That'd be really annoying at school. Excitement and mathematics hold hands together. Imagine hearing that monotonous voice at school. It'd make you sleepy and eventually you'd start to get bored of maths as a subject
Ed's comedy timing is beyond perfect: Ed: So what do you think the sum is? Brady: Well, I would think that the sum would tend to infinity. Ed: Yeah, makes sense doesn't it? Brady: Was I right? Ed: No!
Well he did that because thats the whole concept of analytical continuation setting the value of domain beyond its range and playing with the mathematics its almost like setting sqaure root of -1 to i .
@@subarnasubedi7938 can't i write 1 + 2+ 3+......... = 1+1+1+1... = 1/(1-1) and this becomes infinite, why are we doing this, we can get all sorts of answers by these manipulations 🧐
You really aren't allowed to arbitrarily rearrange a sequence to get a sum unless the series is absolutely convergent. Otherwise you can use such manipulations to prove that a nonconvergent series is equal to any value you chose.
This is a very misleading video. What they don't tell you is that this is a divergent series and has no sum. It's clearly unbounded. However, using a different method of summation, Ramanujan summation in particular, we can assign a number to some divergent series and this happens to be one of them.
Cam GG You say it like it is no big deal. Obviously it is not equal to -1/12 whatever that means. However the idea that you can assign a negative fraction to an infinite sum that goes to infinity it's mind-bending. It don't matter how you look at it.
Cameron Grant The reason why you can't simply sum over the natural numbers is because the complexity of addition is optimistically O(n) and the infinite sum grows faster than linear time. This is why it is difficult to literally concieve a value like -1/12 however, analytic operations are abstract and have no time complexity. This is what I believed allowed Riemann to circumvent the divergence at s=1 he looped around the complex plane. An operation like this has no algorithmic reasoning therefore a solution can only be made via analytical operations. Essentially, there does not exist an algorithmic approach to resolve these class of problems. Q.E.D
I found this: Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to infinite divergent series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties which make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined. ( I still like to think that you can't add all integer number up to infinity)
matt de leeuw first of all: you cant say that without proofing it that easily seccondly: infinity is greater than -1/12 so this proof would be irrelevant
Daryl can you speak clearer, because I'm uncertain as to what your saying. all I was saying is that if you take the right hand limit as x -> -1 that it works for the proof to show that sum sums to -1/12, this isn't really a proof that's required, because it's common knowledge for anyone who's taken at least high school math
As long as 1+1 is 2, If summation is adding, the summation of the series 1,2,3... will always tend to infinity and never have a finite sum. But special "summation" techniques can be applied to ASSIGN values to the series which are useful in physics.
What's the difference between "assigning" a value to a summation using a set of mathematical steps, and a summation "equalling" that value? Isn't that the same thing? Imaginary numbers were invented to be used to "assign" values to the roots of polynomials that otherwise don't have real values. I don't see how this is different. Numbers don't have to "exist" in order to be useful in describing phenomena.
I think that mathematicians over-sensationalize this particular sum because the -1/12 result is not based on traditional sums. You have to treat the number in a certain way and relate it to other functions in order to get this. The problem is that this isn't communicated in this video or by any of the others that talk about this sum. It is mathematically interesting, but it doesn't mean you can literally add up all the natural numbers and come up with a negative fraction.
Oh my goodness. I’ve been watching these, thinking “this guy reminds me of a professor I had at uni, who taught me vector calculus, further calculus, and more; and he took a 4-person study group every week that I was in.” Then I look at my uni records, and.... he was called Ed Copeland!!! I can’t believe it! Mr Copeland, we had great times in Brighton, UK!! BIG UP THE UNIVERSITY OF SUSSEX!!
This theory was proved by Indian mathematician Srinivasa Ramanujan.
11 лет назад+58
Somewhere in the process of adding/substracting infinite sums the equal sign does not mean "equal" anymore. All this demonstration is about getting the Zeta function "extendable" to -1. The trick is that they keep using the sign and word "equal", while they shouldn't (that's why physics is physics... always making nice asumptions but using bad mathematical language). The Zeta function with -1/12 added isn't "equal" to the infinite sum anymore, it is more than that (a new mathematical object). Let's take an example. On both sides of a river there are two similar roads. But you can't get a car above the river until you build a bridge. But the road with the bridge is not the sum of the two previous roads anymore, it is a new road. And while before it would have been absurd to say that you can be driving a car above that river, now you can even stop on this bridge, and keep the car parked here. Brady parked his car at -1 (the river) on the Zeta function, but before that, he made a mathematical bridge appear that "fits" between the two roads so that his car does not sink ;)
My Calculus 2 textbook says that answers like these are "absurd" because the sequence diverges. Don't get me wrong, I love concepts like this and am all for the sum of the natural numbers being -1/12...but what justifies the addition of infinite series?
i dont think you would get a propper answer in a calculus 2 book (specially for this kind of math)... you should look for complex functios and analytic continuation. And i think there you may satisfy your doubt. Im sorry for my English, im from Argentina and its 2 o'clock in the morning... 😁
Chris Bandy I think your misinterpreting your calculus book. They're probalbly talking about the limit of the series, this isn't the limit, its assigning a value to a divergent infinite sum.
Divergence is not really caught in high school mathematics just as complex numbers are not caught in junior school where possibility of square root of -1 is termed absurd. :) There are ways to get sums like this. Each divergent sum like this also has a type and based on that type there are ways to get to the sum. Watch the video on Grandi series (1-1+1-1+1-1.....). There too, the sum (called Cesaro sum) of the divergent series comes out to be 1/2.
Or ist just comes out as basic garbage, because the series 1-1+1-1 has no limit, If you where to apply the Theorem of Limits, you need first to proof that for every epsilon around said limit, you get a certain n, after which the series does not leave an intervall of epsilon around such limit. Which you cannot for the sum of 1-1+1. So if no limit exist, the sum cannot have fixed value. Only if there is proof of a sum being convergent, you can than apply summation of know, convergent sums to determine its value.
In the strictness sense, no you can't. But at the same time then the sum 1 + 2 + 3 + 4 + ... doesn't work either. To give meaning to these diveragent sums you can do things. This is what is called Abel summation; in the fullest rigor you take a limit as x -> -1 which gives you the sum in question. Then you do other trick, such as "analytic continuation" with the zeta function thus you can "uniquely" define the sum 1 + 2 + 3 + 4 + ... = -1/12
The thing is that the series DOES NOT approach -1/12, it IS -1/12, this is why you can't get it by doing a limit or a summation, your result not only will be imprecise but ultimately incorrect, it only makes sense as a whole. For every other application, INCLUDING when it approaches infinity this series does not equal -1/12 and behaves "normally". Another way to see it: no function nor series is continuous at infinity, and we know that in non continuous "points" the limit and the value often don't match, these calculations allow you to find not the limit, but the actual value at infinity.
This video has been completely discredited. I believe even these guys discredited this video. In no way shape or form does 1+2+3+...=-1/12. Infinite sums such as 1+2+3+... are thoroughly discussed in a 2nd semester calculus class in a chapter on sequences and series. 1+2+3+... is an example of a "divergent series." It's an undefined expression something like 1/0.
The way I look at it is that generalization is a tool we use to give meaning to concepts that cannot be described otherwise. Consider x^2=-1. You could say that it can't be solved, because there's no such real number. Or you could imagine such a number, and call it i = sqrt(-1). As it turns out this imaginary concept solves a lot of physical problems, such as describing a rotating vector, and it opens up new ideas, such as frequency domain analysis, which has a real physical analogy. For example, you can sharpen your photos, thanks to the introduction of a new concept that is supposed to be imaginary and not real. There's no such number as infinite. Something is either convergent, when it clearly tends to get ever closer to a real number, or it's divergent. Just like sqrt(-1) didn't exist before, 1+2+3+...+inf doesn't exist, either. However, a genius called Ramanujan from India had invented a special summation, a way of assigning a value to divergent series. It may be terribly confusing, but if it allows us to describe something in the real world, then who's to stop us from doing it? It's certainly done in a logical way, by generalizing something that was meaningful before. For example, the factorial is easy to define for integer numbers, but what about real numbers, or complex numbers? Well, who's stopping us from defining a function that we call Gamma, where Gamma(n) = (n-1)! if n is positive integer? All of a sudden we can solve probability theory problems that we could never handle without it. We just introduced a new concept that solves something real.
Rob Laquiere Generalization is usually applied to definitions (e.g., we can generalize "product" from taking two inputs to taking any finite - or in some cases countably infinitely many - number of inputs). Generalization is not applied to properties, such as the commutative property. Properties of generalizations of definitions must be proven for each generalization of the definition. Also, no one will ever "prove Ramanujan wrong" about Ramanujan summation. What could happen is that someone could come up with a different generalization of the sum of an infinite series which is more useful. But if this happens, it won't prove Ramanujan summation to be wrong; rather, it would simply be a more useful generalization.
MuffinsAPlenty Can you please define "generalization" so that I can be sure we are on the same page? When I first started to learn mathematics, it appeared to me that the basic laws of Algebra were in fact generalizations of Algebra. Just like x*y = y*x is the general form of the law of communicability of multiplication. But just like I stated in my post, this generalization is wrong in certain fields of mathematics! If that is not what you consider a generalization, then I need you to clarify your definition of "mathematical generalization". The properties of mathematics are generalized statements about how the math works out (properties = generalizations). No properties have been proven for all possible cases, as some families of equations are infinite in size (infinity cases cannot be checked). Therefore generalizations are not proven for all cases simply because they yield sensible results for some cases. Like in my example, the generalized form of the law of communicability of multiplication breaks down when vectors are multiplied. It is therefore possible that Ramanujan's generalization here is also wrong (as some generalizations have been proven wrong in the past). For the case of divergent series, the definition of divergence contradicts the result they got. To me, checking the results empirically is more sensible then trying to convince ourselves it's true. As a physicist, I tend to check results to make physical sense first, as that is indeed the goal of mathematics; to count real things and to achieve empirical results. Lastly, as I already stated, the generalization may very well be true... But I need to see this verified in reality (with tangible things), until then, it's just numbers written on a page in a particular way (not proof of anything really).
the problem with the zeta function argument is that the function only works with the condition that s>1 or the series won’t converge. the analytic continuation of the zeta function is ζ(s)=(2^s)(π^(s-1))(sin(πs/2)) Γ(1−s)ζ(1− s). notably not the same as the original function. plugging s=-1 into this will give us -1/12, however because this is not the zeta function, simply a continuation of it, we can’t expect the same result from the original zeta function.
