Does -1/12 Protect Us From Infinity? - Numberphile
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- เผยแพร่เมื่อ 14 ก.พ. 2024
- Ten years later... Professor Tony Padilla returns to the thorny issue of summing the integers arriving at -1/12. More links & stuff in full description below ↓↓↓
Also newly uploaded today about -1/12: • The Return of -1/12 - ...
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Here's the new paper: arxiv.org/abs/2401.10981
Here are our -1/12 videos, including the one that started it all: • -1/12
Here's a blog Tony wrote about the original -1/12 videos: www.nottingham.ac.uk/~ppzap4/...
And here's a blog Brady directs people to when they message about the matter: www.bradyharanblog.com/blog/2...
Interview with Terry Tao: • The World's Best Mathe...
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Also newly uploaded today about -1/12: th-cam.com/video/FmLIGN8ZGdw/w-d-xo.html
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Beautiful video.
if we are able to regularise the most basic summation sum(n) without throwing away infinity, like this video, I guess very soon we would be able to regularise every other infinite sum with some combination of above weighting functions and some other known techniques. May be we misunderstood infinity till now. can someone do it for 1+1+1+1+1+..... ? may be we get a finite sum for this too.
another thing that struck me is that all of this is not for a continuum set like R but for N , and Quantum Theory is about denying the continuum at various places. May be there is a subtle but a fundamental reason why such results find their shadows in bits and pieces in Quantum Theory. Something like you always find a circle somewhere whenever there is pi in some formula.
For some reason, the constant refocusing on this video made me slightly motion sick. PLEASE either stop down the aperture or add a separate overhead PaperCam!
but then you can choose a regulating function at results in the -1/12 being canceled with with some C +1/12. This shows that the series is divergent as you can get any result you like. I would also point out that all the infinitely small numbers are known as infinitesimals and are the basis of how we perform calculus in the first place, so you cannot simple ignore an infinite amount of infinitesimals. This goes along the same lines of you cannot zap an infinite amount of zeroes in an infinite series without changing the value.
Infinity plus 1 is NOT Infinity. Because if it was, then 1 has to be treated as 0 in an infinity equation. There is something called an infinitesimal unit. That unit can represent 1. And that unit is as far from 1 as infinity is from the integer 1. It means that 1 could even represent 1 google, if we count in Google units.
-1/12 is only believed by those that are too lazy to use common sense. With all respect to Ramanujan.
There are 12 fundamental particles that interact with the higgs field that causes them to slow down from the speed of light providing the minus sign🤷
"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
Based
"I saw a tweet that made me so mad" is always true
now that tweeter is called "X" what are tweets called?
someone write a paper about that please
@Paulo_Dirac No one calls it X, we all still call it Twitter and tweets
@@supersuede91 Xitter
xeets
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 😂
Doubt
@@co2metal 🤷♂️
So -1/12 didn't protect you from infinities, quite the opposite.
And now you go to work for corps/state/mill and start removing ppl with fake saiens.
@@Intellllect Some must make the sacrifice of fighting infinity to save the rest of us from it.
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
Sqrt(-1) isn't intuitive either. But it's now well understood, and it wasn't well understood for hundreds of years, and yet it was used since Bernouilli. We are with inifinite series at the same place Bernouilli was when he was using sqrt(-1) without really knowing if that was legitimate.
@@InXLsisDeo No we're not. Infinite series are well understood and the imbecil in the video needs to give it up. The of whole numbers diverges. Always has, always will.
You do realise he didn't tell anyone anything. This entire video is just avoiding the simple fact that he's wrong, with extra steps.
@@InXLsisDeo
But there is no solution in the reals. No real times itself yields a negative product. Complex numbers, problem solved. This question can not be solved that way, because it obviously blows up to infinity.
So what does Tony say that infinity + 1 equals? Less than infinity?
Maybe that's part of why I'm so put off by this one.
Finally, the long-awaited -1/12 redemption arc.
It'd be nice if that's what it was, but it really doesn't redeem anything. Rather, it just n-tuples down on the same falsehood like there's nothing that needs to be qualified, while pretending some other tangentially-related thing backs it up.
@@bjshnog You think you have average IQ but really its room temperature. Ignorance is bliss!
