Why -1/12 is a gold nugget

Why -1/12 is a gold nugget


Is this good? The position?
More like this, right? O, it was you who made that video?
I see. Okay. Well, I think that mathematicians learn this stuff at some point. And I learned it at some point.
And then you kind of forget. So you kind of put it in a box somewhere and you put it in a closet. And you classify it under stuff which you have already understood. But I think actually, we don’t really fully understand what’s happening here. Let me recall what we are talking about. One plus two and so on. And the question is: What is the answer? Well, at first of course you say:
“There is no answer” or “The answer is infinity”. And we say that because this series is what we mathematicians call a divergent series. – It’s blowing up
It’s blowing up, You get larger and larger. So there is no sense of it getting closer to anything. – So traditionally, what do we do with a divergent series? We just ignore them. We just ignore them, we just throw them away. The question is whether this is the right approach? Whether there is actually something we can say about such a series, which is meaningful. In other words. “Can we assign a value to this series which is meaningful?” – Professor, is it not meaningful to say that is blows up and goes to infinity? Is that not meaningful? It is meaningful in the standard context of such a series. The way you can think about it is as follows. Think of this, okay — so we have to resign to the fact that it is infinite somehow. But imagine this whole series as a huge lump. What if there is a way… What if there is sort of a nice scalpel, which will allows us to surgically remove infinity. Kind of a bad infinity out of it.
And then keep kind of a finite part. Then we would say we will assign that finite part as the true answer to this infinite series. Now, we have to realise that that may not be possible for any divergent series. For example, you could do something like 1 + 2 + 3 + 55 + 47 + 6 + 7 + 8 + something else. You know, it’s kind of like a random infinite series, which blows up. It will just blow up. There is no hope or expectation that there would be a way to assign a meaningful value to it. But this is a very special series, because you see it’s very regular. Kind of like 1, 2, 3, 4… We are actually taking the sum of all natural numbers, without any gaps, including each of them exactly once. And what’s interesting is that those kind of sums pop up all the time. In many different branches of mathematics and quantum physics. Mathematicians have thought for a long time about trying to develop a theory in which one could actually make sense of this. And nowadays we have such a theory. And within that theory we could often times say that it is meaningful to think of this sum, or more precisely, that sort of finite piece after you remove the infinite part, that sort of “a regularised” sum. Maybe we should make a distinction between sort of a “naive” sum where it just blows up to infinity and a kind of “regularised” sum, where this regularised sum actually turns out to be minus one over twelve.
– Oh no, -1/12? -1/12. So you see, it’s very counter intuitive, because it’s actually a negative number and you are summing up positive numbers. So it is certainly not the result of summation of these numbers. It is something else, but what is it? So mathematicians have developed ways to come up with this -1/12. And actually the first person to talk about this,
was the great mathematician Leonhard Euler. He was born in Basel in Switzerland. But also spent a lot of time doing research in Russia, my home country. Leonhard Euler was a kind of mathematical outlaw.
A kind of a mathematical gangster. He did things which were unlawful an
illegitimate. And in particular he allowed himself to
manipulate with infinite series like this. In other words he was trying to guess
what could be a possible way to assign a value. In the process of trying to assign
values to this series and other similar series he actually came up with the right
answers which were justified later by other mathematicians, for example Bernhard Riemann. But that was a German mathematician.
But that was like a hundred years later. So, it seems that Euler was way ahead of his time. – You can get to -1/12 in more than one way? That’s right, in more than one way.
So Euler gets to minus one over twelve in a particular way. Riemann explained later; giving a rigorous theory using his zeta function. A theory which involved things like complex numbers. Something which was not yet fully developed at the time of Euler. Although, a lot of this already existed, and Euler himself was considering complex numbers. But also there are other ways. We now know other possible ways of thinking of how to isolate this finite part in this infinite series. Euler was motivated by some questions
and in the process, he not only studied such sums. Here’s an example. What else he studied.
