
f) Subtract that initial 1: 3+6+9+12+... = -5/4
g) Divide by 3: 1+2+3+4+...= -5/12
h) Therefore -1/12 = -5/12.
Ha! Nothing infinite done. Added two zeros, one at a time, and subtracted one once.
Adding or subtracting sequences was permitted, but adding or subtracting an individual number like one wasn't a permitted operation was it? I'm more interested in the fact that a-b implies an infinite sequence of ones sums to zero. Shifting that new sequence again and subtracting from itself easily proves that 1=0, implying that the shift operation isn't allowed for these 'supersums'.
Do you think the sequences of ones supersums to 1? Maybe you mean infinity or do I miss something?
Thanks for this great video! I think there's also another way to reason about this:
Given the infinite series S = 1 − 1 + 1 − 1 + 1… the conclusion was made (by summing it with a shifted copy of itself) that S + S = 1. However a silent assumption is made here that S is an actual number in the first place. It was assumed that S ∈ ℝ (or ℂ if you prefer) from which it follows that the expression S + S is a well-defined mathematical expression that has a meaning, from which one can conclude that S = ½ using the usual manipulations. However if S ∉ ℝ then what is S? Then the expression S + S lacks any definition of what it means and makes as much sense as the expressions "yesterday + the moon" or "the square root of yellow". Thus to complete the proof, one would have to show that symbolic manipulation on S have a meaningful definition and there exists a sequence of valid manipulations on it that lead to S = ½. That could for example be done by showing that S ∈ ℝ, but that is unfortunately not feasible. That is the missing part of the proof. And of course it's invalid to conclude that S ∈ ℝ because ½ ∈ ℝ ∧ S = ½, because that would be begging the question (a circular argument).
Yes you are right .
There are many many problems in which we assume it to be a number by itself in the beginning and solve for that real value
But you should know that "while dealing with real numbers, addition and substraction on them results in real answer" but it is not always true for multiplication and division so as far as the series given in this video fall under this law we can consider them to be equal to s and (s€R)
Ever heard of the square root of a South American abacus?
@@JohnRandomness105 The European Abacus flies faster though because the partial sums of it's constituent states are smaller, right?
@@twobob I never heard of that one before.
I get scared everytime he laughs :(
Funny, I find his laugh charming. Different strokes, and all that.
Hilarious
@@Spathephoros Seems like to some people it was
lol
Don't worry, (unless he is holding a big knife).
The only "mistake" is us assuming that what they mean by an Infinite series being "equal" to something, is that it converges to it
Been teaching cal 2 for several semesters now, and I think this is one of the best videos on youtube explaining the rules about series
You should invite the professor over at Numberphile to a discussion of the topic. You could live stream a hangout, or something. It could be interesting.
But armed with a sharpie, knife and a cleaver.
I don't know if they can overcome this beef
@@koalasquare2145 sad that both those two guys are from Australia
Not likely to happen. When he was called out in the comments/on twitter, he got incredibly defensive and wrote a whole blog post on how “actually this is totally allowed and you’re all wrong”
@@The1DistantFl4pjack who, Brady?
Awesome video! My summary: 1+2+3+... = -1/12 is wrong when we use the standard summation for the infinite series from 1st year calculus, since 1+2+3+... series diverges. However, if we use a very different Ramanujan summation method, then 1+2+3+... = -1/12 is true. The problem with the Numberphile video is that they used the standard summation method incorrectly to prove the -1/12 result. This might give millions of people a false idea that 1+2+3+... = -1/12 is true for standard summation. I think making mistakes is ok. Only those who do nothing don't make them. So good job Numberphile, keep the ball rolling! :)
It should also be said that there are standard meanings for symbols like + and =, and if somebody intends to change their meanings he is socially obliged to explain them in advance. It is much better to retain the standard meanings and use different symbols for operations or things that have nonstandard properties. For example, a right-pointing arrow can represent a transformation that does not preserve equality, as in the string-theory text exhibited on the Numberphile "ASTOUNDING" video.
Well said! And good point about the arrow, it would be more suitable here, since 1+2+3+... is not equal to -1/12 in the usual sense. But we can say that we can use so-and-so method to associate the -1/12 number with the 1+2+3+... series.
Yeah, I wish I knew a rigorous definition for "associating a number with" that supports substituting the number for.
its a interdimensional maths
I disagree about using another method to make a mathematical claim. That’s like saying, I scrambled the letters of the English language into my crypto code, made some sentences and then descrambled it and came up with the perfectly reasonable answer to your question (which is exactly wrong!!). Mathematics is a rigorous scientific method. You obey the rules. It’s that simple.
Nice to see someone do this. I randomly stumbled across someone still in university (I think an engineering program) bringing up these sums, I think as fun puzzles. I quickly put up proofs of their divergence, I might be a chemist but I was taught well enough to test a series for convergence before running off with it in my math classes (that and the sum of all natural numbers is obviously divergent). I was vaguely aware of non-standard summations such as cesaro sums and brought up that those series can be assigned summation values, but struggled to explain the nuance of the difference between being able to assign a value and the sum being that value. If only I could go back in time and have actually studied mathematics instead of science.
23:58 *Turns on Jeopardy background music*
Think, think think!
24:03 *Music fades*
😂🤣I cant get his laugh out my head 😂
I really love your videos and I deeply appreciate and admire your intellect and talent in mathematics. You always explain concepts so well. Phenomenal work.
The way he explained how eta and zeta functions are connected is really great!
So for zeta(-2) = 1 + 4 + 9 + 25 + 36 + 49 + 64 + ... = 0
we can prove easily with this knowledge.
