Does -1/12 Protect Us From Infinity? - Numberphile 

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"I saw a tweet that made me so mad that I disproved it and wrote a paper about it" is the best way to do research

For what it's worth, the original -1/12 video is the reason I went on to study maths and physics in uni and here I am now 😂

Good lord have i really been watchng your videos for 10 years or more? Time sure flies by.

You can genuinely feel Tony going "I told you so!" to everyone by the way he's talking. Man's been brooding for literally a decade

Finally, the long-awaited -1/12 redemption arc.

I was more or less mathematically illiterate and despised everything mathematics, and then I watched this video back in the day and it really intrigued me. Now, a few years later I am a grad student in pure mathematics, and it all started with watching these videos, particularly the one about -1/2! You can say what you want about the rigor of these computations, but for me, this is what started my love for mathematics!

If you plot the graph f(N) = N(N+1)/2, it is a parabola that intersect X axis at points 0 and -1. The area bounded by parabola between 0 and -1 is exactly -1/12

One thing i love about numberphile is how passionate these very intelligent people are about such an awesome topic. Very nice change from the constant bombardment of low level nonsense. Thank you numberphile

I was an undergraduate when the infamous -1/12 video came out and now I’m close to finish a PhD in arithmetic geometry. This made me feel so nostalgic.

I have not seen Tony so excited for years. I like it. Good luck on his way

C N² - 1/12 + O(1/N) such that C happens to be 0 makes way more sense than just a blanket -1/12. This is very cool

For those wondering why e^(-n/N)*cos(n/N) is so elegant, it's the fact that in C*N^2, C is the Mellin transform of the regulator function, which basically amounts to integrating x*e^(-x)*cos(x) from 0 to infinity, and it ends up being 0.

Professor Padilla is a brilliant physicist and mathematician and incredibly skilled at explaining his thinking to us, even when ,like here, it gets into the realm of wonder. Thank you, Brady, for bringing him to us.

Ten years ago I was very unhappy with my career and watched the original video. I was so intrigued and inspired, I quit my job and went back to school for a math degree to better understand this result. I've never regretted this decision and seeing this video feels like everything has come full circle.

Lot more gray hairs on Tony. Cannot believe we have been growing up with this man for more than a decade. On a side note, any more of them big numbers?

Man I love how Brady just goes in with the tough questions and points. Great video!

I watched the first -1/12 video when I was in 7th grade, and I am now a junior in university. I remember in elementary math was my favorite subject and when I reached sixth grade I had a horrible math teacher which made me dislike math. Thanks to you guys for making that video as it absolutely rekindled my love for math when I was in seventh grade.

Watching your videos for many years I sometimes regret I left academia. So much to still be discovered. Thanks for making these intriguing videos that even I can keep track with.

I always hated mathematics, my brain just couldn't handle how numbers work, but I can appreciate people like Dr. Padilla who have devoted their lives to trying to explain things that were previously unexplainable. I admire his humility in saying "I don't want to understand infinity, I want to understand the journey of why we would want to get there in the first place". It may be just numbers, but it's an incredibly profound way of thinking and applying them to help explain how the very fabric of our universe works.

This feels like the most significant and profound numberphile video ever made.

10 years ago, because of your videos, I got really into this -1/12 thing. I kept researching it for quite a while. But this new video is just amazing; what a fascinating result!

Wow can't believe it's been 10 years since that video

Terrence Tao’s formulation is so elegant and beautiful. The first video that was published 10 years ago had a lot of debate around it. This video probably won’t because it’s so convincing. One of the best math videos I have seen

So when you use this method to show the sum of all natural numbers is -1/12 it's fine. Use it to show 2=0 and suddenly the technique isn't valid anymore...

Best Numberphile video I have seen in a while. Thanks Brady.

I loved the "smells of String Theory" line around 13:30, so much of cutting edge mathematics is about feeling out into those intuitive spaces where we don't have models for, in order to get a clearer picture of what we're trying to model.

I feel like I've watched something incredibly profound. What a great discovery! I hope there's something to it.

I feel like the weighing functions should be strictly positive for this example. In the cos example, we are not summing everything anymore. Especially when N->Inf, you don't even know which numbers are summed which are subtracted (I like to think they are added and subtracted at the same time). I also feel like for weighing functions to make sense, they should be strictly decreasing from 1 to 0. In other words, 1>=w(n)>=0 and dw(n)/dn=0 .