Hi five! People all around the world love interesting things. It is this curiosity that had driven us to amazing things. Don't let the loud and disgusting actions of mankind elude the quite and amazing things that people do daily. Captain Enthusiastic For Humanity flies away.
as i understand it, to put it in layman terms. if you added all the numbers together the resulting integer is so huge its dense and it sucks in on itself into another dimension. it then poops out gummi bears. joy
Can somebody riddle me this? At 1:45 that result is only valid if abs(x) < 1 But then at 3:40 ... "let x = -1" Doesn't that break it all? I know I must be missing something obvious here
dazz This is a method in complex analysis called analytic continuation in order to extend the domain of functions. Without that context, yes it breaks everything, but within this context it is okay. Obviously, 1+2+3... is undefined but in order to assign it a value, this framework is used. In the same way that no number squares will equal -1, we create a new framework to allow us to find roots to polynomials with no roots.
+DrDress but as you kept typing 1+2+3+4+... to infinity, with time... lots of time you'd get so bored and go so crazy that you would catch yourself typing -1/12 over and over again.. then realizing that you're now headed towards negative infinity. so then you would try to recompense by typing +1+2+3+4.... all over again and just consider giving up and when you finally did you would find out you gave up at the exact same point you began to start typing -1/12 (the point you lost your mind) and the calculator would read -1/12 . then you'd realize you where still typing. and that number would just haunt you all the way. for eternity..... its the devil....
DrDress The reason why you can't simply sum over the natural numbers is because the complexity of addition is optimistically O(n) and the infinite sum grows faster than linear time. This is why it is difficult to literally concieve a value like -1/12 however, analytic operations are abstract and have no time complexity. This is what I believed allowed Riemann to circumvent the divergence at s=1 he looped around the complex plane. An operation like this has no algorithmic reasoning therefore a solution can only be made via analytical operations. Essentially, there does not exist an algorithmic approach to resolve these class of problems. Q.E.D
+terminator10111 is absolutely right ! Just this small mistaken assumption and you end up with a huge and absurd result. So what does the guy in the video has to say about this ?
Ionel Dragomirescu Yes, but the value approaches 1/2 as you let x approach -1. Since in the framework of all of this stuff everything varies continuously, that is enough.
Colin Smith That's also what upper division math professors tell you, because the series does not converge, because the sequence does not converge, which is easy to prove. If they want to assign number values to divergent sequences, fine, but that's not mind-blowing.
lizokitten2 I didn't say anything about the Taylor series. Anyway, they both converge to the same answer from the left as you approach x=-1. This fact is sufficient by what we know about the continuity of the Riemann Zeta function etc., as I said before.
The functions they are using are not analytic for the values they are plugging in. That's the problem. Some people here are saying that they are simply creating a different, consistent system by ignoring where functions are analytic, which is interesting, but if that's the point of these videos, then what they're doing is misleading, not mind-blowing.
I was wondering the same. I was taught that the sum he uses is only defined for the absolute value of x not x. As in its defined in -1 < x < 1 as you said.
After doing a bit of reading ( and I do mean only a bit) I think its not supposed to be be "equal" to in the usual sense but is instead just meant as a property of the sequence and how the sum of it tends to infinity in a different way than how the sum of other sequences tend to infinity.
Around 17:45, Guy #2 admits that it's a "trick" called "analytic continuation", which apparently involves finding a convenient formula for a function that's valid over a certain range and then arbitrarily extending the range over which you can use it. So apparently it's OK to break basic rules of math as long as you give it a fancy name when you do it. :)
But you can't just remove a divergence! It tells you that there is no limit, no one answer it tends toward! So how is that valid? The obvious way to approach 1+2+3+4+...+infinity is infinity plus any finite number, an infinite number of times--which will continue to be infinity. I just can't accept this one.
I am a 6th year PhD student in Physics and Electrical Engineering, but I double-majored in Physics and Mathematics as an undergraduate, just to give some background on my expertise (or, more likely, lack thereof). A few thoughts: I notice that both Dr. Copeland and Dr. Padilla are physicists, according to the video description. My math professors held up their noses at physicists for their cavalier approach to mathematics, particularly for switching sums and integrals and their handling of infinite series like these without regard for rules and the applicability of formulae. I would be very interested in hearing what a strict mathematician had to say about this result, particularly because we simply "accepted" the insertion of x=-1 into a formula with applicability for |x|
And what about now? still accepting it? I mean after 4 years from your comment.! did you get a mathematician opinion on that? I really would like to know :) Regards
It is very simple: What they present here is wrong. Even reordering convergent series leads to arbitrary results, as Riemann showed. Plus, there is a proper definition of convergence of series, which is that the sequence of partial sums converges, hence their statement: 1+2+3+...=-1/12 is simply wrong. Additionally, the Zeta function is not defined for numbers whose real part is smaller than 1. What actually happens is that you can use analytic continuation to extend the zeta function to numbers with real values smaller than 1 and for this continuation, the value at -1 is -1/12. But this has not much to do anymore with 1+2+3+...
@@adhamm5503 Well the natural numbers with the addition is a group, meaning if you sum natural numbers you can only get a natural number. Moreover a positive one if all the numbers you add are positive. If you find something else, it's either you are wrong, or what you say is addition is in fact something else (which is what Ramanujan meant).
@KarlDeux A bit of a correction, the set of natural numbers is not a group under addition; there's no inverse elements at all! However, the set of natural numbers including 0 is what's called a monoid. That is, it's a set equipped with a binary operation that satisfies all the group axioms except for the inverse axiom. Specifically, this tells us that adding two positive integers yields another positive integer.
I just received the brown papers from this video - thanks Brady :D As soon as I get back home (I'm abroad for work) I'm going to frame them, hang them on a wall and send you a pic ;)
To those who see the problem where he just plugs s=-1, it kinda is wrong but what I think he secretly does is so called Abel summation, where you look at the limit when x approaches -1 it actually gives 1/4 so it's actually true. Although in this video just plugging x=-1 doesn't work but with Abel summation it gives 1/4. Google Abel summation for more information
pantopam You can always justify any BS in retrospect. Mathematics is about being clear and defining what you are doing precisely. The video is just wrong.
I think this type of result should be referred to more as "there is a function f so that f(1+2+3+4+...)=-1/12". It's not a summation in the conventional sense, but we could invent some kind of "divergent sum space" (of which 1+2+3+4+... is an element) which is the domain of this function and the co-domain is the reals, or the complex numbers, or other sets.
+Davy Ker I think you are right. What is bugging me is that this whole process only works if you assume that the limit of the alternating series 1-1+1-1+1.... etc is (1/2) altough it is clear that this series has no limit. And if you watch the Video where he tries show us that the limit of this series is (1/2) you can point out some major flaws in his proof.
The limit of that series is not 1/2, but rather the cesaro convergence of the series is 1/2 --- by taking the limit of the average of the partial sums, you can in a way assign a finite value to an otherwise "normally" non-convergent series. So theyre kind of not giving you the whole picture as they dont say "1-1+1... is not equal to 1/2 in the traditional sense of summation, but we can provide an alternate description based on some characteristic of the series, and say the series is equal to that description."
Brady, please ask the professor the following: In the beginning of the vid, the prof offers to prove the convergence formula for geometric series (a+ax+ax^2+...ax^n= a/(1-x), in our example a=1, so it's just 1/(1-x), as the prof stated) and you said we should just accept it. If he were to prove it, one thing would become very obvious; the convergence formula only holds for -1
Strictly speaking, the series diverges. It does not have a value. However, we can regard the geometric series as a function (this is the whole 1/(1-x) bit) which does converge for some values of x. Then we analytically continue (a process in complex analysis that gives us the unique function that covers a larger domain but agrees with our original function on the smaller domain) this function to other values. We take whatever value our analytic continuation gives and treat the series as having that value. But strictly speaking all the equals signs here aren't really equals signs in the same way that 2+2=4
Not really. We can treat the series as having a value of -1/12 in many calculations, but this is why in the Polchinski book that they show, Polchinski uses '->' instead of '='.
It doesn't even tend toward -1/12. It tends toward +infinity. Again, the series does not converge. Its just that there is a consistent way of assigning this series a value and that value happens to be -1/12.
I guess the thing that makes this the most confusing is that I assume that if you add an infinite amount of numbers the answer would be infinity. And if the answer is -1/12, then what happens if I add the"next" term in the sequence, as the sequence can never end?
No idea why this video is still up. It is easily the numberphile video with the most wrong statements I have ever seen. Maybe someone should have told the guy that an implication with a false assumption is true, regardless of the truth of the conclusion. You can make a video about this, but then do it properly and introduce analytic continuation.
+xxBIGBIRDxx he took the derivative of 1/(1-x), he had already established that the sum of an infinite geometric series is a/(1-r) [which in this case is 1/(1-x)]
I find it really interesting that 1 - 2 + 3 - 4 + 5 -... = 1/4 which ( i realized since I just came from some videos on it) is the point at which real numbers begin to "escape" from the mandelbrot set or whatever. Makes me wonder, where does -1/12 fall on the mandelbrot set? I'm certainly no mathematician but its fun to think about these things.
Numberphile, I've got a question. It might even be a silly one. At 1:45, Ed says that the sum of a geometric series equals to 1/(1-x) strictly when x < 1. However, in some textbooks, we see abs(x) < 1 instead of x < 1, which would render that formula not defined for x = -1 also. It would give as domain, the open interval -1 < x < 1. Why is it that makes it possible to consider all values x
+IsItHardToBeStupid? And in that case, the sum would be convergent. Eventually, the fraction ^ n would counteract the coefficient and it would approach a finite number. Ignoring those boundaries makes it divergent and not calculable. The Riemman Zeta function is a nice trick once you've ignored the very first restriction that was placed on your equation.
+IsItHardToBeStupid? That doesn't make any difference. And no, the sum does not suddenly converge. Euler is notorious for doing things that are completely unjustified, such as putting (x=-1) into a formula that is not valid there.
But that is *not* the part where this "proof" falls apart. The problem is where the -2.2^(-s) line is split up. Instead of -2(2^(-s)+4^(-s)+...) it becomes -2(0+2^(-s)+0+4^(-s)+0+...) which could have a different value. For example 1-1+1-1+1...=1/2, but 1+0-1+1+0-1+1...=2/3.