We already know from this video that
zeta(z) = eta(z) / (1 - (2 / 2^z))
This means in our case z = -2 so the denominator becomes -7.
So we can now just calculate the eta function.
1 - 4 + 9 - 16 + 25 - 36 + 49 - ... = ?
To do this we double the value.
1 - 4 + 9 - 16 + 25 - 36 + 49 - 64 + ...
0 + 1 - 4 + 9 - 16 + 25 - 36 + 49 - ...
1 -3 + 5 -7 + 9 - 11 + 13 - 15 + ...
Since we still have no result let's double it again!
1 - 3 + 5 - 7 + 9 - 11 + 13 - 15 + ...
0 + 1 - 3 + 5 - 7 + 9 - 11 + 13 - 15 + ...
1 - 2 + 2 - 2 + 2 - 2 + 2 - 2 + ...
Now we recongize the Grandi series.
1 - 2 * (1 - 1 + 1 - 1 + ...) = 1 - 2 * 1/2 = 0
4 * eta(-2) = 0
eta(-2) = 0
zeta(-2) = eta(-2) / -7 = 0 / -7 = 0
That's it!
When the numberphile guys said "so this series alternates between 1 and 0, so the sum must be 0.5" I was like, "what, no, it doesn't work like that", but since I only have 'high school' maths and they're professors, I went along with it. I am feeling relieved and validated now that youtube has recommend me this. I'll be honest, started to struggle to follow around the zeta/eta part, but at least thanks to the first half of this vid I can rest assured the 0.5 thing was indeed nonsense
One can do all kinds of stuff with the Grandi's series, for example I can make it equal to 1 by writing
1-1+1-1+... = 1 - (1-1) - (1-1) - .. = 1 + 0 + 0 + .. = 1
and I can take it even further and make it equal to any number X by writing
1-1+1-1+... = (1-1) + (1-1) + .. = 0 + 0 + ... = (X-X) + (X-X) + ... = X - (X-X) - (X-X) = X - 0 - 0 -.. = X
This series is actually the most profound counter example for unjustified arithmetic operations with infinite series. It's one of the first things a math major learns in the theory of infinite series. It's incredible how dishonest that Numberphile video was in that regard.
This reminds me of the story the little engine that could. Gotta have some faith in yourself. Be a fucking Zaibatsu.
Wow, did not know about the sequence 1-1+1-1… not having a sum. Though it makes sense when u consider that one cannot evaluate oscillating functions, e.g. sinx or cosx, as they go to infinity.
Indeed. The first thing i thought of when i saw that sequence was sin(x) which has no limit according to calculus.
I thought it would be s={0,1}
@@fifty784well sum should be a single value so you can't say it has a sum if it gives 2 different values
That's what we call adherence points. These are points for which there exists an infinite subsequence with that point as its limit.
only near the end of this video when i'd gotten long lost in the proof did my brain suddenly pull out the much shorter summary of all of it from my college calculus class:
"the method that gives -1/12 is for predicting an output at the very end of a function, but not every function you find a limit for calculates a cumulative sum of things"
(edit: or in this case one very specific sum of things versus others)
The beauty of math is that "alternate facts" are not possible. When someone misunderstands, their misunderstanding can always be analyzed and exposed unambiguously, and forced into regeneracy.
You mean 'alternative facts' which means 'replacement or substitute facts'
'Alternate facts' is a command to contantly swap 2 facts, which is a totally different meaning.
@@SteveNicholasLondon No, that's the verb which is pronounced /ˈɔl.tɚ.neɪt/ (or some analogous pronunciation). But the spelling "alternate" also represents an adjective, pronounced /ˈɔl.tɚ.nət/ (or some analogous pronunciation), which in some English vernaculars is synonymous with "alternative". So Sara did, in fact, mean "alternate facts", with the intended meaning the same as what you would write/say as "alternative facts".
@@Veeno00 Only in America. Why not use alternative and not alternate if alternative is what is meant?
It's this American degradation of the English language that typically leads to American pilots shooting their own planes out of the sky and landing in rivers.
Why not use the correct, unambiguous words?
@@SteveNicholasLondon Oh, you're one of those people. 🙄 In retrospect, I should have guessed that you weren't merely ignorant but just intentionally wrong.
The modern English word "deer" comes from the Old English word "dēor", which meant generally "wild animal", not specifically one particular species. The Old English word which carries the meaning of the modern English "deer" is "heorot". If you want to stick to your principle, stop calling heorotas "deer". Why not use "heorot" instead of "deer" if "heorot" is what is meant? Why not use the correct, unambiguous words?
Language change is not degradation. If it were, then English itself should already be rolled back to Proto-Germanic, or perhaps even Proto-Indo-European, or maybe even further back?
The way you phrase it, one would think that British English is somehow a pristine language devoid of ambiguity, which is most certainly not the case - for example, in every single variety of English, the -ing forms of verbs can often ambiguously represent either an action noun or an adjective ("flying aeroplanes" - is this what pilots do, or is this a collection of aeroplanes that are currently flying?).
In any case, the ambiguities already present in English are certainly worse than the potential ambiguity of the modified meaning of "alternate", which can easily be disambiguated from context. To answer your question, the reason to use the word "alternate" is because "alternate" IS what is meant - the word is being used appropriately according to its meaning. By pretending that the only valid meaning of this (or any other) word is prescribed by your vernacular of English, all you accomplish is that you make it seem like you just don't know English well enough.
Alternate facts are indeed possible in mathematics sometimes people who ate using to conventional myths choose what to consider as facts
Unlike the previous times I watched this video, I managed to understand convergent vs divergent series, and basically understood everything up to the Zeta function. Now I'm watching just trying to understand xD