"falsehood"@@bjshnog
@@bjshnogyou are firmly in the middle of the bell curve meme
@@whiskeyecho3523 I've left another comment on this video; see if you can find it. If you still feel that way after reading through the convo, I don't know what to tell you, as in that case you must love this channel so much that the idea of thinking critically about what it presents never crosses your mind.
Good lord have i really been watchng your videos for 10 years or more? Time sure flies by.
i had the same thought.
@@dagudelo88Likewise!
same!
I've been watching these videos for -1/12 years. That's negative one month.
It’s been 12 years… I can’t imagine
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!
haha that's a great story!
That is great, thanks for sharing it. The -1/12 videos also rekindled my interest in maths, as did the superb video of hackerdashery on the p vs. np problem
That's awesome! I still do that hacky "proof" in my junior-level high school math classes sometimes just to rope students in in pursuit of exactly that response.
I subscribed to numbphile 10 years ago because of -1/12 video
Wow, that's a story! Hahaha. What a surprise.
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
well said.
I have not seen Tony so excited for years. I like it. Good luck on his way
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
My dad showed me that just after the first -1/12 video!
@@L9MN4sTCUk ... An integral is not a partial sum but it is the limit of a partial sum as n goes to infinity. int[a,b] f(x) dx = lim [n to inf] sum [i=1 to n] f((b-a)/n*i)*(b-a)/n). It's the Riemann sum of the area of rectangles f(x)*Δx as Δx goes to zero (and hence n, the number of rectangles, goes to infinity)
I figured that was just a coincidence. But I looked at the sum of cubes, fifth powers, 7th powers and 9th powers, and it always worked. Note, the 5th and 9th powers have extra roots outside [-1, 0], but I still only integrated from -1 to 0 to get the Riemann zeta result.
The graphs are really nice, I wish I could post them (with a zoomed in vertical axis). But search Faulhaber polynomials if you care to check yourself.
Area can't be negative
@@B3Band In math yes, area can absolutely be negative.
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.
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.
We played that video in my calc 2 class when we got to infinite series. The professor basically said nah it's divergent vid is wrong. 💀
@@RingxWorld I asked my calc professor about that video at the time. Turns out his PhD advisor is the same as Terry Tao’s, and he gave a talk on the dept colloquium about the summability methods and how to make sense of this (which is half of the topic of this video). This was such a flashback 😂
@@RingxWorld yeah that's what it is simply wrong...
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
I remember feeling unsatisfied with the way -1/12 was justified with the zeta function. That's just one arbitrary function, who's to say if you used a different function you wouldn't get a different finite value, you know? This is much more convincing!
@@khaduopha2640Analytic continuation is kind of weird. There is only one correct continuation for any function
Whatever the method of regularization, if it allows shifting, adding, etc., its value must be -1/12.
That is what the original video showed.
There are many such regularization methods, some of which have physical realizations.
But even if they don't have a physical realization, the math still exists.
But what about the O(1/N). How do you get rid of it? Or what does its existence imply?
@@vezokpiraka O(1/N) is something that converges to zero as N goes to infinity "by definition". The expression CN^2 - (1/12) + O(1/N) is telling us than *any* analytic continuation that gives us "1+2+3+4+..." at a non-singular point *must* have value -1/12 at that point.
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.
Beautiful!
Thanks!
YOu explained it very well, now i only need a maths phd to understand it:)
I'm not sure I see that beauty you speak about. This regulating function is a periodic function (which binds any value between two maxima) times a negative exponential, which makes the two maxima converge. It's not surprising at all it results in a finite value. It amounts to doing +1-1+(0,5)+(-0,5)+(0,2)+(-0,2).... Which eventually converges to 0. These kinds of functions are used everywhere in NMR spectroscopy to regulate noise in the output signals.
I have no idea what my entry point to begin understanding that is.
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.
What do you do for work now?
@@sbnwncfishes for likes on TH-cam...
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.
Man I love how Brady just goes in with the tough questions and points. Great video!
Me too, that's one of the reasons I like Brady's videos so much. Once he groks it, he throws curveballs at it. Sometimes people miss things that he sees pretty quickly.
But the answeer was. "Of course i reverse engineerd it..." = fraud.
@@YEC999 Almost the entirety of maths is developed this way, bro. You reverse engineer an answer *and then* check to see if it breaks stuff. Then you slowly prove that the answer couldn't be any other. Quite the opposite of a fraud. It's the most sure-fire way to prove things.