Allow me to share my understanding of this subject. In complex analysis there is something as a Laurent series. It is much like a Taylor series in real analysis where you try to approach a certain function using a series or a sum. For example in real analysis the function f(x) = sin(x) can be approached around the origin (x=0) by the taylor expansion x - x^3/6 + x^5/120 +... This expansion which we call a Taylor series does not equal the function sin x but it very much looks like sin x around the point x = 0. However when you go far away from x=0 this series won't even be close to where the actual function is. We call the place where the series looks like the function the radius of convergence. Now that we got this out of the way let's talk about the Riemann Zeta function. Riemann never defined this function as the sum we see in this video. The sum we see in this video is a mere approach of the Riemann Zeta function on certain parts of the complex plane. This is what we call a Laurent Series. Again we can talk about where this series converges. It does on most of the positive real number axis in the complex plane, but NOT on the negative number axis. So we can say that the Riemann Zeta function of square root 2 for example will be the same as the sum over n of n^sqrt(2). THIS IS NOT TRUE FOR NEGATIVE NUMBERS SINCE THE LAURENT SERIES IS DIVERGENT THERE AND IS NOT A GOOD APPROACH OF THE RIEMANN ZETA FUNCTION!! So Riemann Zeta of -1 DOES NOT EQUAL the sum of all natural numbers, however Riemann Zeta of -1 does equal -1/12, and this is where the mistake has been made. The sum of all natural numbers does NOT EQUAL -1/12. In certain branches of physics however we try to make sense of divergent series like the sum of all natural numbers. When we get such sums in our calculations what we can do is REPLACE this sum with a certain MEANINGFUL value and get the right answer. In this sense the sum of all natural numbers is equivalent with -1/12. We can get to these equivalences by 'bending' mathematics and going outside the well defined 'rules', however we haven't yet found a good explanation to get to these equivalences. I hope this brings light to some shades created by this video. :)
Thank you for this explanation. Without it this video doesn't make any sense and is misleading. Which is probably done for sensation. It's just click bait and unworthy of this channel
At about 12:50, Dr. Copeland refers to "removing the divergence". The problem is that the only way to do this is to not use infinite series. Infinite sums diverge or converge. About a minute later, he asserts "you've got to deal with divergent numbers- divergences very carefully", but the way he "deals" with infinite series wouldn't be considered "careful" even in Euler's time, as Wanner & Hairer note in Analysis by its History. Of another divergent infinite series that Euler wrote equaled a finite number, they state "its mathematical rigor was poor even by 18th century standards", So why are we repeating mistakes made in the 1700s? Even a defender of Euler's ridiculous equations, Sir Roger Penrose, admits "..,the rigorous mathematical treatment of series did not come about until the late 18th and early 19th century...Moreover, according to this rigorous treatment [Euler's equations] would be officially classified as 'nonsense'." While a rigorous treatment of the zeta function (or any other topic in complex function theory) is obviously beyond the scope of a RUclips clip like this, that doesn't entitle one to make patently false claims while ignoring one's own advice just to be able to make a mathematical claim that's seems mind-blowing (to the mathematician, it's only "mind-blowing" for its fundamentally flawed basis).
I used to harbor a huge dislike for Maths when I was in high school and I simply feel the need to say that through the years this dislike has turned around 180 degrees and I found the maths in this video simply beautiful. The Universe is so stunning.
Willans' Formula for primes: 2 to the n part = vertical asymptote. 1/n part = vertical tangent. Factorial part = vertical line. These tensors from differential calculus determine singularities in stable matter as represented as primes.
2:51, you also need the sum rule of derivatives (f+g)'(x)=f'(x)+g'(x), on top of the power rule f(x)=x^n (n is constant with respect to x), f'(x) = nx^(n-1).
In a mathematical "proof", the first erroneous argument nullifies (neither refutes or supports) all subsequent arguments, and thus the conclusion is unsupported.
I would like to hear more examples in quantum field theory where this property plays out. String theory has a bit of a reputation for being nice in theory, but unprovable (from what people who have researched it more of told me). Quantum mechanics is something we're getting experimental results out of, and I'd love to hear about one (or more) which were predicted by a theory built on this sum.
As a physicist, I have to ask, why do you need to see such things? This result is a mathematical fact that can be made totally rigorous provided you have the years of training to talk about analytic continuation and zeta functions. Adding physics is just muddying the waters.
Another Numberphile video has a demonstration that really helped me understand this. First he explained the sum of 1+1/2+1/3+1/4 etc, which is Infinity. Imagine an ant standing on the edge of a rubber band one meter in diameter. Every second the band grows one meter and the ant advances one centimeter. By the time the ant gets all the way around, the band will be infinity large, but it will eventually get there. Conversely, for the sum of 1+2+3+4+5 etc, which is -1/12, imagine the rubber band is infinity in diameter with the same ant moving one cm. per second while the band shrinks one meter. The closer the ant gets to going all the way around, the smaller the band gets, until it has to shrink into itself, i.e. become a negative number. Once I envisioned that, it made perfect sense, despite seeming like it should be the other way around :)
Based on 3:38.. 1+ 2x^2+ 3x^3 + 4x^4 +... = 1/(1-x)^2 Then why not set x = 1 to make series as 1+2+3+4+.. and its answer as 1/(1-1)^2 which is infinity.. Series sum as -1/12 doesn't make any sense. How can you keep adding something and end up deficit??
That's actually how Romanian came up with his "prove" because he was not formally educated in math. This concept is similar to imaginary numbers. Square roots of negative numbers do not exist but we pretend that they do. For sanity, sum of natural numbers is not equal to -1/12, it is only a concept.
It's like saying √-1 is i. It's breaking the rules in a clever way to do what you want anyway, and yet, you get real results out of it that actually work and can be experimentally verified (and have been).
@@Invisifly2 yes, it is called ramanujan summation in case of the sum, not a conventional number. Imaginary number is another convention, and cannot be experimentally verified.
@@Invisifly2 how have they been "experimentally verified?" can you show me an instance where the sum of natural numbers = -1/12 has been useful for describing things in the real world?
This proof is still incorrect. The faulty part is where he says that 1-2+3-4+5.. =1/4 based on the power series 1+2x+3x^2+...=1/(1-x)^2. If x = -1, then the LHS does not exist. In fact in the limit the power series oscillates around approx. 0 for say x = -.999.. But saying that for x=-1, that the RHS = LHS = 1/4 is totally preposterous, and is equivalent to dividing by zero. In fact the limit DOES NOT EXIST at x = -1 and saying that this is 1/4 is just pure lying!!
The left hand side doesn't exist under the simple rules of limits taught in precalculus (the series' partial sums diverge, so it has no limit). Fortunately, people like Euler and Cesaro and Borel figured out how to sum divergent series hundreds of years ago, thus generalizing things like the power series limit for cases that had previously been undefined.
It may exist, however, when the function is expanded to the complex plane, as the actual [formal] proof of this involves the reiman-zeta function where s is a complex number.
Question: We have a test to see if a series is divergent or not r=(An/An-1) where An is any number in the series and An-1 is the value before An. when r < 1 it converges to a number when r > 1 it diverges when r=1 we don't know when we plug in this series, r is clearly greater than 1 which means it diverges to infinity why doesn't this series follow the test?
He notes that this (1+2+3...= -1/12) is critical in string theory. That's not a reason to believe it's correct. Has there been any observations to predicted by string theory that depend on this? If not, maybe we have a problem with string theory. Euler...well, he was pretty smart. But what did he think of his result? Was he serious? Was it an erudite joke of his? And he was a genius at convergent sequences... For this to be serious, we must believe that an infinite series may have a finite sum even if the series does not "converge" in the classical sense. What would Gregor Cantor think?
The first series given by Ed (that being 1-2+3-4+...=1/4) is actually divergent. The expansion of 1/(1-x) is invalid for x = -1. It is in fact only valid for -1 < x =< 1. But I have a feeling that I don't need to say this.
There is nothing in the sum that would ever turn it into anything but a huge number, it's an infinite sum of ever increasing positive numbers, there is no way it can do anything but go towards infinity. Something in those manipulations HAS to be wrong. Maybe in the future it will have the same status as Zeno's paradoxes have now, something interesting and clever to ponder, but ultimately wrong.
It's no less correct than 1+1-1+1-1+1-1+1-1+1-1+1.... = 1/2 The problem isn't that the math is wrong, it's simply that it can't be explained by simple algebraic rules. There is lots of correct math that can't be explained by simple algebra. It's why we have higher level mathematics.
Greg Vanderpool That I can accept, the answer alternates between 1 and 0, infinity is not even or odd so you average both values, it makes some sense, but for an infinite sum of positive numbers to result in -1/12 makes no sense whatsoever to me
Gerardo Lozano Except you don't actually take the average. That's a simplification to make it more understandable, not what's actually happening in the math. You can't divide by infinity, which you would need to do in order to take the average.
CrazedPorcupine Sorry, but string theory is not part of the standard model, the whole theory could be abandoned at any time, so the fact that it's used there adds little to it's credibility for me. As for it's role in more solid phenomena like the casimir effect, I don't know how vital this is to that, so I can't just accept it because of it.
Thank you for the second explanation! Analytic continuation... I can buy that because it's not saying that this is actually valid, but that it is a 'what if' situation (just like imaginary numbers, where we said 'what if the square root of -1 existed?', and it has been a HUGE help in physics). Really wasn't happy with the first part, at 6:35, when we just started multiplying bases together because they shared a common exponent, but kept watching hoping for someone to make-up for that.
This is actually making me a bit angry. The sum of any two integers is an integer and the sum of any two positive numbers is a positive number. How an infinite series of positive integers can be equal to a negative fraction is beyond me. Tony's explanation starts with something that doesn't really have an answer or has two answers, so lets average them, and then builds on that as if the average were really the answer. How is it not undefined like 1/0? It's mad, and it's maddening.
To be formal, it to the existence of the zeta function ζ（z), such that if you evaluate it at z = -1 that gives you -1/12 and if you analytically continues the expression on the positive real plane to -1, you get the summation of natural numbers To be less formal, it means that if you get -1/12 in a certain context, like manipulating the zeta function at -1 and doing Taylor theorem, your calculation in complex analysis is wrong and by the uniqueness of Analytic Continuation of ζ(-1), the correct answer should be the summation of the natural numbers.
By extending the zeta function through analytic continuation to negative integer values you forfeit the ability to write that the sum of positive natural numbers equals zeta(-1) and therefore the sum of positive integers in no way equals -1/12. It is an absolutely divergent sum. The methods in which are used in this video to prove this result are simply incorrect. This video is extremely misleading and condescending towards its audience.
