The Return of -1/12 - Numberphile

to try to guess what kind of things are out there that we haven't discovered yet, but uh in the end we need to justify things with rules and here both calculations are not really justified because of this problem of this not converging, it's really like oscillating, It's like it starts at one, goes to zero and then to one and zero. and so on, the way to justify what the value should actually be is to say: well, this here is again the specialization um of this type of expression when R is Nega 1, so when R is negative 1, this 1 + r plus r s and so on will be this expression here with alternating signs and our recipe tells us that we should think of it as 1 over 1 minus 1 so that's 1/2 so in fact the correct way to regularize this type of infinite sum uh at least in this context type is going to produce the answer of 1/2 how much is halfway between the two? exactly yes, it's halfway between these two extremes it's just a coincidence that it's halfway between the two extremes or um, it would certainly have to be between these two extremes, but there's definitely no rule like oh, it will always be, you know , half of the if you have two different answers, it will be the average of them, there is no rule like that, okay, now I'm going to discuss um, a little more complicated example that is related to things that mathematicians today today they care a lot and it also has some historical significance, so I'm going to consider a sum that looks like this 1 + 2 + 3 + 4 plus and so on and this, like the others we've seen, certainly doesn't converge, you're adding numbers whose increments are getting bigger, so they don't converge towards some specific goal, however, it's really interesting to think about when we have this expression, if we were to assign it some numerical value, what that value would be and then at the end, also How can we justify it?

How are we going to analyze this? So I'm going to do something that is mathematical doodles where We're going to do some operations, at first we won't worry so much, at least we'll break some rules like assigning this numerical value and then we'll try to justify the conclusion later if I look at this. Here let's say I look at -4x so I can figure out what this is, but I want to position it a certain way, so I'm going to think of it as -4 * this 1 and then I have to add -4 * this 2. but I'm going to put that in here , so I'm doing things in steps of two here, so minus 12 and so on, and the point of writing it this way is that now if I add the X and the... 4x on one side I'll get, of course, -3x, but then on the other side I'll get 1 - 2 + 3 - 4 and so on, so it will be the kind of alternate sign version of the original X. now I'm going to do something similar again where I just say well -3x is and now I'm going to shift the position by one so it's like 1 - 2 + 3 - 4 and so on, so I'm writing the same expression but shifted by one and the reason I did it is because then I'm going to add these things and I'm going to get -6x and it's going to be equal to this 1 - one + 1 - one and so on, did you break any rules by making that little change by moving them around that was just for visual reasons?

Wasn't that just a help? Yes, yes, the change should not do anything good because it is the same expression I am referring to. addition and subtraction don't matter, position doesn't really matter of course. I broke the rules when I wrote the equals sign and I was treating which was done by an 18th century mathematician named Oiler and it turns out that it produces the correct conclusion, um, so it's kind of interesting, but we have to, of course, in mathematics we have to go back and justify, like, why? okay, so minus 6X gives us something that we are familiar with, you showed us, we saw that, yes, exactly, we saw that before, we saw that this is 1/2, so the conclusion is that X actually sells 1 out of 2, except for that naughty thing. at first, well, so what does this kind of conclusion really mean?

How is it justified and what is the meaning of that? Previously we were talking about this function 1 + r + r 2 plus r cub. and so on as a function of r and then we said wait as a function of r is equal to 1 1 - R uh at least when R is less than one in absolute value and it is 1 over 1 minus r actually that has less restrictions what you can plug in for R you can plug in any R that is not equal to one, you get a sensible answer, similarly here is something called a remon zeta function, so it is a function of a variable s and is 1/1 to the S. + 1 2 to the S + 1 3 to the S S Plus 1 4 to the S and so on, you'll see each positive counting number here, so the way we would write this in shorthand is the total sum n of 1. /n the S we need to ask ourselves where this expression actually makes sense and it will only make sense when s is large enough because you need to make sure you add very, very small increments to get this quantity. to converge to a to a fixed value, then it turns out that this is fine if s is greater than one.