@@Voshchronos If you think to multiply whole number with a cos function is the most "sure-fire way" to prove that the sum of whole numbers=-1/12. than i would never believe anything scientific you say.Literally if i knew your name i would not believe anything you say. You should really play with a cos function before you make claims like this...
Another thing that you don't understand: To manipulate to reverse engineer a weighing fucntion that it outputs what you want to have is a brutal fraud. It is like the definition of fraud: You use an manipulative function to get the result you wanted. The equivalent in economics is balance sheet manipulation like Enron or Bernie Madoff where people went 30 years to jail for.That is all that is: a manipulative weighing function to get to a sum that you wanted to have. in Case of Enron: lower debt than it would be. In case of this: lower sum to get to -1/12
And no nothing was invented in maths with manipulated weighing functions. No Complex numbers (they came through a very tanngible problem) no negative numbers no reals no nothing..
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.
Hey! Lovely seeing you here :)
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...
You can actually set this up in Excel by using a "SEQUENCE" function to create the series and then wrapping a simple "SUM" function around it. If you set n to be sufficiently large compared to N, it does converge on -0.08333. So, the infinite sum if you're using that weighting function does actually equal -1/12. No weird math tricks or anything. Just changing the way you weight it.
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.
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?
Besides his age?
Somewhere between Grahams number and Tree(3)?@@ashleylawrence2110
my exact thought when I saw the thumbnail. Been heavy watching numberphile 10-12 years ago just when they were starting out, but then abruptly stopped following besides casual video here and there, and was kind of shocked and sad to realise so much changed
Rude hahaha it’s graceful.
Also I think he’s got kids.
Best Numberphile video I have seen in a while. Thanks Brady.
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.
this is so insightful and to see how Terrance approached the problem with regulator functions and Tony's way of explaining it
Wow can't believe it's been 10 years since that video
Right!!
And they're still wrong.
I feel like I've watched something incredibly profound. What a great discovery! I hope there's something to it.
I'm completely math illiterate, I understand the individual concepts, but I can't put them together even when explained. I still felt the profoundness.
It's like watching a foreign movie. I don't know what's going on, but the effects and action is awesome 😊
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 .
I was thinking exactly the same till I looked more closely: n/N is always a positive number below 1 and therefore cos(n/N) is also always positive 😊
@@volkeru2718 While you are correct, there is a caveat. The sum is infinite and cos(n/N) is not. They calculate the result of that infinite sum and only then set N to infinity. If they start with sum of N numbers and then looked at the limiting case N->Inf, I think your assertion would be valid.
In other words, the initial sum has a period of T, which repeats infinitely within the sum already. Increasing its period to infinity voids the initial solution because It (probably) requires infinite repeat of that period (I assume this is correct. Else, we could just repeat the entire thing with sum up to N, then do N goes to infinity.).
I am however wrong in my other assertions. We are not increasing frequency but decreasing it. However, the assumption that cos cycles infinite times within the sum remains even when N->Inf.
Thanks! I indeed overlooked the infinite sum. It is a pity that they did not even mention that the weighing function gets negative and why they are allowed to do so... Or maybe, as before, they do not ask these questions... They simply do it, as before when treating the sum of a non converging row as if it was just a normal number 😬
I am still reflecting about this one. And I Excel-led it... Guess I am much more a financial analyst / controller than a mathematician by now. 😅 However currently I tend to say that if they first proved that the result is proportional to C*N^2-1/12+epsilon independent of the weighing function then they are thereafter free to choose any weighing function, even a negative one. And indeed it is amazing that the results using the function shown in the video converge to -1/12 though nothing in the function itself hints in this direction. However interpreting the result as the sum of all integers would clearly be wrong. Allmost all weighing functions will result in a C ≠ 0 and therefore diverge to infinity which is the intuitive result.
@@volkeru2718 Lol my instinct after watching this video was to do the exact same thing--create an excel spreadsheet that approximates taking each component to infinity. Must be the engineer in me. 😆 Unfortunately, Excel limits any sequence to being about 10^6 items long, but that still gives you some leeway to play around. What's really interesting is if you try to apply this to other, established, convergent series. The ones I've tried it with so far still worked perfectly fine.
Awesome to see the joy and excitement in the eyes of Tony when he talks about this. ❤
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.