He also studied things like one cube plus two cube plus three cube plus four cube. That seems to diverge even faster than this one, right? Because you’re now taking the sum of cubes. You see, so again within this context of divergent series, you just approach
this the way, you know, we approach it when we study, you know, first year calculus. Definitely this blows up. Definitely divergent.
This is infinite, the answer is infinity. Period. There’s no way around it. But Euler allowed himself to do some manipulations with such series and came up with a different answer, which was… I don’t even know exactly but I think it was 1 minus 1 over 120. I don’t remember exactly I think
it was that that was the answer You might ask why did they skip squares? Actually you can do that as well, but the answer is even more surprising. You actually get zero, within that scheme and so on. So he was actually studying all possible integer powers. Powers by natural numbers like here. You know, you can think of this as one to the first power, two to the first power, three to the first power and so on Here these are the square, the cubes, into the fourth powers and so on. And so the funny thing happens, that for all even values you actually get zeros, and for odd values you get some
rational numbers. – On the original videos everyone got really upset and said you cannot do this divergent series. – Are you saying you can? – And you’re saying they broke the rules and then you say they didn’t. What is the rules here? Well the rules, it depends. Rules within
which context? Okay. So let me ask you, to illustrate what I mean by this. Let me ask a question. Does the square root of -1 exist? Come on Brady. – Well, I know that we call it “i”. We have these imaginary numbers. Right. But does it really exist? Does it exist? Does it make sense to speak of the square root of -1? One possible answer is that: Absolutely not! Right. Because if we think about real numbers we know that a square of any real numbers is positive. So the square root of a positive number
is well-defined. Square root of 0 is also well defined: zero. But there’s no square root of a negative number So we can stop there and say: the square root of -1 doesn’t exist. Anyone who uses square root of -1 is an outlaw. Right? Because that’s not legitimate. This is some dirty tricks.
But actually we now understand… And that’s how people viewed it for a long time. But actually now we understand that there is a rich and consistent theory which includes a
square root of -1. That is the theory of complex numbers. And this theory provides us with a much more interesting, much richer contex, much more fruitful context in which in fact we could solve a lot of
problems about real numbers. So in other words we have to go outside of the realm of real numbers Often times to get the best, most optimal or sometimes the only possible solutions about real numbers. So it is in that sense that now no mathematician in their right mind would say the square root of -1 does not exist. Yes, it exists, in the sense that we can add it to the real numbers. We obtain a well-defined numerical system, which is called complex numbers, which is just as legitimate as the system of real numbers. – Are you saying that manipulating a divergent series is in the same category as that? Well, I would say that’s a good analogy. Because… It shows that sometimes in a different context in which you can discuss different things. So in the case of square root of -1 there’s a context of real numbers, where square root of -1 surely doesn’t exist. Or there’s a context of complex numbers where it does exist. And it’s actually very useful.
And likewise here, there is also obvious context of, you know, the rules of analysis, the rules of calculus, the rules of infinite series in which none of these series are well-defined.
And therefore all the manipulations we do with such infinite series are not well defined. But then there is another context in which we replace this series by their sort of regularised values and I really like to think all these regularised values as sort of like, you know, removing sort of like —
imagine, like a piece of gold which is surrounded by this infinite
amount of dirt and you kind of throw away this dirt and you’re left
with this little piece of gold. So what I’m trying to say is that each
of these infinite series contains inside, it seems, this little piece of gold. And then we can say well that little piece of gold is the value, is a true value of that infinite series. And on the rest of it, it is kind of useless and we can just throw it away. If you say that, and there is a rigorous mathematical framework for doing that, then some of the manipulations, in fact all the manipulations that Euler did, become legitimate. Because what you’re doing is you’re kind of carrying with you those little valuable pieces on each side of the formula as well as those infinite things and you, kind of, you can throw the infinite things away and then whatever relations you find between the infinite series will also be the value relations between
