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 + ......
I think the best way to understand this is that the sum obviously diverges. But there is a well defined operator $T:S \to R$ mapping a set of sequences $S$, including many divergent series, such that for a given $s \in S$, in many mathematical and physical contexts it is meaningful to replace the limit of the divergent series with $T(s)$. Furthermore, there is a consistent set of manipulations one can do beyond the standard, safe manipulations of infinite series, that preserve the value of $T(s)$.
Numberphile has been treating this subject much better recently. Their original video with proof was pretty abysmal. Now that they've covered the Reimann hypothesis, complex numbers, and have been much more careful about the naive sum versus the "nugget" sum I think they are doing a great job at explaining why this isnt just magic or nonsense.
As someone who isn't exactly a math heavy person, these -1/12 videos have been fascinating. While I'm no mathematician, I do like to look at 'interesting' math and see if I can get a vague sense of what is being done.
So what I'm getting here, is that "equals" is not being used in the common sense. Prof. Frenkel says right out that the literal summation of natural numbers is divergent, which is intuitively obvious and why most people take issue with the -1/12 result. However through some rigorous manipulations they are able to consistently assign -1/12 to this summation.
I've looked at the Riemann Zeta function, and while it's largely over my head I could see that what was actually being done to arrive at -1/12 is different from the strict summation of natural numbers but in a way the Zeta function and the summation of natural numbers can be made analogous and then you can swap in -1/12 as a result. Kind of like saying this particular infinity of course looks like infinity, but it acts as if it were instead -1/12 disguised as infinity.
As if there is a core identity to the summation of natural numbers which is -1/12, like a gold nugget covered by infinite dirt.
I knew it. Euler was an OG.
Would it be valid to say that the two sides of the equation aren't actually related by the standard equality = but by some other relationship say ~ which assigns useful(?) numbers to various divergent series?
Great video! As usual, I'm going to promote the Wikipedia article on 1 + 2 + 3 + 4 + ⋯ for anyone who wants more detail.
Well that was really interesting.
I didn't realize there was a little mystery surrounding the subject.
Does anyone here disagree with that?
I'm probably about to take up an unpopular opinion but I really don't like the format in which concepts are presented by Grime in his Numberphile videos. I've watched 6 or 7 to get a feel for the series and I find them ebullient to the point of distraction. Frenkel's approach was much more organized and easier to follow.
I like this explanation the best