Thanks! I love your channel, and this particular video is one of my all-time favorites. It explained a complex topic in a clear and compelling way. Amazing.

I am absolutely entranced with this linking of such a bizarre result from number theory and the physical workings of our universe. Tony Padilla is such a treasure.

I remember getting upset at the original -1/12 video. I enjoyed this one much more and it was the first I'd heard of these regulators as a way of studying series! Also, 10 years since the original? I feel very old.

That video was 10 years ago?? Feel like the biggest mathematical mystery here is where that time went.

It’s an endless debate. People will still be arguing about about this 1 month ago.

Your videos were one of the things that really got me into mathematics, and now I'm in graduate school for my physics PhD. Looking in the comments, I'm not alone in this either! Thanks for ten years of accessible math education -- even if some of it isnt entirely rigorous :p This sum regulation method is fascinating... I wouldnt buy putting an equals sign anywhere, but it's still extremely compelling that -1/12 is just as important here as it is via analytic continuation. Edit: That connection to physics -- wow! It makes me think that there must be some universally "preferred" method of renormalizing sums, some way that's "natural" in both mathematics and physics for reasons that arent entirely clear. I'm very curious why certain regularizations produce that behavior but not others...

I personally think this is genuinely a momentous find. The fact that infinities can be numerically represented and define behavior of systems without having to be aggressively gotten rid of can possibly reconstitute many of the existing conjectures and hypotheses that are stuck due to infinities... not all, but even if one, that'd be huge.

The best way my math professor described infinity is that it's not a number, but more like a direction. It's like traveling east, you never reach the end because you can always go more east. So, treating infinity like a number doesn't make sense since there's always more east.

"Lemme give you another example of another choice you could make" Best words to hear in math video 😊

Love your content, presented in a very accessible way. Coming from St. Helens, hearing someone who comes from the Northwest providing such great insights into mathematics feels comforting and even more relatable (and hopefully inspiring for kids growing up in the region). Thanks!

Oh my how the time has passed. 10 years ago I was in high school, watching these videos with astonishment and a genuine feel of adventure and love for all these awesome discoveries. A decade and two degrees later, and nothing has changed, except maybe for greying Professor Padilla, but still as enthusiastic as ever. Thank you, sincerely.

Great video! I would love for it to become public and viewed by more people as this really helps explain where the "-1/12"th-ness of the sum is. To use Ed Frenkel's terminology, this is how to extract the gold nugget from the infinite dirt!

This is astounding! Relating a childhood fascination of mine with a current intrest of mine. What a delight! I'm going to read this paper

But then ... integers _do_ make for a rather sharp cutoff. As far as making it -1/12, I would agree with Tom Petty that the weighting is the hardest part.

this is so insightful and to see how Terrance approached the problem with regulator functions and Tony's way of explaining it

Well, I agree that our intuition isn't necessarily right, but neither is doing something BECAUSE it gives the answer you want it to give you.

I love Tony’s passion in how he presents things. And a bit of string theory thrown in for good measure!!

I love that his takeaway was "it got people interested in mathematics, so I'm glad we did it." It's something that resonates with me so much. Sometimes in education and outreach, it's not about teaching the right answer, but getting people interested.

Using the cos function to "smooth" the cutoff point seems a much more physical way of doing this. Any RF engineer will tell you there's no such thing as a pure step function; that rising edge is always a collection of ever increasing frequencies so that it just looks like a step.

Wow, thanks for an excellent revisit to -1/12. This was just beautiful! Pure partial and infinite sums, to end up where we all know we would end up... love it!❤

Without a doubt the most important topic ever covered on Numberphile. People probably laughed the first time someone said “let’s just call SQRT(-1) i and see what happens”. Now, instead of just ignoring the fact that -1/12 keeps popping up, a few courageous mathematicians are seeing where the clues lead. Kudos to Dr. Padilla and others for boldly going where no mathematicians have gone before.

I'd hazard that arriving at -1/12 tells us more about these weighting functions than it does about the concept of infinity. So in a way we're still "sweeping infinity under the rug."

Astounding result. It really does seem like something quite fundamental has been uncovered, and it comes out of such a simple question that seemed unanswerable for the longest time. I wonder if this is what it felt like when people started taking imaginary numbers seriously and realised they gave real, verifiable answers to questions. It seems super weird, but if the maths gives you useful answers then you just have to follow it.