those valuable pieces. – Professor it seems it’s very… My understanding of mathematics is: – It’s very rigid and rigorous and it’s never arbitrary. – How can you just throw away dirt and keep the gold? It doesn’t seem… That’s right. Well, in a way, it’s a great question because I think that it’s a misconception to think of mathematics as a sort of linear process, where we are only doing things which are legitimate, which are allowed. If we were doing that, we would never discover square root of -1. We would never even discover the square root of two. For a long time people did not believe that
the square root of 2 actually was a legitimate number, because it cannot be expressed as a fraction. Right, that cannot be expressed by a fraction and for a long time people thought the only legitimate numbers were fractions. Right, so actually, every once in a while, there are people like Leonhard Euler likely Riemann and others who actually… Ramanujan is another example, who kind of jumped into the abyss of the unknown, and break the rules and try to… to kind of push the veil over the unknown and try to understand more. And sometimes they are actually doing
something maybe illegitimate at the time. Maybe they’re ahead of their time.
But one thing which is important in mathematics, is that we can never leave, sort of, these things as “loose ends”. We have to find a justification.
So you’re right mathematics is rigorous. And at the end of the day, we are
looking for a rigorous justification, a rigorous explanation of everything. In other words we’re not content with just saying that there’s some magic over there. There is magic. But we always want to explain it and that’s what has happened to some extent with these series. The work of Riemann gave us a tool to analyze this, sort of, the golden
parts of these infinite series. But I still think that the last word on the subject has not been said, because we still don’t fully understand. Because I don’t fully understand why every time such a series pops up in mathematics or in physics. We get the right result by replacing it by precisely that value or by this value for this one. In physics for example these kind of calculations are done all the time. And in fact maybe, it is the best kept secret maybe in physics and quantum physics is that most of the calculations that physicists do today are like this. That the answer they get is infinite at the outset. On the face of it, it looks infinite, but they find ways to assign meaningful values to this
infinity, so to speak. And they… and that is really..
I think it’s a good analogy to think about it is, kind of like surgically removing some kind of infinite part which is redundant and superfluous and throwing it away and replacing the answer with this… What you might find a remainder.
And the interesting thing is that in physics… Physicists are still kind of waiting for their own Riemann to come and sort of… to justify these calculations but they’ve been incredibly successful in getting results which they can then test experimentally. And some of this has been tested with an astonishing degree of accuracy. – Is there a sum is assigning the value the same as the sum? Is that where things have gone wrong? – Or where things have become confused?
I would say it’s a it’s not exactly the sum because …the exact sum is, you see, you
know it blows up it is infinite. It is a kind of regularised sum.
But I am surprised for example why is it that every time we encounter such a sum in mathematics, and there’s so many
places where we do that, where we do encounter these kind of sums. Every time we encounter these sums we always have this sort of reactions like Oh, we should replace it by -1/12. And every time we do that, we get the right answer and then maybe later on mathematicians find an alternative way. You know, because in mathematics you can often… you know, there are different approaches. There are different solutions. You have a problem but you can solve it in many different ways. So that’s an indication I think that if we come up with such an infinite sum… It’s an indication that maybe we’re doing something not quite right. We’ re kind of applying maybe kind of a naive approach. But interestingly, every time it happens if we replace it by that -1/12 we get the right result and then later on we can justify and choose a different route and have a different explanation. So what does it mean?
Does it mean that in some sense it is. There is a context in which this sum, this infinite sum, is mysteriously -1/12. I’m not sure. It’s clearly, there is something in there which we still don’t fully understand. For now, our understanding is that -1/12 is a sort of golden part. It is this sort of finite part in this infinite lump which you get by throwing away some infinite dirt. I’m gonna give you an astounding result. – Astounding?
An astounding result. So I was going to write down a little sum.
I’m just going to see what answer it gives. 1 + 2 + 3 + 4 + ……