I have to object. the answer to s=1-1+1-1+1... has to be either exactly 1 or exactly 0. The average of those two is 1/2, and the median of the possible answers would be 1/2, but the actual amount must necessarily be EXACTLY 0 or EXACTLY 1. The number is quantized. A similar example is that the average number of children in a household might be 2.3 children, but the actual number of children will always necessarily be a whole number. Since the entire solution is based on the incorrect assumption that the expression can have a solution that is not either 0 or 1, the solutions for the other expressions are necessarily wrong as well.
It seems that a lot of people (myself included) have a problem with this bit of mathematics, but for now I am willing to accept it for the sake of asking another question I have. Is there another possible answer for the sum 1+2+3+4+... ? What I mean is, if we tried to use a different method to "simplify" the infinite sum, could we not get an answer other than -1/12? Also, does every divergent infinite sum like this have a finite equivalent solution? Like what is 1+3+5+7+9+... for example? And is there only one possible finite value for this sum?
@@jdw6925 True, it does tend toward infinity in the sense of a limit of partial sums, but there are plenty of logical holes in this video that I wish they would’ve covered in more detail (I’m acknowledging this even as someone who loves infinite sums and analytic continuation); there are plenty of proofs (or “proofs”, I suppose one might want to say) that shed more light on this topic, and I would implore you to check on those that go over more rigorous proofs because they become truly fascinating.
Some people here seem to have tremendous difficulty understanding how mathematics progresses. They would probably be happy saying x^2 = -1 has no solution, end of story. Hell, many of them might even call game over just with x^2 = 2, I mean, solving that would be completely irrational right?
+Veon1 Although this response is 4 years old, I would just like to explain why this is wrong. Saying the sum of all natural numbers is -1/12 due to the zeta function is just objectively wrong. -1/12 is the evaluate of the anayltic continuation of the zeta function. Meaning, whatever is causing -1/12 is now due to something other than the 1/n^s equation. Analytic continuation is what you get when you assume what the graph would look like if you asked yourself "What WOULD the graph look like if it was defined for values below 1?" You get it through a transformation.
X^2+1 has no solution in the real numbers. If you had ever attended a maths lecture, you'd know that things like real and complex numbers are defined precisely and justified rigurously. You don't just suddenly throw all of your definitions out of the window.
If instead of setting x = -1 into that intermediate derivative series you set x = 1 you get sum(n=1, inf) n = 1/0 which runs directly counter to the alleged result. Also there are very real mathematical problems with evaluating any series that can be rearranged to infinity - infinity.
+Anthony Purcell No. 1/0 isn't infinity, it's undefined. Those are different things. You can say that for 1/x, as x tends to 0, 1/x tends to infinity. But that's NOT the same as saying n=1/0= infinity. Learn some math.
My mistake, I had assumed that the mathematically literate would not need me to go into such pedantic detail. I should have remembered that this is the internet. Allow me then to elaborate. If you substitute x = 1 into the first equation you get a quantity that is larger than any finite number, though perhaps not as large as ***** 's dickishness.
The more people argue, the more laws of math they violate. He also gives wrong information about zeta function. (such as using the given representation for s=-1) He also assigns values to alternating sums and changes orders of the numbers like it is very acceptable. If that proof is valid, 1 = 0, and numbers mean nothing.
That is the opposite of how math works. Math is entirely based on making assumptions and then working out what the consequences are. That's why there are different systems of axioms, ZFC is just the one we are familiar with. You can make any assumptions you want in math and try to build up from them, and you should because otherwise your mathematical framework will never advance.
According to Wolframalpha I am right: wolframalpha.com/input/?i=derive+sum+x^n+%2C+n%3D0+to+infinity (^ doesnt seem to work in links, just copy and paste) Is the proof wrong?
Very strange that you would be ashamed to be a physics professor doing correct math on a math channel. Are you also frequently embarrassed for doing other things right? Maybe you should talk to a professional about this.
@@eoinlanier5508 Apparently you neither read my comment nor understood the math being discussed. When such a highly respected group puts out something that violates several mathematical priciples and well known limits they should feel embarrassed. Edited to correct auto correct errors.
PLEASE PLEASE PLEASE do a video on the proof of the Basel problem (1/1sq + 1/2sq + 1/3sq + ...). I know the answer is (pi sq) / 6, but I don't know how. Thanks
Do you know "Proofs from the BOOK" by ziegler and aigner? There are (I believe) 3 different elementary proofs, which only use the geometric series and integration by substitution.
At around 2:25 the rate of change of distance as a function of time is speed and not velocity ; velocity would be the rate of change of displacement as a function of time.. sorry for being picky, but that's what it is considering the vectors.. i understand that you might be speaking in loose terms just to illustrate an example... great proof though
A blog with more links and info - www.bradyharanblog.com/blog/2015/1/11/this-blog-probably-wont-help
Ed's voice is the auditory equivalent of getting a hug from a feather pillow.
One that isn't overly-warm, even.
Post op Physio for shoulder decompression
And a pillow that makes you learn things.
most apt statement i have ever read
Nice
"1+2+3+4+5... = ∞?"
"Yeah that makes sense"
"So.... am I right??"
"No."
No. Ethan
Ethan Martin
tell me how the sum become -1/2
it didnt
Every lecture in history of math
0:25 Yeah. I like that part.
Someone in the US government is trying this with the national debt.
Unless the debt is infinite,the result will be a big number,never -1/12
@@emilcatar4620 r/woosh
@@ravenlord4 r/whooosh
@@emilcatar4620 it's not -1/12 even if its infinite
this video is just bunch of bs
@@ccv3956 it is. Atleast in real maths it's fake
Professor has the most calming voice ever to listen to him describing this proof.
He does, doesn't he?
The Bob Ross of numbers.
That'd be really annoying at school. Excitement and mathematics hold hands together. Imagine hearing that monotonous voice at school. It'd make you sleepy and eventually you'd start to get bored of maths as a subject
But really calm voice indeed xd
Ed's comedy timing is beyond perfect:
Ed: So what do you think the sum is?
Brady: Well, I would think that the sum would tend to infinity.
Ed: Yeah, makes sense doesn't it?
Brady: Was I right?
Ed: No!
An ordinary sum WILL approach infinity, though.
Mat Hunt wow
@_@
If the sum is infinite, the sum will be =1.
You can never reach infinity with finite steps.
Laurelindo
*An ordinary sum WILL approach infinity, though.*
By how much then?
Bullsh!t, bullsh!t, bullsh!t!!!!!!!
Says only converges for |x| < 1
Proceeds to set x to -1
Well that was his trick. He only said the geometric sum converges for x
Well he did that because thats the whole concept of analytical continuation setting the value of domain beyond its range and playing with the mathematics its almost like setting sqaure root of -1 to i .
Wasnt that |x|
its the idea of extending summation of divergent series to give meaning to them. This is called Abel summation
@@subarnasubedi7938 can't i write 1 + 2+ 3+......... = 1+1+1+1... = 1/(1-1) and this becomes infinite, why are we doing this, we can get all sorts of answers by these manipulations 🧐
You really aren't allowed to arbitrarily rearrange a sequence to get a sum unless the series is absolutely convergent. Otherwise you can use such manipulations to prove that a nonconvergent series is equal to any value you chose.
Addition is commutative.
Euler: Reality can be whatever I want.
@@andyjabez9780 Yes normally, but with infinite sums it isn't unless the sum is absolutely convergent
That's what I've been thinking too. What is the reasoning behind shifting the series one term to the right.
@@ahmadrahimisudin8364 just as 3x4 is the same as 3+3+3+3 or 4+4+4. You just write it that way because it makes it easier to understand.
"That would tend towards infinity"
"Yeah, that makes sense doesn't it"
"Am I right?"
"No"
I don't know why thats so funny to me xD
...I need more paper Brady... Get me ma damn paper Brady...
I'm glad to live in a world where numbers stop at 2^64-1
I'm looking for a bit more precision...
@@jon_collins what a negative comment! 😁
Laughs in python.
LAUGHS IN YO MOMMA LANGUAGE BINARY AHAHAHHA
Where's your point?
If this is fundamental to string theory no wonder they're having a hard time working it out..
This is how we get the 26th dimensions in string theory.... Grrrrreat 😅
This is a very misleading video.
What they don't tell you is that this is a divergent series and has no sum. It's clearly unbounded.
However, using a different method of summation, Ramanujan summation in particular, we can assign a number to some divergent series and this happens to be one of them.
Riemann put a complex number into the function. What happens if you put in a quaternion?
Cam GG You say it like it is no big deal.
Obviously it is not equal to -1/12 whatever that means. However the idea that you can assign a negative fraction to an infinite sum that goes to infinity it's mind-bending. It don't matter how you look at it.
Daniel Andrade
Wrong. They claimed it to be the (normal) sum. Would you excuse it if a student made the same mistakes?
Cameron Grant The reason why you can't simply sum over the natural numbers is because the complexity of addition is optimistically O(n) and the infinite sum grows faster than linear time. This is why it is difficult to literally concieve a value like -1/12 however, analytic operations are abstract and have no time complexity. This is what I believed allowed Riemann to circumvent the divergence at s=1 he looped around the complex plane. An operation like this has no algorithmic reasoning therefore a solution can only be made via analytical operations. Essentially, there does not exist an algorithmic approach to resolve these class of problems. Q.E.D
wrong , even under other summation methods, this series is still divergent
I found this: Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to infinite divergent series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties which make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined. ( I still like to think that you can't add all integer number up to infinity)
But 1 + 2x + 3x^2 + 4x^3 +... = 1(1-x)^2 is only valid for |x|
you can see the binomial expansion and can see that it is true
Yes. I cannot rely on this proof. You cant put -1 in that equation
but you can put x->-1, this proof shows that 1 + 2 + 3 + 4 + ... ->-1/12
matt de leeuw
first of all: you cant say that without proofing it that easily
seccondly: infinity is greater than -1/12 so this proof would be irrelevant
Daryl can you speak clearer, because I'm uncertain as to what your saying. all I was saying is that if you take the right hand limit as x -> -1 that it works for the proof to show that sum sums to -1/12, this isn't really a proof that's required, because it's common knowledge for anyone who's taken at least high school math
As long as 1+1 is 2, If summation is adding, the summation of the series 1,2,3... will always tend to infinity and never have a finite sum.
But special "summation" techniques can be applied to ASSIGN values to the series which are useful in physics.