You can express a partial sum of an infinite series in terms of the sum of the infinite series.... WHAT? I had to rewatch 3:30 to 5:30 again and then pause on this to ruminate how amazing what is being expressed here. Excellent videos. This feels like you all are mass producing the discovery of one person down for the rest of us to understand. Just as important, to me, than the discovery itself. Thank you.

Surprisingly compelling, even for finitists!

This is the most exciting science news I've heard for a long time!

Ask the Computerphile guys to explain. They'll tell you it's common or garden integer overflow to sum positives and end up with a negative.

Do we need to start a gofundme to buy Tony a new phone charger

What a delightful video! Finally an honest way to do renormalization, and great to see more of the amazing work that Terry's been doing since we were grad school room mates :)

The sum of strictly positive integers from 1 to N, where N goes to infinite, is of course infinite. 1+2+3=6. Adding more strictly positive integers to this sum will clearly increase the sum. QED.

What a Valentine's day treat XD

Agree with the others, this is one of my favorite Numberphile viedos in a long while (and a good redemption of Dr Padilla :))) ). It's fun and interesting and it communicates a lot of high-level ideas without being too bogged down by details. It reminds me of being in undergrad and having much smarter people than me explain deep ideas over lunch, or during "office hours".

There is something very metaphysical about this episode.

-1/12 is the error code you receive when part of your formula tries to treat an infinity as a number.

Awesome to see the joy and excitement in the eyes of Tony when he talks about this. ❤

My layman understanding of the need for a weighting function is that you only see the sharp cutoff at N if you stop the partial sum at N, but you're interested in what happens when N goes to infinity so you will never see a cutoff: it's infinitely "far away".

seeing the phone cord at 0:22 worsened my asthma and shortened my life by 30 minutes

Even though i enjoy what I'm studying this video honestly made me regret not studying maths or physics

Saw this on day1, now watching again. Very enjoyable.

Thank you for fighting the hate and discovering this exciting connection to sting theory as a result!

I really like this weighting function explanation.

I honestly think, that this was the way, that Ramanujan approached these problems. That was his talent, to be open minded to techniques. I'm not surprised, that Terence Tao was the one to decode parts of this secret...

This is incredible. You should really consider doing this as a full release video (not unlisted).

I was screaming that this is exactly the basis for why renormalization works in QFT.(once I saw the power series come out of the sum) But I had no idea that only the regulators with convergent answers preserve the symmetry. I'm reading this paper for sure.

"That's what you're really doing here," is gaslighting.

What Tony says in the end is actually a request to *redefine* infinite sums, because the very definition (being the limit of a finite sum upto N, where N goes to infinity) has built this sharp cutoff in. Like there's a Cesaro sum etc, we will be having the Padilla Sum.

This could be told like a story by H.P.Lovecraft. The protagonist thinks the -1/12 is not a real answer, it is just an analytic continuation, a trick. But it haunts him, he sees it in his dreams more and more often, then he gets visions during the day. He can no longer ignore it, he has to scratch the itch, searching deeper. And he finds it. The answer. -1/12 was real all along, he can no longer protect his mind by pretending that it is just a trick. And he goes insane. Oh good old infinity. We cannot understand you. But how sweet can ignorance be if you face the monster, the old god of chaos. Twisting, shouting, breaking what we meager humas tought to be the well ordered little world we inhabit.

I understood this about as well as the original -1/12 video.

The cosine function does make half of the terms negative, so it is not that unexpected that the result is close to zero.

So in another 10 years I hope to see another follow-up on how -1/12 solved string theory. Looking forward to that one! :)

My problem is the statement "I am removing the regulator because I'm taking N to infinity". The weights are still there even as N goes to infinity. With W(n,N) = exp(-n/N)cos(n/N), even though for a fix n, W(n,N)--->1 as N--> infty, those weights are still N. For example W(2N,N) ~ -0.056, no matter how large N is. That means, even taking N to infinity ("removing the regulator"), the weighted sum always includes the term -0.056*2*N. It seems to me that these weights being considered are too general. The weighted sums they yield don't have much to do with the original sum 1+2+...+n+...

This how math moves forwards - craziness for a long time and denial and then a little piece of wisdom from a genius like Terry, Euler or Ramanujan and we sheeple get a little further.

