Why -1/12 is a gold nugget

Is this good? The position? More like that, right? Oh, were you the one who made that video? I see. Well. Well, I think mathematicians learn these things at some point. And I learned it at some point. And then you forget. So you put it in a box somewhere and put it in a closet. And you classify it into things that you have already understood. But I think we don't really fully understand what's going on here. Let me remember 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 infinite." And we say that because this series is what mathematicians call a divergent series. - It's exploding. It's exploding. You get bigger and bigger. So there is no feeling that it is close 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. If there's really anything we can say about a series like this, let it be meaningful. In other words. "Can we assign a value to this series that is meaningful?" - Professor, doesn't it make sense to say that it explodes and goes to infinity?

That makes no sense? It is significant in the standard context of a series of this type. The way you can think about it is this. Think about this, okay: then we have to resign ourselves to the fact that it is somehow infinite. But imagine this entire series as one huge lump. What if there is a way? What if there was some kind of pretty scalpel that would allow us to surgically remove infinity? A kind of bad infinity. And then keep a kind of finite part. So we would say that we will assign that finite part as the true answer to this infinite series.

Now, we must realize that this 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 like a random, infinite series that explodes. It will just explode. There is no hope or expectation that there is a way to assign any meaningful value to it. But this is a very special series, because as you see, it is very regular. Something like 1, 2, 3, 4... We're actually taking the sum of all the natural numbers, without spaces, including each of them exactly once. And the interesting thing is that those kinds of sums appear all the time.

In many different branches of mathematics and quantum physics. Mathematicians have long thought about trying to develop a theory in which they could actually make sense of this. And today we have that theory. And within that theory, we could often say that it makes sense to think of this sum, or more precisely, that kind of finite piece after eliminating the infinite part, that kind of "regularized" sum. Maybe we should make a distinction between a kind of "naive" sum, where it just explodes to infinity, and a kind of "regularized" sum, where this regularized sum actually turns out to be minus one over twelve. - Oh no, -1/12? -1/12.

You see, it's very counterintuitive, because it's actually a negative number and you're adding positive numbers. Therefore, it is certainly not the result of the sum of these numbers. It's something else, but what is it? So mathematicians have developed ways to arrive at this -1/12. And in fact, the first person to talk about this was the great mathematician Leonhard Euler. He was born in Basel in Switzerland. But I also spent a lot of time researching in Russia, my home country. Leonhard Euler was something of a mathematical outlaw. Some kind of math gangster. He did things that were illegal and illegitimate.

And in particular he allowed himself to manipulate with infinite series like this one. In other words, he was trying to guess what could be a possible way to assign a value. By trying to assign values ​​to this series and other similar series, he came to find the correct answers, which were later justified 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. - Can you get to -1/12 in more than one way? That's right, in more ways than one.

So Euler arrives at negative one over twelve in a particular way. Riemann later explained; him giving a rigorous theory using his zeta function. A theory that involved things like complex numbers. Something that was not yet fully developed in Euler's time. Although much of this already existed and Euler himself was considering complex numbers. But there are other ways too. Now we know other possible ways to think about 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 is an example. What else did he study?

He also studied things like one cube plus two cubes plus three cubes plus four cubes. That seems to diverge even faster than this one, right? Because now you are taking the sum of cubes. You see, again within this context of divergent series, you just approach this the way, you know, we approach it when we studied, you know, freshman calculus. This definitely blows up. Definitely divergent. This is infinity, the answer is infinity. Period. There is no way to avoid it. But Euler allowed himself to do some manipulations with those 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. That was the answer. You might wonder why squares were skipped? Actually, you can do it too, but the answer is even more surprising. You actually get zero, within that scheme and so on. So I was actually studying all the 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 are the square, the cubes, to the fourth power and so on.

And then the funny thing happens: for all even values ​​you actually get zeros, and for odd values ​​you get some rational numbers. - In the original videos everyone got very angry and said that this divergent series cannot be made. - Are you saying you can? - And you say they broke the rules and then you say they didn't. What are the rules here? Well the rules, it depends. Rules within what context? Well. So let me ask you to illustrate what I mean by this. Let me ask you a question. Does the square root of -1 exist?

Come on Brady. - Well, I know we call it "i." We have these imaginary numbers. Good. But does it really exist? Exists? Does it make sense to talk about the square root of -1? A possible answer is: Of course not! Good. Because if we think about real numbers we know that a square of any real number is positive. So the square root of a positive number is well defined. The square root of 0 is also well defined: zero. But there is no square root of a negative number. So we can stop there and say: the square root of -1 does not exist.

Anyone who uses the square root of -1 is an outlaw. Good? Because that is not legitimate. These are some dirty tricks. But we actually understand it now... And that's how people saw it for a long time. But in reality we now understand that there is a rich and consistent theory that includes a square root of -1. That is the theory of complex numbers. And this theory gives us a much more interesting, much richer, much more fruitful context in which we could, in fact, solve many problems about real numbers. In other words, we have to get out of the realm of real numbers.