I agree "useful" rather than true.
Thanks
And that's the disclaimer they should have included in the original video, instead of misleading people to farm views.
What's the difference between "assigning" a value to a summation using a set of mathematical steps, and a summation "equalling" that value? Isn't that the same thing? Imaginary numbers were invented to be used to "assign" values to the roots of polynomials that otherwise don't have real values. I don't see how this is different. Numbers don't have to "exist" in order to be useful in describing phenomena.
Ed Copeland's voice is so soothing. Math with his voice is so beautiful. I love it. :)
I think that mathematicians over-sensationalize this particular sum because the -1/12 result is not based on traditional sums. You have to treat the number in a certain way and relate it to other functions in order to get this. The problem is that this isn't communicated in this video or by any of the others that talk about this sum.
It is mathematically interesting, but it doesn't mean you can literally add up all the natural numbers and come up with a negative fraction.
Oh my goodness. I’ve been watching these, thinking “this guy reminds me of a professor I had at uni, who taught me vector calculus, further calculus, and more; and he took a 4-person study group every week that I was in.” Then I look at my uni records, and.... he was called Ed Copeland!!! I can’t believe it! Mr Copeland, we had great times in Brighton, UK!! BIG UP THE UNIVERSITY OF SUSSEX!!
I hope he got fired after this abomination of a “proof”
@@ethanwilliamson782 only because u cant understand it?
@@snoxh2187 it's clearly wrong, nothing to "understand" here
This theory was proved by Indian mathematician Srinivasa Ramanujan.
Somewhere in the process of adding/substracting infinite sums the equal sign does not mean "equal" anymore. All this demonstration is about getting the Zeta function "extendable" to -1. The trick is that they keep using the sign and word "equal", while they shouldn't (that's why physics is physics... always making nice asumptions but using bad mathematical language). The Zeta function with -1/12 added isn't "equal" to the infinite sum anymore, it is more than that (a new mathematical object).
Let's take an example. On both sides of a river there are two similar roads. But you can't get a car above the river until you build a bridge. But the road with the bridge is not the sum of the two previous roads anymore, it is a new road. And while before it would have been absurd to say that you can be driving a car above that river, now you can even stop on this bridge, and keep the car parked here.
Brady parked his car at -1 (the river) on the Zeta function, but before that, he made a mathematical bridge appear that "fits" between the two roads so that his car does not sink ;)
that example is just really bad
This makes a certain kind of sense to me; thanks.
Beautiful handwriting!
My Calculus 2 textbook says that answers like these are "absurd" because the sequence diverges.
Don't get me wrong, I love concepts like this and am all for the sum of the natural numbers being -1/12...but what justifies the addition of infinite series?
Thank you!
i dont think you would get a propper answer in a calculus 2 book (specially for this kind of math)... you should look for complex functios and analytic continuation. And i think there you may satisfy your doubt.
Im sorry for my English, im from Argentina and its 2 o'clock in the morning... 😁
Chris Bandy I think your misinterpreting your calculus book. They're probalbly talking about the limit of the series, this isn't the limit, its assigning a value to a divergent infinite sum.
Divergence is not really caught in high school mathematics just as complex numbers are not caught in junior school where possibility of square root of -1 is termed absurd. :) There are ways to get sums like this. Each divergent sum like this also has a type and based on that type there are ways to get to the sum. Watch the video on Grandi series (1-1+1-1+1-1.....). There too, the sum (called Cesaro sum) of the divergent series comes out to be 1/2.
Or ist just comes out as basic garbage, because the series 1-1+1-1 has no limit, If you where to apply the Theorem of Limits, you need first to proof that for every epsilon around said limit, you get a certain n, after which the series does not leave an intervall of epsilon around such limit. Which you cannot for the sum of 1-1+1. So if no limit exist, the sum cannot have fixed value. Only if there is proof of a sum being convergent, you can than apply summation of know, convergent sums to determine its value.
1:47 shouldn't the range of x be -1
In the strictness sense, no you can't. But at the same time then the sum 1 + 2 + 3 + 4 + ... doesn't work either. To give meaning to these diveragent sums you can do things. This is what is called Abel summation; in the fullest rigor you take a limit as x -> -1 which gives you the sum in question. Then you do other trick, such as "analytic continuation" with the zeta function thus you can "uniquely" define the sum 1 + 2 + 3 + 4 + ... = -1/12
You're right - the range of x has to be -1 < x < 1 for the summation to converge. End of.
"But 1 + 2x + 3x^2 + 4x^3 +... = 1(1-x)^2 is only valid for |x|
The thing is that the series DOES NOT approach -1/12, it IS -1/12, this is why you can't get it by doing a limit or a summation, your result not only will be imprecise but ultimately incorrect, it only makes sense as a whole. For every other application, INCLUDING when it approaches infinity this series does not equal -1/12 and behaves "normally".
Another way to see it: no function nor series is continuous at infinity, and we know that in non continuous "points" the limit and the value often don't match, these calculations allow you to find not the limit, but the actual value at infinity.
This video has been completely discredited. I believe even these guys discredited this video.
In no way shape or form does 1+2+3+...=-1/12. Infinite sums such as 1+2+3+... are thoroughly discussed in a 2nd semester calculus class in a chapter on sequences and series. 1+2+3+... is an example of a "divergent series." It's an undefined expression something like 1/0.
The way I look at it is that generalization is a tool we use to give meaning to concepts that cannot be described otherwise. Consider x^2=-1. You could say that it can't be solved, because there's no such real number. Or you could imagine such a number, and call it i = sqrt(-1). As it turns out this imaginary concept solves a lot of physical problems, such as describing a rotating vector, and it opens up new ideas, such as frequency domain analysis, which has a real physical analogy. For example, you can sharpen your photos, thanks to the introduction of a new concept that is supposed to be imaginary and not real.
There's no such number as infinite. Something is either convergent, when it clearly tends to get ever closer to a real number, or it's divergent. Just like sqrt(-1) didn't exist before, 1+2+3+...+inf doesn't exist, either. However, a genius called Ramanujan from India had invented a special summation, a way of assigning a value to divergent series. It may be terribly confusing, but if it allows us to describe something in the real world, then who's to stop us from doing it? It's certainly done in a logical way, by generalizing something that was meaningful before.
For example, the factorial is easy to define for integer numbers, but what about real numbers, or complex numbers? Well, who's stopping us from defining a function that we call Gamma, where Gamma(n) = (n-1)! if n is positive integer? All of a sudden we can solve probability theory problems that we could never handle without it. We just introduced a new concept that solves something real.
One of the best comments I've ever seen on a RUclips video. You truly "get it" when it comes to mathematics.
*****
I agree with muffins. This is by far the best comment I have ever seen in a youtube video. Thank you :)
***** You sir, made my day! Thank you for the best insight. You would make a great teacher. :-)
Rob Laquiere Generalization is usually applied to definitions (e.g., we can generalize "product" from taking two inputs to taking any finite - or in some cases countably infinitely many - number of inputs).
Generalization is not applied to properties, such as the commutative property. Properties of generalizations of definitions must be proven for each generalization of the definition.
Also, no one will ever "prove Ramanujan wrong" about Ramanujan summation. What could happen is that someone could come up with a different generalization of the sum of an infinite series which is more useful. But if this happens, it won't prove Ramanujan summation to be wrong; rather, it would simply be a more useful generalization.
MuffinsAPlenty
Can you please define "generalization" so that I can be sure we are on the same page?
When I first started to learn mathematics, it appeared to me that the basic laws of Algebra were in fact generalizations of Algebra. Just like x*y = y*x is the general form of the law of communicability of multiplication. But just like I stated in my post, this generalization is wrong in certain fields of mathematics! If that is not what you consider a generalization, then I need you to clarify your definition of "mathematical generalization".
The properties of mathematics are generalized statements about how the math works out (properties = generalizations). No properties have been proven for all possible cases, as some families of equations are infinite in size (infinity cases cannot be checked). Therefore generalizations are not proven for all cases simply because they yield sensible results for some cases. Like in my example, the generalized form of the law of communicability of multiplication breaks down when vectors are multiplied. It is therefore possible that Ramanujan's generalization here is also wrong (as some generalizations have been proven wrong in the past).
For the case of divergent series, the definition of divergence contradicts the result they got. To me, checking the results empirically is more sensible then trying to convince ourselves it's true. As a physicist, I tend to check results to make physical sense first, as that is indeed the goal of mathematics; to count real things and to achieve empirical results.
Lastly, as I already stated, the generalization may very well be true... But I need to see this verified in reality (with tangible things), until then, it's just numbers written on a page in a particular way (not proof of anything really).
"Yeah, that makes sense"
"Am I right?"
"No."
the problem with the zeta function argument is that the function only works with the condition that s>1 or the series won’t converge. the analytic continuation of the zeta function is ζ(s)=(2^s)(π^(s-1))(sin(πs/2)) Γ(1−s)ζ(1− s). notably not the same as the original function. plugging s=-1 into this will give us -1/12, however because this is not the zeta function, simply a continuation of it, we can’t expect the same result from the original zeta function.
finally some more difficult stuff :) thanks numberphile!
The fact that an unlisted maths video on RUclips can overwhelm the view counter gives me some faith in humanity...
Hi five! People all around the world love interesting things. It is this curiosity that had driven us to amazing things. Don't let the loud and disgusting actions of mankind elude the quite and amazing things that people do daily.
Captain Enthusiastic For Humanity flies away.
Well that video presents absolutely shitty mathematics.
It is not just wrong, but much worse than that.
but its not even wrong
as i understand it, to put it in layman terms. if you added all the numbers together the resulting integer is so huge its dense and it sucks in on itself into another dimension. it then poops out gummi bears. joy
lol
Can somebody riddle me this? At 1:45 that result is only valid if abs(x) < 1
But then at 3:40 ... "let x = -1"
Doesn't that break it all? I know I must be missing something obvious here
dazz This is a method in complex analysis called analytic continuation in order to extend the domain of functions. Without that context, yes it breaks everything, but within this context it is okay. Obviously, 1+2+3... is undefined but in order to assign it a value, this framework is used. In the same way that no number squares will equal -1, we create a new framework to allow us to find roots to polynomials with no roots.