I feel like I'm in a Numberphile speakeasy

Adding integers and arriving at a fraction. Adding positive numbers and arriving at a negative number. Yeah, I don't see anything in this method that would make it the "preferred" method unless I purposely want to get unreasonable results.

Great video!! It truly felt like this "coincidence" of weighting functions, unique pathways to the minefield of the infinite and particle physics is telling us something. Like a jab of the prospector pick, hitting straight into the gilded veins flowing from the inner most reality and up into our realm.

I like Numberphile. In other places, I sometimes don't understand the explanation. Here I don't understand the question, much refreshing.

As a layperson, I feel like the "weighting" was unduly glossed over. I would have loved to see what happens to the actual numbers in the series when you apply some of those complex weighting functions.

Many have said this before but numberphile is the channel that I credit for sparking my interest in and subsequent love for mathematics. And without that I wouldn't have started studying computer science. I also remember the original -1/12 video being absolutely mind blowing and as one of the key factors that sparked this love.

this shouldn't be unlisted. amazing video

Why doesnt this video show up on your channel?

Thanks! Incredibly cool stuff!

An instant classic. Feels like they are about to have a breakthrough

This kinda thing feels like it's an indication we're playing with a broken toy; there's one or more warped/cracked/broken gears hidden in there somewhere, the malfunctions are small enough to not be noticed in most cases, but when you hit just the right spot things act really weird....

Wow I finally feel like I have some intuitive grasp on how this -1/12 makes sense. Amazing explanation, thank you.

I commented this on the original video in a reply to one of the naysayers. To me, Reimann regularisation seems like the invention of complex numbers. Of course complex numbers aren't "real" but they turn out to be a very useful tool in the physical world. I look forward to seeing what more research turns up in this little corner of mathematics.

I remember back when I was obssesed about both the Numberphile video about -1/12, and Death Battle's video about Goku vs Superman. Ten years later and they both revisited the same subjects

Definetly one of your most interesting videos in recent memory! This could turn out to be the first step in something big after all

This really brings me back. I have really watched Numberphile for 10 years!

I don't feel like this weighting is fair. He draws the step weighting first (hard transition from 1 to 0) and then pretends that e^(n/N) is a smoothed-out version of that transition. It is not. He draws the graph as if its starts at 1 horizontally for n=0 and curves down when n approaches N and then it curves up again as n goes from N to infinity. Almost nothing of that is true. It odes start from 1 for n=0, but it already has a very marked negative slope regardless of the value of N (d(e^(-n/N))/dn = -1/N * e^(-n/N) which, for n=0, is -1/N), it never ever approaches horizontal at any point before n=N, and it never ever curves down (the second derivative is always positive so it is always concave up). So it is constantly diminishing and the rate of diminishing is greatest at n=0, at the beginning, not as n approaches N), by the time n is approaching N, tit already lost 63% of its weight. And, again, is not that it loses a significant part of that weight as it approaches N, most of the weight is lost in the beginning where the derivative has its minimum (i.e max negative slope). So with that kind of weighting it is not surprising that you turn an initially diverging series in a converging one. Except that even with all that you don't. You still gen N^2-1/12 which as N approaches infinite it is still infinite - 1/12 which is infinite. But wait there is more: Then he adds another weighting term: cos(n/N). He describes it as "this still goes between one and zero with some transitioning around N, it does a bit of wiggling around". That is profoundly deceptive, and I can only thing that it is intentionally deceptive. Why didn't he attempt to do a graph of this weighting function? Did he just forget to mention that, for any value of N, that weighting factor is NEGATIVE for HALF of the values of n?????? Little detail, hu? So it doesn't go between one and zero. It OVERSHOOTS zero and goes NEGATIVE... HALF OF THE TIME!!!!! So ware not adding NATURAL numbers with a weighting anymore. -1,000,000 is not a natural number. I wonder what would happen if you take |cos(n/N)| instead. I bet it doesn't converge.

This feels like one of the most exciting videos Ive watched on youtube. All those years ago numberphile inspired the sense in me that there was something myserious going on with -1/12. A decade later Brady goes full circle, showcasing new maths to explain it and maybe change our fundamental understanding of reality!

10:17 this weighting function does not go between 1 and 0 as stated, as cos(n/N) will have negative values for π/2 +2kπ < n/N < 3π/2 + 2kπ. (integer k) the ne^(-n/N) factor is always positive, so it oscillates.