To be fair, String Theory is been shown to be pretty much a hoax as well. Michio Kaku is basically a Charlatan. Dr. Angela Collier (@acollierastro) has an excellent video about it.
I know, it blew my mind. More so than Eulers identity which was the big thing that blew my mind in math
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.
Wow I love this
Exactly. I think what we're seeing is that there might be different ways of travelling east (one step at a time or two steps at a time for a simple example), and there might be ways of quantifying those different destinations other than "they blow up to east".
You can only go east as far as the international date line. From there, there's only north, south, west, up, and down.
@@andrewj22 love when people are pedantic about a metaphor, that doesn’t deride the point at all
@@Varooooooom Of course the point stands. It's just a bad analogy.
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
"Lemme give you another example of another choice you could make"
Best words to hear in math video 😊
This feels like the most significant and profound numberphile video ever made.
New knowledge! Very exciting.
Or least! Who knows!
Indeed, the optimist in me is already looking forward to a numberphile mention at a future nobel ceremony, this seems like it could be a potientialy fundamental insight once properly understood, was left very excited after watching this.
It really does. Full circle on a topic that kicked off controversy, excitement, and many peoples' math careers. But this time, it comes back with new insights. Amazing 😍🤩
Yeah, but string theory....
That video was 10 years ago?? Feel like the biggest mathematical mystery here is where that time went.
I hear that.
No. It was like three Tuesdays ago, MAX.
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!❤
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.
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!
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.
It makes me wonder if it is that our descriptions are wrong since infinities pop up that we have to throw out or is that since we can, in a sense, wrangle the infinity to finite using regularization, that nature at the quantum level is infinite but has a way of regulating itself back to finite.
@@henryjm1943Well you are right in at least one way: in quantum physics there are infinities that are reduced back to finite. The values for any property of a particle is a superposition of all the values this property could have, and the possibilities are indeed infinite, yet they all add up in a way to a single usable physical value.
So is this process more generalized that we are expecting or is it inherently only tied to possibilities ? Hard to tell
The separation between mathematics and reality is the distinction between the computable and the infinite/continuous
Processes in nature are computationally bounded in the same way that your computer does not have enough time in the universe to calculate all the digits of pi
@@discotecc A circle doesnt need to compute the value of pi to have it's diameter as the exact value.
Your argument is ridiculously wrong
i find this explanation so much more insightful than any other treatment of this topic ive seen before!
I started following you about one year before that -1/12 video. And then gone in other area. So happy that TH-cam pushed me this 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.
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
EPIC. This was really amazing, all the more so as I think I got a notion I might actually understand what was going on!
Brilliant work both!!
The graph reminded me a lot about boundary layer theory in fluids and mass transfer adsorption in a packed bad too - even more links from the wonderful world of maths.
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!
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.
Regularisation has been used for about a century... and there are a number of experimentally verified results that 'use' it. The novel intuition is that there may be a 'best' or 'preferred' way to regularise certain types of expressions.
but isn't this basically the gist of many of the videos on the topic 10 years ago?
This is pretty much the point of another video, "-1/12 is a Gold Nugget"
The Chonotop conjecture
He selected a weighting function that, for any value of N, goes NEGATIVE for half of the values of n. That's not something "between 1 and zero" as he claims and totally denaturalizes the addition of NATRUAL numbers which are NEVER negative.
After a decade+ of hating -1/12, this video has me finally appreciating it.
Same felling.. Have to get the next one that do the same with the 0.999… =1
I appreciate it considerably less now
@@NaR00W why?
@@sepikometry it's still not -1/12
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.
AMAZING video. I can't wait for more research to come of this.
I love Tony’s passion in how he presents things. And a bit of string theory thrown in for good measure!!
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!
No this is the dirt. Infinity is just OK.
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...
Let me guess, pointwise (this series minus that one) vs. (this series missing early terms)? Tao showed if you put in explicit cutoff factors their arguments give away when it's illegitimate.
I was rewatching the original series and wondered if you'd ever revisit it. I'm so glad you did.
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.
Yeah, but that may only be our "idealized" perception of integers. Perhaps there's a better way to grasp that which involves weightings.
@@radoskan I see it not as a way of _perceiving_ integers, but of extending them for injection into the squishy world of ℝeal numbers.
@@radoskanit’s not idealized or a perception, it’s just the definition
Is this a joke? Let's redefine integers as squishy real numbers?