100 Comments

  1. In some way I feel like there is a lack of definition of the domain of the function itself. If the function were y=1/x then we would exclude x=0. Why is the zeta function any different? It is clear that the domain would not represent continuity over the entire dependent axis.

  2. I'm not listening to anything else you have to say mom! I'm going to be a mathematical gangster… AND NO ONE CAN STOP ME!

  3. would these be just ways to describe 'infinity' then? Because no two infinities are the same value, at least not as a subsection. I still don't know what -1/12 is indexed to or describing, whether its some kind of rate or finite product but anyway… why not then =-1/12 (infinity) and replace the bracket with some other symbol that actually makes it make sense?

  4. Cantor is another example of the prof.'s outlaws. The thing most non-mathematicians don't know is that the math that gets presented (such as in published papers, lectures, classrooms, etc.) is the "finished product." It seems clean and neat and perfect and irreducible – precisely because some mathematician (or more than one) toiled away at it (sometimes for years – even decades) to figure it out; it's clearest, simplest form, it's implications, etc. I've heard the process of doing math likened to a restaurant. Most of us sit in the dining area; we only get the finished meals. We never see the chaos that goes on in the kitchen! Can you imagine what a chef goes through? Experimenting with different ingredients, failing time and time again – until, finally, AHA! The perfect dish! That's what doing math 'feels' like. We aren't simply given the answers! We have to work at it – often fumbling around, until we get it right. Of course we're only going to present the world (that is, other mathematicians) with the completed, perfect recipe! Anything would be an embarrassment! Doesn't mean it's easy! And, as often as not, we STILL don't understand ALL the implications of our own discoveries; we may never do so in our own lifetime. That's the way it goes, kids. tavi.

  5. The square root of -1 actually does exist in the real world, and shows up every time we use capacitors or inductors. The existence of i implies that time isn't a single dimension, but two dimensions; that is, there is a dimension of time at right angles to the dimension of time that we experience.

  6. Okay why am I mysteriously attracted to this guy and I am straight as an arrow. I think I’m. A numberphile

  7. This was not invented by Euler this was invented by ramanujan
    If you don't know then go and watch the film the man who knew infinity

  8. 7:11 "A square of any real number is positive." is not correct. The correct one is "The square of any real number is not negative."

  9. Mathematicians- We can't write sum of naturals as infinity we need something sensible
    Other mathematician- Ohh right yeah -1/12

  10. 3:05 yes make a distinction.
    -1/12 is a regularized sum
    Infinity is the naive sum…
    This is what folks had difficulty with.

  11. would have loved more details about the way these seemingly arbitrary numbers get derived from these series.
    But then again, i think 15 minutes might not have been enough for this 😉

  12. Can these identities be treated as regular numbers? Like, is (1+2+3…N) + (1+2+3…N) equal to (-1/12) + (-1/12) ?

  13. If the camera man is going to speak it should have its own mike. Nothing like empty classroom echo mto improve the production quality. Other than that amateur oversight a very interesting video. Thank you.

  14. You might argue, that anything but the natural numbers is illegitimate. Once you've crossed that line, why should you even consider drawing another?

  15. i love him breaking the rules to make his points more clear to the unexperienced people like me. i know u should not say "imaginary" numbers and u should not .. and i totally get why .. but it really helps to get glimpse of what it really means. he can explain this very well imo

  16. See Ramanujam in You tube famous mathematician from INDIA invent _1/12 for the addition answer.He invent on 1919

  17. I can't grasp the limp-brained underbite numbskull who would give this presentation a thumbs down . the guy is articulate, oozes intelligence, and glows with passion for his subject.

  18. I think what professor Frenkel is trying to say is that the number-1/12 is more related to the structure of the series than to its actual value.

  19. Listening to his explanation about mathematics rigorously finding loose ends makes me hopeful that one day we’ll have some sort of answer for dividing by 0 other than “error” or “you can’t” or “it doesn’t fit.”

  20. Anyone tried scaling 2 to 2 million or some higher number to search for a more solid fraction or decimal expression of root? Scale it back and there it is….

  21. Math fascinates me but I'm not very math-y, so maybe I’m missing something obvious to everyone else… but why is there no mention of how Euler arrived at the number -1/12? Why is there nothing to say he didn’t just pick it out of a hat? Is it the only value that works for the sum of squares, or are there others?

  22. This means a growing business eventually meets bankruptcy? A rich person who became richer and richer actually ends up being a beggar?. I am not getting its physical sense.

  23. I have been thinking about the Gold Nugget for a while, and I think I have come to terms with what it actually is; namely, it is the differentiator that makes one infinite series different from another.

    The number 8 and the number 6 are different. But what makes them different? They are both even, and divisible by 2. But 6 is different from 8, despite both being even numbers.