@@HWEWSWEW finally now I can sleep
13:08
If you press 1+2+3+4+5 ect for ever you'd never press the equal sign now would you? If you did that that means that you'd hve stopped
+DrDress but as you kept typing 1+2+3+4+... to infinity, with time... lots of time you'd get so bored and go so crazy that you would catch yourself typing -1/12 over and over again.. then realizing that you're now headed towards negative infinity. so then you would try to recompense by typing +1+2+3+4.... all over again and just consider giving up and when you finally did you would find out you gave up at the exact same point you began to start typing -1/12 (the point you lost your mind) and the calculator would read -1/12 . then you'd realize you where still typing. and that number would just haunt you all the way. for eternity..... its the devil....
+halofan313
What an d thing to say...
+halofan313 I find ur explanation really convincing
Yeah that makes sense
DrDress
The reason why you can't simply sum over the natural numbers is because the complexity of addition is optimistically O(n) and the infinite sum grows faster than linear time. This is why it is difficult to literally concieve a value like -1/12 however, analytic operations are abstract and have no time complexity. This is what I believed allowed Riemann to circumvent the divergence at s=1 he looped around the complex plane. An operation like this has no algorithmic reasoning therefore a solution can only be made via analytical operations. Essentially, there does not exist an algorithmic approach to resolve these class of problems. Q.E.D
Abandon all common sense, ye who enter here. This is realm of logic alone.
Brady, could you please explain how he assumed x=-1 at in that infinite series (3:56). as far as i know its -1
The error in this proof happens at 1:48. He says the polynomial sum is valid for x
+terminator10111 is absolutely right ! Just this small mistaken assumption and you end up with a huge and absurd result. So what does the guy in the video has to say about this ?
The formula 1 + x + x^2 + x^3 + x^4 + ... = 1 / (1 - x) work only for -1 < x < 1. So, x cannot be -1. Ionel
Ionel Dragomirescu Yes, but the value approaches 1/2 as you let x approach -1. Since in the framework of all of this stuff everything varies continuously, that is enough.
Colin Smith That's also what upper division math professors tell you, because the series does not converge, because the sequence does not converge, which is easy to prove. If they want to assign number values to divergent sequences, fine, but that's not mind-blowing.
lizokitten2 I didn't say anything about the Taylor series. Anyway, they both converge to the same answer from the left as you approach x=-1. This fact is sufficient by what we know about the continuity of the Riemann Zeta function etc., as I said before.
The series you're referring to is the Taylor series.
The functions they are using are not analytic for the values they are plugging in. That's the problem. Some people here are saying that they are simply creating a different, consistent system by ignoring where functions are analytic, which is interesting, but if that's the point of these videos, then what they're doing is misleading, not mind-blowing.
how have you put -1 into the formula for sum to infinity? the limits of that formula are -1< x
Well technically, the limits of the formula for the sum of a geometric series are not -1
I was wondering the same. I was taught that the sum he uses is only defined for the absolute value of x not x. As in its defined in -1 < x < 1 as you said.
Exactly what I was wondering. Can someone explain this?
After doing a bit of reading ( and I do mean only a bit) I think its not supposed to be be "equal" to in the usual sense but is instead just meant as a property of the sequence and how the sum of it tends to infinity in a different way than how the sum of other sequences tend to infinity.
Around 17:45, Guy #2 admits that it's a "trick" called "analytic continuation", which apparently involves finding a convenient formula for a function that's valid over a certain range and then arbitrarily extending the range over which you can use it. So apparently it's OK to break basic rules of math as long as you give it a fancy name when you do it. :)
People this isn't a finite sum, it is infinite. The second you put this in a finite realm, it ceases to be true.
+Logan Hollis You're bad at math.
+evilkillerwhale 2+2=5
It doesn't need to be true if it's useful.
But you can't just remove a divergence! It tells you that there is no limit, no one answer it tends toward! So how is that valid? The obvious way to approach 1+2+3+4+...+infinity is infinity plus any finite number, an infinite number of times--which will continue to be infinity. I just can't accept this one.
Much better explanation than the first video. Thank you!
I am a 6th year PhD student in Physics and Electrical Engineering, but I double-majored in Physics and Mathematics as an undergraduate, just to give some background on my expertise (or, more likely, lack thereof). A few thoughts:
I notice that both Dr. Copeland and Dr. Padilla are physicists, according to the video description. My math professors held up their noses at physicists for their cavalier approach to mathematics, particularly for switching sums and integrals and their handling of infinite series like these without regard for rules and the applicability of formulae. I would be very interested in hearing what a strict mathematician had to say about this result, particularly because we simply "accepted" the insertion of x=-1 into a formula with applicability for |x|
And what about now? still accepting it? I mean after 4 years from your comment.! did you get a mathematician opinion on that? I really would like to know :)
Regards
It is very simple: What they present here is wrong.
Even reordering convergent series leads to arbitrary results, as Riemann showed. Plus, there is a proper definition of convergence of series, which is that the sequence of partial sums converges, hence their statement: 1+2+3+...=-1/12 is simply wrong. Additionally, the Zeta function is not defined for numbers whose real part is smaller than 1.
What actually happens is that you can use analytic continuation to extend the zeta function to numbers with real values smaller than 1 and for this continuation, the value at -1 is -1/12. But this has not much to do anymore with 1+2+3+...
@@adhamm5503 Well the natural numbers with the addition is a group, meaning if you sum natural numbers you can only get a natural number.
Moreover a positive one if all the numbers you add are positive.
If you find something else, it's either you are wrong, or what you say is addition is in fact something else (which is what Ramanujan meant).
@KarlDeux A bit of a correction, the set of natural numbers is not a group under addition; there's no inverse elements at all!
However, the set of natural numbers including 0 is what's called a monoid. That is, it's a set equipped with a binary operation that satisfies all the group axioms except for the inverse axiom. Specifically, this tells us that adding two positive integers yields another positive integer.
@@xavierstanton8146I did not want to have the fish drowned but thx to have done this for me.
Loved this. But Brady, you desperately need a parfocal zoom lens.
I need lost of stuff…. if you want the unedited footage as "extras" there is a price to be paid! :)
+Numberphile lots*
+Numberphile lol - love to see the video of you guys working out who gets what, from the RUclips payment for the views.
I bet the maths is extreme.
The person in the pink parka at 16:44 gets £0.5772156649
labibbidabibbadum lol
I just received the brown papers from this video - thanks Brady :D
As soon as I get back home (I'm abroad for work) I'm going to frame them, hang them on a wall and send you a pic ;)
Look forward to seeing the pics!!!
+Numberphile it's been over 2 years, have you seen any pictures?
its over two years an two months!! still waiting? :P
elave16 for what?
For picture confirmation!
Ed: for this abs(x) must be less than 1
... few moments later
Let x = -1
Wtf???
pitreason -1
@@viannedemirel yeah but abs(-1)=1
N word
@@ViratKohli-jj3wjAfrican-American is the proper term.
It's so fascinating to listen to Ed Copeland!
To those who see the problem where he just plugs s=-1, it kinda is wrong but what I think he secretly does is so called Abel summation, where you look at the limit when x approaches -1 it actually gives 1/4 so it's actually true. Although in this video just plugging x=-1 doesn't work but with Abel summation it gives 1/4. Google Abel summation for more information
pantopam
You can always justify any BS in retrospect.
Mathematics is about being clear and defining what you are doing precisely.
The video is just wrong.
I think this type of result should be referred to more as "there is a function f so that f(1+2+3+4+...)=-1/12". It's not a summation in the conventional sense, but we could invent some kind of "divergent sum space" (of which 1+2+3+4+... is an element) which is the domain of this function and the co-domain is the reals, or the complex numbers, or other sets.
actually it is f (-1)= 1+2+3+4+5+6...=-1/12
Muhammad Abdullah i think you misunderstand me
+Davy Ker I think you are right. What is bugging me is that this whole process only works if you assume that the limit of the alternating series 1-1+1-1+1.... etc is (1/2) altough it is clear that this series has no limit. And if you watch the Video where he tries show us that the limit of this series is (1/2) you can point out some major flaws in his proof.
The limit of that series is not 1/2, but rather the cesaro convergence of the series is 1/2 --- by taking the limit of the average of the partial sums, you can in a way assign a finite value to an otherwise "normally" non-convergent series. So theyre kind of not giving you the whole picture as they dont say "1-1+1... is not equal to 1/2 in the traditional sense of summation, but we can provide an alternate description based on some characteristic of the series, and say the series is equal to that description."
thats exactly what i said on another video!!
Brady, please ask the professor the following: In the beginning of the vid, the prof offers to prove the convergence formula for geometric series (a+ax+ax^2+...ax^n= a/(1-x), in our example a=1, so it's just 1/(1-x), as the prof stated) and you said we should just accept it. If he were to prove it, one thing would become very obvious; the convergence formula only holds for -1
...
hes not claiming that its true,he just says that it is the thing that euler did
He plugs it into a differentiated formula to which the limits no longer hold true, and it's a hypothesis that euler used.
Look into Borel integral summation. You can rigorously write it in a form where instead of it being convergent at |x|
Please make a video explaining how a series that diverges can equal some value that's not infinity.
Strictly speaking, the series diverges. It does not have a value. However, we can regard the geometric series as a function (this is the whole 1/(1-x) bit) which does converge for some values of x. Then we analytically continue (a process in complex analysis that gives us the unique function that covers a larger domain but agrees with our original function on the smaller domain) this function to other values. We take whatever value our analytic continuation gives and treat the series as having that value. But strictly speaking all the equals signs here aren't really equals signs in the same way that 2+2=4
Thanks Anytus2007! So 1+2+3+4+5..... doesn't really equal -1/12?
Not really. We can treat the series as having a value of -1/12 in many calculations, but this is why in the Polchinski book that they show, Polchinski uses '->' instead of '='.
So it is tending towards -1/12 but does not really equal -1/12?
It doesn't even tend toward -1/12. It tends toward +infinity. Again, the series does not converge. Its just that there is a consistent way of assigning this series a value and that value happens to be -1/12.
What amazing (and unexpected) results. Great video by the way, and the Dutch angle makes the video even cooler haha
I guess the thing that makes this the most confusing is that I assume that if you add an infinite amount of numbers the answer would be infinity. And if the answer is -1/12, then what happens if I add the"next" term in the sequence, as the sequence can never end?
No idea why this video is still up. It is easily the numberphile video with the most wrong statements I have ever seen.
Maybe someone should have told the guy that an implication with a false assumption is true, regardless of the truth of the conclusion.
You can make a video about this, but then do it properly and introduce analytic continuation.