@@CalifornianViking You're right - I said it wrong. I should have said real _world_ - as in "protecting us from infinity"
This is the most exciting science news I've heard for a long time!
This is a cracking video. Best one for a few years !
This really brings me back. I have really watched Numberphile for 10 years!
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".
I really like this weighting function explanation.
These are very convincing arguments. Good job Tony and friends!
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 :)
What a Valentine's day treat XD
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.
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+...
The fact that the form
CN^2 - 1/12 + o(1) is independent of weight, still says there's something very special about -1/12. Wheter you think of it as "the" sum or 'the finite residue' is splitting hairs for me at that point.
@@johngalmann9579 Seeing this form makes me wonder if the result comes from an assumption that a 'C' that evaluates to zero can 'overpower' an N^2 that goes to infinity. Similar to how 0^0 has different limits when approached from different directions, 0 x infinity has different values depending how the 0 and infinity are arrived at. If I take a limit of a sum of 0x1 + 0x2 + 0x3 etc it is easy to say that all terms are zero, then extend to N->infinity, but if I take the sum of N first and then multiply, will I get a different answer?
It does change them. At 3:40 he tells you quite open. That he did everything he could to get to the -1/12 he is clearly not a scientist. He wanted that result.
The summation function is from n=0 to n=N. You do not add the terms for n>N because the summation function explicitly tells you to stop at n=N.
When you say the weights are still less than 1, I can't help myself to compare it with the incorrect claim
0.9 repeating < 1
Infinity challenges our intuition, we must be very cautious when dealing with it
Definetly one of your most interesting videos in recent memory!
This could turn out to be the first step in something big after all
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.
"I wonder if this is what it felt like when"
No, this is _exactly_ what it felt like
that is the result of Terrence Tao I believe. First time I kinda understand something he wrote, and you can really see a glimpse of his genius with this.
This unanswerable question was solved in the 16th century. The sum diverges. Sorry, but It is not and will never be -1/12.
@@herbie_the_hillbillie_goat"the *sum* diverges". You didn't heard (or understood) the whole point of the video.
You wouldn't have said "it was solved since". Not it wasn't "solved". And you need to understand clearly why by reflecting some more time on the topic.
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.
Best video in a long while. Really fantastic!
exciting stuff, looking forward to seeing how this pans out.
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.
Yep. And this is all undoubtedly connected to the p-adics in some way. We just don’t know how. This is the profound mathematical and scientific mystery of our time.
@@LookToWindwardhow are the p-adics involved in this? Please explain a little more.
@@4984christian It is easy to represent and manipulate these sums in -adic notation. For example, the powers of tens sum mentioned in the video is just ….1111 in the 10-adics. Now I know 10 is not prime but that is the general idea. These infinite sums in the -adics act a lot like repeating decimals do in the ordinary reals (technically the reals are a subset of each n-adic number system.)
@@LookToWindward So you could represent the infinite series over the integers by a lot of infinite "power-series" over some p? Like the powerseries over p = 10 would yield 11111111...?
@@4984christian yes but the dots are on the left. Adic numbers can extend infinitely to the left just like the reals can extend infinitely to the right.
Do we need to start a gofundme to buy Tony a new phone charger
Came here to say this!
I saw that too!!😬😬
That charger does not do a smooth transition.
That's a wonderful way to look at series!!
For anyone curious about the formula at 7:54, this can be gotten nicely using generating functions! For simplicity of notation, let A = e^(-1/N).
Let G(x) be sum from n=0 to infinity of A^nx^n. Then that inner term (the summand) can be written as (Ax)^n, so G(x) is a geometric series equal to 1/(1-Ax). Taking the derivative of the series G(x) gives a sum from n=1 to infinity of nA(Ax)^(n-1). So xG'(x) has summand xnA(Ax)^(n-1) = n(Ax)^n and setting x = 1 gives our desired sum.
So taking the derivative of 1/(1-Ax) gives A/(1-Ax)^2, and setting x = 1 gives A/(1-A)^2, which they rearranged as A/(A-1)^2 because the square cancels out the negative anyway.
That second one with cosine...I'd have to think on!
Actually, it looked so similar to the last problem and it made me think of that nice trick where you use Euler's identity e^(ix) = cos(x) + isin(x) to get info about a sum multiplied by cosine and sine at the same time. Then taking the real part gives the cosine sum. Let's try!