Many times to obtain the best, most optimal or sometimes the only possible solutions on the real numbers. So it is in that sense that now no mathematician in his right mind would say that the square root of -1 does not exist. Yes, it exists, in the sense that we can add it to the real numbers. We get a well-defined number system, which is called complex numbers, which is as legitimate as the real number system. - Are you saying that manipulating a divergent series is in the same category as that? Well, I'd say that's a good analogy.

Because... You notice that sometimes in a different context in which different things can be discussed. So in the case of the square root of -1, there is a real number context, where the square root of -1 surely does not exist. Or there is a complex number context where it does exist. And it's actually very useful. And in the same way here, there is also an obvious context of, you know, the rules of analysis, the rules of calculation, 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's another context where we replace this series with its kind of regularized values ​​and I really like to think that all these regularized values ​​are like, you know, removing something like... imagine, like a piece of

gold

that's surrounded for this infinite amount of dirt and, as you throw away this dirt, you are left with this little piece of

gold

. So what I'm trying to say is that each of these infinite series contains within it, apparently, this little piece of gold. And then we can say well that that little piece of gold is the value, it is a true value of that infinite series.

And the rest is useless and we can throw it away. If you say that, and there is a rigorous mathematical framework for doing so, then some of the manipulations, in fact all the manipulations that Euler did, become legitimate. Because what you're doing is like carrying with you those little valuable pieces on either side of the formula, as well as those infinite things, and you, in a way, can throw away the infinite things and then any relationships that you find. Among the infinite series there will also be the value relationships between those valuable pieces. - Professor, it seems that it is very...

My understanding of mathematics is: - It is very rigid and rigorous and is never arbitrary. -How can you just throw away the dirt and keep the gold? It doesn't seem... That's right. Well, in a way, it's a great question because I think it's a mistake to think of mathematics as some kind of linear process, where we only do things that are legitimate, that are allowed. If we did that, we would never discover the square root of -1. We wouldn't even discover the square root of two. For a long time people did not believe that the square root of 2 was really a legitimate number, because it cannot be expressed as a fraction.

Well, that can't be expressed with a fraction and for a long time people thought that the only legitimate numbers were fractions. Well, actually, from time to time, there are people like Leonhard Euler, probably Riemann and others who actually... Ramanujan is another example, who jumped into the abyss of the unknown, broke the rules and tried it. .. to lift the veil on the unknown and try to understand more. And sometimes they're actually doing something that may be illegitimate at the time. Maybe they were ahead of their time. But one thing that is important in mathematics is that we can never leave these things as "loose ends." We have to find a justification.

So you're right, math is rigorous. And at the end of the day we look for a rigorous justification, a rigorous explanation of everything. In other words, we are not content to say that there is some magic there. There is magic. But we always want to explain it and that is what has happened to a certain extent with these series. Riemann's work gave us a tool to analyze these, more or less, the golden parts of these infinite series. But I still think that the last word has not been said on the subject, because we still do not fully understand it.

Because I don't fully understand why every time a series like this appears in mathematics or physics. We get the correct result by replacing it with precisely that value or with this value for it. In physics, for example, these types of calculations are done all the time. And in fact, perhaps the best kept secret perhaps 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 from the outset. At first glance it seems infinite, but they find ways to assign meaningful values ​​to this infinity, so to speak.

And they... and that's really... I think it's a good analogy to think about, kind of like surgically removing some kind of infinite part that is redundant and superfluous and throwing it in the trash and replacing the answer with this... . What you might find is a remainder. And the interesting thing is that in physics... physicists are still waiting for their own Riemann to come along and somehow... justify these calculations, but they have been incredibly successful in getting results that they can then test experimentally. And some of this has been tested with a surprising degree of precision. - Is there any sum that you are assigning the same value as the sum?

Is that where things have gone wrong? - Or where things have become confusing? I would say it's not exactly the sum because... the exact sum is, you see, you know it explodes, it's infinite. It is a kind of regularized sum. But it surprises me, for example, why every time we find a sum like that in mathematics, and there are so many places where we do that, where we find these kinds of sums. Every time we findthese sums, we always have these kinds of reactions like Oh, we should replace it with -1/12. And every time we do that, we get the right answer and maybe later mathematicians will 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 get to such an infinite sum... It's an indication that maybe we're doing something that's not quite right. We are applying a perhaps somewhat naive approach. But interestingly, every time it happens, if we replace it with that -1/12, we get the correct result and then we can justify it 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. Clearly there is something there that we still don't fully understand. For now, we understand that -1/12 is a kind of golden part. It's this kind of finite part in this infinite mass that you get by throwing away a little bit of infinite dirt. I'm going to give you a surprising result. - Amazing? A surprising result. So I was going to write down a small sum. I'm just going to see what answer he gives. 1+2+3+4+......