Mathematicians care about this Romanian zeta function because its properties are actually closely intertwined with the distribution of prime numbers, so if I have heard of the REM hypothesis, one of the longest and most prominent uh connectors in mathematics, then it is about of the properties of this function. It's great, so it's a big deal, yeah, so I said this makes sense if s is greater than one. So for example there is the famous Zeta of 2, you can plug in S = 2 and here you will get pi^2 over 6, if s is not greater than one the problem is that your denominators become less than one and therefore , the numbers become the real numbers become larger the integers become larger and it just becomes this Whopper again exactly yes, so when you have these infinite sums so that they have a well-defined objective value, then the increment that you're adding each time it has to go down fast enough, but actually when s is less than one the increment is not small enough and for example if s is -1 then what you would see in this expression is something like Z of1, you know what you were trying to say is equal to I'm going to put this in quotes um because now you think it's a negative sign so it's actually 1 + uh 2 + 3 + 4 there's this expression that we saw before that uh we said it should be equal a um -12 cool, it's a similar story to here where this Zeta of s I've defined it in a way that only makes sense when s is greater than one, but you can come up with some other description that makes sense, actually, when s is less than one and as long as s is not equal to one then it will have a well defined objective number, let's say Z of s makes sense as long as s is not equal to one, this makes sense means there is some analytic continuation Therefore, There is another perspective on this definition that you are allowed to connect, for example, negative 1 and two, and it is true that Z of 1 is 12, that is, arrive at -12 using a significantly different technique, as if it were the technique of doodles that you showed me and that you said was on slightly shaky ground is the technique you would use in this second form.

Is different? I think you can make the doodles legit, so there are different ways to approach this, but the doodles can be changed. in a real test, but there are certain scribbles that I didn't show that could lead you to the wrong conclusion, so be careful about the ones that can't be, so it's a bit of a fine line at the end, um, you definitely have to do it. Do more work to justify it, but basically the scribbles are part of maybe like the tip of the Iceberg in some mathematically rigorous approach, so if someone tells you that the sum of all positive integers, all of them up to Infinity, Whatever that means, it's minus a 12, what do you say, what do you tell them, you say, yeah, oh, I'm okay with that?

Well, my first instinct would be: Do you know what you're talking about? It's obviously ridiculous, but I guess after studying the theory. so you know, so you think I know what you really mean, but I'll just say it's a funny anecdote that I think is kind of compelling is that sometimes you're doing some real calculation, like in physics, there's this thing. It's called the Casmir effect in quantum mechanics and you're trying to know that you're trying to predict some experimental result and you're trying to calculate energy and you come across this kind of thing that you're like, "oh, it looks like physics." it's telling me that energy is something like 1 plus 2 plus 3 plus four and so on um and of course that doesn't make sense um but then you say okay so maybe the right way to make sense of it is in this sense of the function zeta this is called Zeta regularization where it's a kind of procedure whereby when I see certain types of Infinities that look like this kind of thing I'm going to replace them with the value of the zeta function and it turns out that this This leads to a kind of experimentally verifiable result, so you could have the point of view that, like when you see these kinds of Infinities, it's Nature's way of actually telling you that you know it should be this other kind of negative 112 thing, okay?

I just wanted to give a little more context and information on today's video because, as you know, we covered minus 12 before, exactly 10 years ago, when I was in Berkeley and met with Tony, who was in today's video , I didn't know what he wanted to talk about and he hadn't seen all the number five videos, I didn't know we had covered minus 12 before when I realized what he was going to be talking about. I thought it might be an interesting experiment to hear someone else explain the whole concept and Tony was fine with that, so I hadn't seen the previous video and I thought it might be interesting for you to watch this and compare it to what's been said. before going back to the original video 10 years ago, one of the people in that video also named Tony was Tony Padilla and he has been interested in minus 12 since following him and recently he and a contributor wrote a new article. all about this, how it links to physics, how minus a 12 could be a shield that nature uses to protect us from Infinity, if you want to know more about that, I also uploaded that video recently to the number archive. include a link in the description and I will also include a link to a playlist for all of our negative videos or 12, knock it out, it's because of that abrupt transition that says you don't have to do it like that, there are no others. ways to regulate this sum now I am not doing anything strange there is no strange analytical continuation anything like that ter to say that there are other ways to regulate some do not make the abrupt transition make a smoother one something that takes you between one and zero in a much softer way, it explodes, you get bigger and bigger, so there's no sense of approaching anything so traditionally, what do we do with the Divergent series?

We just ignore ours, we just ignore them, we just throw them in the trash, eh, but the question is whether this is the right approach, whether there is really something we can say about such a series.