In this case, we let B = e^((i-1)/N), so that Sum(B^n) = Sum(e^(-n/N)cos(n/N)) + iSum(e^(-n/N)sin(n/N)).
Then writing G(x) = Sum((Bx)^n) gives two infinite sums C(x) = Sum(e^(-n/N)cos(n/N)) and S(x) = Sum(e^(-n/N)sin(n/N)), so that G(x) = C(x) + iS(x).
So our desired sum is again xG'(x) at x = 1, which is equal to B/(1-B)^2. Writing bar(z) for the complex conjugate of a complex number z, the real part of z is (z + bar(z))/2. Since bar(B/(1-B)^2) = bar(B)/(1-bar(B))^2, we can use this to calculate the real part C(1). And we immediately get S(1) as well!
I remember seeing the -1/12 back in the day. Like most of your other videos, I loved it and it made me love math as well ❤
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
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".
Worse than that. He selected a weighting function that, for any value of N, goes NEGATIVE for half of the values of n. That's not something "between 1 and zero" as he claims and totally denaturalizes the addition of NATRUAL numbers which are NEVER negative.
@@adb012 you copy this comment like 20 times already. Make a yt video proving this is wrong, don't create spam
@@dexterPL ... I copied it only 2 or 3 times as responses to other comments where it was relevant. In some of these places it triggered some constructive discussion. I also wrote a long comment myself (not in response to another comment) explaining my position more formally and in more detail. You don't need to like what I do and I don't need to care whether you like what I do or not. I will not make a yt video.
Hey, maybe you were the guy from the tweet he talked about @1:08 ;-) @@adb012
@@adb012 I don't know why you're so concerned about negative numbers. Using just e^(-n/N) as the "weighting function" produces a series where _not one single term_ of the series is a natural number. So I find it a bit odd that you are outraged that he uses a regulator which allows for negative terms but aren't equally outraged by the fact that other regulators produce series which have no natural numbers at all.
But maybe this point has already been made in another discussion. I don't know where that discussion is, though.
Amazing connections here! One of my favorite numberphile videos in recent memory
An instant classic. Feels like they are about to have a breakthrough
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.
Modular arithmetic (which is one easy way to describe integer underflow/overflow) is distinctly different from zeta regularization (which is what they are doing here).
One does not directly inform understanding of the other.
You can think of zeta regularization like squishing a function so that the annoying parts don't cause you problems and then taking the answer for the squished function instead of the original function. Like, what is the "value" of sin^2(x) at infinity?
That depends entirely on where infinity "stops". (Which doesn't make sense, right?)
Since the sin wave never converges, it doesn't have a value in the limit as x approaches infinity. Regularization allows us to assign a knowingly false, but still useful number to the nonsense equality.
sin^2(x) does NOT equal 0.5, but there is a way of thinking about the function where you can wave your hand and say "for all intents and purposes, it's 1/2".
Very useful for physicists that don't know why a problematic infinity can be ignored but are happy to ignore them, but extremely unhelpful to a mathematician trying to better describe mathematical systems.
"I don't know why, but it's useful even if it's not necessarily true" seems to be the hallmark of modern maths and I am not a fan. (I'm looking at you Axiom of Choice...)
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.
You looked so hard into the distance for the monster, only to see it was right behind you!
think this is one of my fav numberphile vids i've seen so far if not the fav
keep coming back to it
Extremely well described by Tony. Thanks!
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."
Yes but now we are doing it with a Roomba instead of a broom.
That would be my conclusion too. Introducing a weighting function of anything other than 1 or 0 is just sleight of hand, ie a deception
I thought that as well, but I think the idea is, that as N goes to infinity, these weighting functions all go to w(n/N)=1, so you're back at the original infinite sum. The only thing that wasn't really clear to me, is if there are weighting functions that go to a different finite constant. He says he reverse-engineered the cos(n/N) one, so is it possible to reverse-engineer a w that makes the sum go to eg. -1/42?
@@ComaVN the thing is, you're not "back at the original sum". fundamentally, infinite sums are not sums at all, they are limits of sequences. the original sum is a limit of N(N+1)/2. and that's not because we just chose to "cut it off", it's because that's the way the idea is formally defined. and that same definition is used even in these other regularization methods.
when you introduce the weights, you're taking some limit of infinite sums (which are themselves limits of sequences). you're changing the sequence under the limit, so you're changing the result. and yes, by the way, you could get any result you want. any of the methods that will give you -1/12 are basically just asking "what if we treated these sums as some other object". because the idea of "adding to infinity" is meaningless without a formal definition, we're allowed to explore alternatives like this. you just have to be careful not to conflate the different definitions.