4:10 this is incorrect. for the sum of an infinite geometric series to be 1/(1-x), |x|
+xxBIGBIRDxx it's the Taylor series expansion, nothing to do with what you have said
+OneSixteenMike same im curious
+xxBIGBIRDxx he took the derivative of 1/(1-x), he had already established that the sum of an infinite geometric series is a/(1-r) [which in this case is 1/(1-x)]
+OneSixteenMike It's basic calculus. Look it up and prove it to yourself. Don't be an idiot.
+crazymuthaphukr theres always that one guy that decides to take it personally... congrats crazymutha
I find it really interesting that 1 - 2 + 3 - 4 + 5 -... = 1/4 which ( i realized since I just came from some videos on it) is the point at which real numbers begin to "escape" from the mandelbrot set or whatever. Makes me wonder, where does -1/12 fall on the mandelbrot set? I'm certainly no mathematician but its fun to think about these things.
So if instead of setting x at -1, you set it at +0.73(it's still less than 1), would you still get -1/12, or am I breaking math again?
Numberphile, I've got a question. It might even be a silly one.
At 1:45, Ed says that the sum of a geometric series equals to 1/(1-x) strictly when x < 1.
However, in some textbooks, we see abs(x) < 1 instead of x < 1, which would render that formula not defined for x = -1 also. It would give as domain, the open interval -1 < x < 1.
Why is it that makes it possible to consider all values x
+Akira Uema He did say that we will push the boundaries. Anyways, onstead of -1 you can think of -0.999999999999999999999999999999999999999999999999
+IsItHardToBeStupid? And in that case, the sum would be convergent. Eventually, the fraction ^ n would counteract the coefficient and it would approach a finite number. Ignoring those boundaries makes it divergent and not calculable. The Riemman Zeta function is a nice trick once you've ignored the very first restriction that was placed on your equation.
ala4sox02 and it would converge to something very close to 1/4.
+IsItHardToBeStupid? That doesn't make any difference. And no, the sum does not suddenly converge. Euler is notorious for doing things that are completely unjustified, such as putting (x=-1) into a formula that is not valid there.
But that is *not* the part where this "proof" falls apart. The problem is where the -2.2^(-s) line is split up. Instead of -2(2^(-s)+4^(-s)+...) it becomes -2(0+2^(-s)+0+4^(-s)+0+...) which could have a different value. For example 1-1+1-1+1...=1/2, but 1+0-1+1+0-1+1...=2/3.
Allow me to share my understanding of this subject.
In complex analysis there is something as a Laurent series.
It is much like a Taylor series in real analysis where you try to approach a certain function using a series or a sum.
For example in real analysis the function f(x) = sin(x) can be approached around the origin (x=0) by the taylor expansion x - x^3/6 + x^5/120 +...
This expansion which we call a Taylor series does not equal the function sin x but it very much looks like sin x around the point x = 0.
However when you go far away from x=0 this series won't even be close to where the actual function is.
We call the place where the series looks like the function the radius of convergence.
Now that we got this out of the way let's talk about the Riemann Zeta function.
Riemann never defined this function as the sum we see in this video.
The sum we see in this video is a mere approach of the Riemann Zeta function on certain parts of the complex plane. This is what we call a Laurent Series.
Again we can talk about where this series converges.
It does on most of the positive real number axis in the complex plane, but NOT on the negative number axis.
So we can say that the Riemann Zeta function of square root 2 for example will be the same as the sum over n of n^sqrt(2).
THIS IS NOT TRUE FOR NEGATIVE NUMBERS SINCE THE LAURENT SERIES IS DIVERGENT THERE AND IS NOT A GOOD APPROACH OF THE RIEMANN ZETA FUNCTION!!
So Riemann Zeta of -1 DOES NOT EQUAL the sum of all natural numbers, however Riemann Zeta of -1 does equal -1/12, and this is where the mistake has been made.
The sum of all natural numbers does NOT EQUAL -1/12.
In certain branches of physics however we try to make sense of divergent series like the sum of all natural numbers.
When we get such sums in our calculations what we can do is REPLACE this sum with a certain MEANINGFUL value and get the right answer. In this sense the sum of all natural numbers is equivalent with -1/12.
We can get to these equivalences by 'bending' mathematics and going outside the well defined 'rules', however we haven't yet found a good explanation to get to these equivalences.
I hope this brings light to some shades created by this video. :)
Thank you for this explanation. Without it this video doesn't make any sense and is misleading. Which is probably done for sensation. It's just click bait and unworthy of this channel
It looks kinda like the student in the background at 20:20 is walking by with a light-saber in his/her hand...
Some people were quite upset that the material in these extras wasn't mentioned in the original video. I guess editing cut some key definitions.
At about 12:50, Dr. Copeland refers to "removing the divergence". The problem is that the only way to do this is to not use infinite series. Infinite sums diverge or converge. About a minute later, he asserts "you've got to deal with divergent numbers- divergences very carefully", but the way he "deals" with infinite series wouldn't be considered "careful" even in Euler's time, as Wanner & Hairer note in Analysis by its History. Of another divergent infinite series that Euler wrote equaled a finite number, they state "its mathematical rigor was poor even by 18th century standards", So why are we repeating mistakes made in the 1700s?
Even a defender of Euler's ridiculous equations, Sir Roger Penrose, admits "..,the rigorous mathematical treatment of series did not come about until the late 18th and early 19th century...Moreover, according to this rigorous treatment [Euler's equations] would be officially classified as 'nonsense'."
While a rigorous treatment of the zeta function (or any other topic in complex function theory) is obviously beyond the scope of a RUclips clip like this, that doesn't entitle one to make patently false claims while ignoring one's own advice just to be able to make a mathematical claim that's seems mind-blowing (to the mathematician, it's only "mind-blowing" for its fundamentally flawed basis).
I used to harbor a huge dislike for Maths when I was in high school and I simply feel the need to say that through the years this dislike has turned around 180 degrees and I found the maths in this video simply beautiful.
The Universe is so stunning.
The maths in that video are BS though.
Willans' Formula for primes:
2 to the n part = vertical asymptote. 1/n part = vertical tangent. Factorial part = vertical line. These tensors from differential calculus determine singularities in stable matter as represented as primes.
What if you reverse the sign of the sum so that it's 1 - 2 - 3 - 4...? Does it go to some interesting finite number?
Lets see, that would be (-1/12)*(-1) + 2 = (2 and 1/12)
1/12 + 2?
maybe it's -1 ( 1 + 2 + 3 + 4 . . . ) = -1( -1/12 ) = 1 / 12
No, it's - S + 2. 1 - 2 - 3 etc. = 2 - (1 + 2 + 3)
if you reverse the sing you would get -1-2-3-4. Just a little edit here...
"Yeah, that makes sense doesn't it..."
"Am I right?"
"No."
2:51, you also need the sum rule of derivatives (f+g)'(x)=f'(x)+g'(x), on top of the power rule f(x)=x^n (n is constant with respect to x), f'(x) = nx^(n-1).
Ok seriously.
What the shit?
Maths are broken, I want a refund.
Does this work in any number base, or would you get a different answer if you did it in, say, base 8?
In a mathematical "proof", the first erroneous argument nullifies (neither refutes or supports) all subsequent arguments, and thus the conclusion is unsupported.
I would like to hear more examples in quantum field theory where this property plays out. String theory has a bit of a reputation for being nice in theory, but unprovable (from what people who have researched it more of told me). Quantum mechanics is something we're getting experimental results out of, and I'd love to hear about one (or more) which were predicted by a theory built on this sum.
As a physicist, I have to ask, why do you need to see such things? This result is a mathematical fact that can be made totally rigorous provided you have the years of training to talk about analytic continuation and zeta functions. Adding physics is just muddying the waters.
Another Numberphile video has a demonstration that really helped me understand this. First he explained the sum of 1+1/2+1/3+1/4 etc, which is Infinity. Imagine an ant standing on the edge of a rubber band one meter in diameter. Every second the band grows one meter and the ant advances one centimeter. By the time the ant gets all the way around, the band will be infinity large, but it will eventually get there.
Conversely, for the sum of 1+2+3+4+5 etc, which is -1/12, imagine the rubber band is infinity in diameter with the same ant moving one cm. per second while the band shrinks one meter. The closer the ant gets to going all the way around, the smaller the band gets, until it has to shrink into itself, i.e. become a negative number.
Once I envisioned that, it made perfect sense, despite seeming like it should be the other way around :)
Based on 3:38.. 1+ 2x^2+ 3x^3 + 4x^4 +... = 1/(1-x)^2
Then why not set x = 1 to make series as 1+2+3+4+.. and its answer as 1/(1-1)^2 which is infinity..
Series sum as -1/12 doesn't make any sense. How can you keep adding something and end up deficit??
Firstly he stated that 1/1-x is true for x
i am agree with you.
That's actually how Romanian came up with his "prove" because he was not formally educated in math. This concept is similar to imaginary numbers. Square roots of negative numbers do not exist but we pretend that they do. For sanity, sum of natural numbers is not equal to -1/12, it is only a concept.
It's like saying √-1 is i. It's breaking the rules in a clever way to do what you want anyway, and yet, you get real results out of it that actually work and can be experimentally verified (and have been).
@@Invisifly2 yes, it is called ramanujan summation in case of the sum, not a conventional number. Imaginary number is another convention, and cannot be experimentally verified.
@@Invisifly2 how have they been "experimentally verified?" can you show me an instance where the sum of natural numbers = -1/12 has been useful for describing things in the real world?
This proof is still incorrect. The faulty part is where he says that 1-2+3-4+5.. =1/4 based on the power series 1+2x+3x^2+...=1/(1-x)^2. If x = -1, then the LHS does not exist. In fact in the limit the power series oscillates around approx. 0 for say x = -.999.. But saying that for x=-1, that the RHS = LHS = 1/4 is totally preposterous, and is equivalent to dividing by zero. In fact the limit DOES NOT EXIST at x = -1 and saying that this is 1/4 is just pure lying!!
Watch the main video, they explain perfectly why that function will equal 1/4, using algebra.
The left hand side doesn't exist under the simple rules of limits taught in precalculus (the series' partial sums diverge, so it has no limit). Fortunately, people like Euler and Cesaro and Borel figured out how to sum divergent series hundreds of years ago, thus generalizing things like the power series limit for cases that had previously been undefined.
It may exist, however, when the function is expanded to the complex plane, as the actual [formal] proof of this involves the reiman-zeta function where s is a complex number.