    I see the Gold Nugget as simply the concept of the factor, so to speak, that makes one infinity different from another infinity. If you could divide an infinite series with something that magically makes it finite, you get the Gold Nugget. And that every infinite series have something that is unique to them, which is normally hidden by its infinite nature.

    If you can "cancel" the infinity from an infinite series, you get something finite that defined that number. And that finite number is usable for mathematics, because infinity itself isn't actually useful and is something that is muddling the waters.

    Or in other words, every infinite series is "Infinity multiplied a Gold Nugget" (Knowing full well that Infinity is not a number). And despite the difficulty in isolating that gold nugget, it doesn't mean it isn't there. After all, two different infinite series are usually not equal.

  24. Wow he is the first person explaining infinite number series and also telling that the result is not due to addition

  25. I respect this professor’s mastery of my language (English), his second language, in addition to his mastery of mathematics.

  26. My theory is that infinity in a given dimension actually breaks into the next dimension as a non-infinite value of that higher dimension. Then when we try to view that value in the original dimension you can get all kinds of absurd values. Just like when viewing an object of higher dimensions in a lower dimension, you get absurd and seemingly non-deterministic observations due to how you can only view lower-dimensional cross sections of the object as it passes through the lower dimension.

    This even makes sense in our reality as well. You can think of a cube as infinitely many 2d squares stacked on each other. But in the higher (3rd) dimension, there is nothing infinite about that cube. It is a finite object, but it looks weird when it passes through the 2d world. Just like why the -1/12 looks weird as it passes back through the lower dimension, because the infinite value of the divergent series requires another dimension in order to comprehend its complete structure.

    Now give me my PhD please.

    Also, the different sizes of infinite series (countable, uncountable, etc…) that sound preposterous in this dimension, simply create different size/shape objects in the next dimension, and make complete sense in that next dimension. Just like different types of infinite 2d shapes create different 3D objects.

  27. I don't believe in Mathematical gangsters. I do however believe in Mathematical con artists. Where does -1/12 appear in nature? I have this idea that higher theoretical mathematics are a confidence scam, the main goal of which is to establish intellectual chauvinism and occupy academic positions in western academia, producing things that nobody understands and have no practical utility. These people just promote themselves to the point where they dominate the ivy leagues. Once established they introduce new and useless areas of study, like gender and ethnic studies, to delude and divert young people.

  28. but for sqrt(-1) we say it's a number that's not equal to any Real number, i, rather than saying it is equal to 1. In this case you're saying it is equal to a Real number instead of saying it's equal to, say, -z/12

  29. 1 + 2 + 3 + … is just the sum of the ordinals of counting events, if there is a unit of sums of terms of infinite counts labelled 'z' then we can say inficount( 0, + ) = -z/12, but what's sqrt( inficount( 0, + ) )?

  30. 3,6,9. Key to universe
    salvage 50 ways 2 leave your lover 5150 I can't drive> (55). LIGHTSPEED= 1234,000 MPS

  31. One way I try to understand this is through the following analogy
    Mathematics always isn't about why and how
    Just as complex numbers
    Sometimes it is about what if ..
    What if I agree to a point
    Where does that lead me to

  32. 12:05 "the last word on the subject has not been said" – is there really a concept of last word in mathematics/physics/science at all? If there was – the square root of -1 would have never been even allowed, because the belief that it didn't exist would have been that last word . If mathematicians like to approach problems from different angles, then some of the angles might be "illegal" and turn out revolutionary.

  33. C'mon..be brave..for a time acknowledge the contribution of Ramanujam who first came to the value of infinite sum.

  34. Sum of series of 2-gonal numbers (i.e. natural numbers) = -1/12 (= Riemann zeta function at -1)
    Sum of series of 3-gonal numbers (i.e. triangular numbers) = -1/24
    Sum of series of 4-gonal numbers (i.e. square numbers) = 0 (= trivial zero of Riemann zeta function at -2)
    Sum of series of 5-gonal numbers (i.e. pentagonal numbers) = +1/24
    Sum of series of 6-gonal numbers (i.e. hexagonal numbers) = +1/12
    etc.

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