@@ComaVN No either C=0 which makes the limit -1/12 or C≠0 and makes the limit blow up. There are no other finite limit only -1/12.
I understood this about as well as the original -1/12 video.
but the math in the first video was HORRIBLY wrong. You can't attach a fixed value to a infinite series that is not convergened.
Saw the original video when I started accelerated study for math, seeing this video one year after finishing my masters in mathematics is so nostalgic and awesome
Absolutely loved this - can't wait to here more!
Even though i enjoy what I'm studying this video honestly made me regret not studying maths or physics
It's never too late!
@@numberphile If only I could still pay my rent while being back at school.
I’ve literally had lectures taught by this guy and this still doesn’t make much sense to me
This is incredible. You should really consider doing this as a full release video (not unlisted).
It will be - just when we release two videos at once we keep one unlisted for a few hours.
@@numberphile oh no. It smells of math-drama now.
The best kind of drama! 🤗
This is insanely awesome, it really seems like the connection with QFT is very deep! It’s like catching a glimpse of an insanely grandiose and beautiful mathematics. I obviously don’t know, and won’t claim to know, but that feeling it always striking to me.
Ive never seen somebody so happy to teach 😊
Great to see you back 😀
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.
This is probably the most exciting TH-cam video I've ever seen.
That is quite intuitive way to sum the numbers to infinity , never thought this can be done to be quite honest.
There is something very metaphysical about this episode.
That's a flaw, not a feature.
I personally don't want math videos to feel or sound like a horoscope.
New ideas always start off sounding like this. Calculus (as Newton and Leibniz knew it) sounded even more like a horoscope for over a hundred years; people like George Berkeley in 1734 criticized the whole subject as being nothing but the study of "ghosts of departed quantities." It wasn't until the 1800s that mathematicians like Weierstrass and Dedekind came around in the to formalize what was going on. You can't expect a new mathematical idea to sound like it came straight from a textbook the moment people start thinking about it.
@@japanada11 Anyone that believes the natural numbers actually sums to -1/12 is a fool.
Period.
Believe whatever fantasies you wish. The rest of us will be doing real math while you twiddle your thumbs with neo-sacred geometry.
It's because they crammed metaphysics into the video without any reason to do so, and in a very philosophically unsound manner.
“Of course I reverse engineered it” 😂
Amazing, this is way more compelling than the analytic continuation motivation for -1/12, and than other regularization schemes for renormalization in QFT !
The shield merch is hilarious!!! Love it ❤️
I feel like I'm in a Numberphile speakeasy
?
@@codycastthis video was previously unlisted
So in another 10 years I hope to see another follow-up on how -1/12 solved string theory.
Looking forward to that one! :)
All because Tony Padilla saw a tweet that made him so angry he had to go and disprove it and in the process solved all of math.
@@efulmer8675 The guy changed the problem. The sum of all whole numbers still diverges.
I think I might have gotten a little bit more insight into why it is -1/12. But only at the last minute when Tony talked about it in a more general sense in the "outro". It is like a very tiny click happened in my brain. I am neither a mathematician nor a physicist but for some reason this finally made a very very tiny amount more sense to me. So thank you for that!
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.
the "limit of partial sums" definition is much more ad hoc than it might seem because it's essentially mathematical induction used outside of it's domain of applicability. i.e. induction only works for a countably finite number of steps, whereas here we're DEFINING the value of the sum such that it keeps the induction valid past infinity. though ad hoc, this could potentially be a reasonable approach. but it turns out there are - arguably - better ways of assigning unique values to divergent sums, as shown in the video.
@@surrendherify induction is not limited to finite ordinals
@@toxicore1190 Sure but if you want to go "past" a limit ordinal (like the infinity of natural numbers), the proposition has to be proved separately at the "limit ordinal"th step, it's truth doesn't follow via the usual induction principle.
@@toxicore1190 when assigning the value infinity to divergent sums whose finite partial sums blow up, no such proof is involved, instead it's just a definition.
@@surrendherify Since when is induction limited to a finite number of steps?