Question:
We have a test to see if a series is divergent or not
r=(An/An-1) where An is any number in the series and An-1 is the value before An.
when r < 1 it converges to a number
when r > 1 it diverges
when r=1 we don't know
when we plug in this series, r is clearly greater than 1 which means it diverges to infinity
why doesn't this series follow the test?
He notes that this (1+2+3...= -1/12) is critical in string theory. That's not a reason to believe it's correct. Has there been any observations to predicted by string theory that depend on this? If not, maybe we have a problem with string theory.
Euler...well, he was pretty smart. But what did he think of his result? Was he serious? Was it an erudite joke of his? And he was a genius at convergent sequences...
For this to be serious, we must believe that an infinite series may have a finite sum even if the series does not "converge" in the classical sense.
What would Gregor Cantor think?
The first series given by Ed (that being 1-2+3-4+...=1/4) is actually divergent. The expansion of 1/(1-x) is invalid for x = -1. It is in fact only valid for -1 < x =< 1.
But I have a feeling that I don't need to say this.
Someone please answer this cause that's what I was thinking too.
I thought the same..
1:47 minutes in and there already something wrong : this is |x| < 1and not x < 1.
Which makes the "oh let's go with x=-1" at 3:40 already a Game Over
There is nothing in the sum that would ever turn it into anything but a huge number, it's an infinite sum of ever increasing positive numbers, there is no way it can do anything but go towards infinity. Something in those manipulations HAS to be wrong. Maybe in the future it will have the same status as Zeno's paradoxes have now, something interesting and clever to ponder, but ultimately wrong.
It's no less correct than 1+1-1+1-1+1-1+1-1+1-1+1.... = 1/2
The problem isn't that the math is wrong, it's simply that it can't be explained by simple algebraic rules. There is lots of correct math that can't be explained by simple algebra. It's why we have higher level mathematics.
Greg Vanderpool That I can accept, the answer alternates between 1 and 0, infinity is not even or odd so you average both values, it makes some sense, but for an infinite sum of positive numbers to result in -1/12 makes no sense whatsoever to me
except that physics and observations in the natural universe has actually proven these sums.
Gerardo Lozano Except you don't actually take the average. That's a simplification to make it more understandable, not what's actually happening in the math. You can't divide by infinity, which you would need to do in order to take the average.
CrazedPorcupine Sorry, but string theory is not part of the standard model, the whole theory could be abandoned at any time, so the fact that it's used there adds little to it's credibility for me. As for it's role in more solid phenomena like the casimir effect, I don't know how vital this is to that, so I can't just accept it because of it.
Thank you for the second explanation! Analytic continuation... I can buy that because it's not saying that this is actually valid, but that it is a 'what if' situation (just like imaginary numbers, where we said 'what if the square root of -1 existed?', and it has been a HUGE help in physics). Really wasn't happy with the first part, at 6:35, when we just started multiplying bases together because they shared a common exponent, but kept watching hoping for someone to make-up for that.
At 1:50 the result we get is sum of an infinite GP, which as he says is only true for 0
"strictly true for x
This is actually making me a bit angry. The sum of any two integers is an integer and the sum of any two positive numbers is a positive number. How an infinite series of positive integers can be equal to a negative fraction is beyond me. Tony's explanation starts with something that doesn't really have an answer or has two answers, so lets average them, and then builds on that as if the average were really the answer. How is it not undefined like 1/0? It's mad, and it's maddening.
To be formal, it to the existence of the zeta function ζ（z), such that if you
evaluate it at z = -1 that gives you -1/12 and if you analytically
continues the expression on the positive real plane to -1, you get the summation of natural numbers
To be less formal, it means that if you get -1/12 in a certain context, like manipulating
the zeta function at -1 and doing Taylor theorem, your calculation in complex analysis is
wrong and by the uniqueness of Analytic Continuation of ζ(-1), the correct answer should be
the summation of the natural numbers.
"this result is critical to getting the 26 dimensions of sting theory to pop out" made me smile.
By extending the zeta function through analytic continuation to negative integer values you forfeit the ability to write that the sum of positive natural numbers equals zeta(-1) and therefore the sum of positive integers in no way equals -1/12. It is an absolutely divergent sum. The methods in which are used in this video to prove this result are simply incorrect. This video is extremely misleading and condescending towards its audience.
I have to object. the answer to s=1-1+1-1+1... has to be either exactly 1 or exactly 0. The average of those two is 1/2, and the median of the possible answers would be 1/2, but the actual amount must necessarily be EXACTLY 0 or EXACTLY 1. The number is quantized.
A similar example is that the average number of children in a household might be 2.3 children, but the actual number of children will always necessarily be a whole number.
Since the entire solution is based on the incorrect assumption that the expression can have a solution that is not either 0 or 1, the solutions for the other expressions are necessarily wrong as well.
WhereWhatHuh Umm... wrong video.
It seems that a lot of people (myself included) have a problem with this bit of mathematics, but for now I am willing to accept it for the sake of asking another question I have.
Is there another possible answer for the sum 1+2+3+4+... ? What I mean is, if we tried to use a different method to "simplify" the infinite sum, could we not get an answer other than -1/12?
Also, does every divergent infinite sum like this have a finite equivalent solution? Like what is 1+3+5+7+9+... for example? And is there only one possible finite value for this sum?
If you apply enough mathematical slight of hand, you can make it equal anything you want.
1 + 2 + 3 + 4 +.... tends to infinity.
@@jdw6925 True, it does tend toward infinity in the sense of a limit of partial sums, but there are plenty of logical holes in this video that I wish they would’ve covered in more detail (I’m acknowledging this even as someone who loves infinite sums and analytic continuation); there are plenty of proofs (or “proofs”, I suppose one might want to say) that shed more light on this topic, and I would implore you to check on those that go over more rigorous proofs because they become truly fascinating.
Some people here seem to have tremendous difficulty understanding how mathematics progresses. They would probably be happy saying x^2 = -1 has no solution, end of story. Hell, many of them might even call game over just with x^2 = 2, I mean, solving that would be completely irrational right?
+Veon1
Although this response is 4 years old, I would just like to explain why this is wrong.
Saying the sum of all natural numbers is -1/12 due to the zeta function is just objectively wrong. -1/12 is the evaluate of the anayltic continuation of the zeta function. Meaning, whatever is causing -1/12 is now due to something other than the 1/n^s equation. Analytic continuation is what you get when you assume what the graph would look like if you asked yourself "What WOULD the graph look like if it was defined for values below 1?"
You get it through a transformation.
X^2+1 has no solution in the real numbers.
If you had ever attended a maths lecture, you'd know that things like real and complex numbers are defined precisely and justified rigurously.
You don't just suddenly throw all of your definitions out of the window.
If instead of setting x = -1 into that intermediate derivative series you set x = 1 you get sum(n=1, inf) n = 1/0 which runs directly counter to the alleged result.
Also there are very real mathematical problems with evaluating any series that can be rearranged to infinity - infinity.
+Anthony Purcell No. 1/0 isn't infinity, it's undefined. Those are different things. You can say that for 1/x, as x tends to 0, 1/x tends to infinity. But that's NOT the same as saying n=1/0= infinity.
Learn some math.
My mistake, I had assumed that the mathematically literate would not need me to go into such pedantic detail. I should have remembered that this is the internet.
Allow me then to elaborate.
If you substitute x = 1 into the first equation you get a quantity that is larger than any finite number, though perhaps not as large as ***** 's dickishness.
+Anthony Purcell LOL NERD THINKING THAT LOGIC IS USED ON RUclips COMMENTS! YOU HAVE BEEN SEVERELY MISTAKEN!!! Sorry I had to lol
+Blake T It's an occupational hazard
+evilkillerwhale Would you mind terribly pointing out the bit where I claim 1/0 is infinity?
3:43 "OMG, did he actually just set x to -1???"
lololol
What
The more people argue, the more laws of math they violate. He also gives wrong information about zeta function. (such as using the given representation for s=-1) He also assigns values to alternating sums and changes orders of the numbers like it is very acceptable. If that proof is valid, 1 = 0, and numbers mean nothing.
John Baez makes a surprise appearance at 15:50!
Rule 1 math: don't make assumptions and go wild with it into nonsense
That is the opposite of how math works. Math is entirely based on making assumptions and then working out what the consequences are. That's why there are different systems of axioms, ZFC is just the one we are familiar with. You can make any assumptions you want in math and try to build up from them, and you should because otherwise your mathematical framework will never advance.
Is there a proof to show you can use -1 which is outside of the allowed range?
Correct me if I am wrong, but:
1 + x + x^2 + .... = 1/(1-x) for |x| < 1
If we derive it, we cannot use the equation for x = -1, can we?
According to Wolframalpha I am right:
wolframalpha.com/input/?i=derive+sum+x^n+%2C+n%3D0+to+infinity
(^ doesnt seem to work in links, just copy and paste)
Is the proof wrong?
I also want to see this clarified.
JustMush
Yeah, we cannot, as |-1| = 1 is not < 1.
i believe u are right, i think for x>=1 or for x
I would be ashamed were I to put up this video. I'd be doubly ashamed if I had the stellar body of work you had before posting it.
Very strange that you would be ashamed to be a physics professor doing correct math on a math channel. Are you also frequently embarrassed for doing other things right? Maybe you should talk to a professional about this.
@@eoinlanier5508 Apparently you neither read my comment nor understood the math being discussed. When such a highly respected group puts out something that violates several mathematical priciples and well known limits they should feel embarrassed.
Edited to correct auto correct errors.
@@blargblarg5657 I understand the math, but you seem to have some inaccurate preconceptions about it; what mathematics does it violate?
Let's start with the fact that their first series is only convergent when |x|
@@blargblarg5657 What rule does that break?
"I would think.....that it tends towards infinity"
"Yeah that makes sense doesn't it"
"Am I right?"
"No"
Brutal......
PLEASE PLEASE PLEASE do a video on the proof of the Basel problem (1/1sq + 1/2sq + 1/3sq + ...). I know the answer is (pi sq) / 6, but I don't know how.
Thanks
Do you know "Proofs from the BOOK" by ziegler and aigner? There are (I believe) 3 different elementary proofs, which only use the geometric series and integration by substitution.
Rene Roundthecorner Thanks. I will buy the book.
At around 2:25 the rate of change of distance as a function of time is speed and not velocity ; velocity would be the rate of change of displacement as a function of time.. sorry for being picky, but that's what it is considering the vectors.. i understand that you might be speaking in loose terms just to illustrate an example... great proof though
This all vindicates good old Bill Clinton: it all depends on how you define "is".