@@iPlayDotaReligiously I mean, what he says in the video is not that. Dude says not that it's The Answer, but that it is A Useful Answer, and that we could really use you do a lot more math to understand why exactly is a useful answer.
@@NormReitzel Frenkel is no charlatan like PT Barnum. He’s one of the greatest mathematicians alive. He has an incredible mastery of a wide variety of subjects, and is a leading researcher in the Langlands program, with dozens of brilliant papers within it. Keep in mind that the Langlands program is already one of the deepest, hardest subjects to study within mathematics, and Frenkel is at its forefront. Not only this, but he has created many of the links between Langlands duality and mathematical physics. I can’t speak of any specifics (I don’t study the subjects he does), but I know enough to have a tremendous respect for him.
"One thing which is important in mathematics is that we just can never leave [...] loose ends. [...] Mathematics is rigorous, and at the end of the day we are looking for a rigorous justification of everything. In other words, we are not content with just saying that there is some magic over there. There is magic, but we always want to explain it." Edward Frenkel, Professor of Mathematics I just... this... wow...
I am a physicist and I deal with quantum-electro-dynamics (QED). I want to share with you (or show you ) the existance of a finite value for divergent sums. In QED we calculate vertices (meaning something like electron interacts with another electron via one photon). And we can calculate the magnitude of this interaction. But if one includes fluctuations (which are known to be omnipresent) we always get infinity. The vertex "electron interacts with itself via a photon" is for instance always infinite and can be added to any interaction. And now the above video comes into the game. We can renormalize the infinte sums just the way described above. and the outcome of the theory is absolutly impressive. QED has given values of physical constants up to the 14 spot behind 0 correctly, compared with experiments (which is unreached in any other physical theory). Therefore the existance of finite values for divergent sums is not a mathmatical fantacy, it is the TRUE reality.
@@michaelzimmermann9804 Don't know if it is okay to bother you with subject 4 years after your work, but i really wish you can add more precisions and details about your explanation
6:52 "Does the square root of -1 exist". I laughed so hard (in delight) when he said that because his point was made blindingly obvious by asking such a simple question. Elegance at it's best. I'd sit through one of his lectures any time any where.
@@paulbennett7021 exactrly, we can concecve -1/12 describe it, and use it, its not the sum to the series, but its a special result. he called it regularized sum, although i dont like using the word sum. i would say it is a special result, and this is in fact the rieman zetta function. -1/12 is an important output of the series, kind of like a function, but its not the sum but still a very important number.
@@rohangeorge712 I don't really understand. He keeps referring to -1/12 as the infinite sum. But he says you get there by removing the infinite part. So -1/12 can't be the infinite sum, right? So what is it? Is it like a function, and you're just taking a slice of the series?
Numberphile has managed to take the subject I disliked the most in school and turned it into one of my favourite and inspiring subjects on UA-cam! Bravo!
James Flyleaf y u no like math... Unlike all the language and humanity subjects, math and science have a definite answer, as in if you are right, you are right. If you are wrong, you are wrong. There is no such thing in math and science where your are half right or wrong, which is what I love.
It's much easier to accept the answer on an intuitive level, when you think of the result as a number that describes the series, or describes a certain property of the series, rather than the actual sum. Whether or not it is the actual sum, is like asking whether or not the square root of -1 is a number. Mathematics is so crazy, I like that nature can spring up essentially patterns that exist, that blow our intuition out of the water.
Well, there is an = there, so by mathematic law it means that the mathematic operations on the other side of = equal the number on the left. If we had a "describes as" symbol I would be fine with this, but what he is showing here is nonsense. And yes, I get it, "a root cannot be negative" and such have held us back and its important to push boundaries. But, if you are gonna push boundaries, then do it well.
The sum of any infinite series is a matter of definition. If we adopt the definition of the Ramanujan summation, then the sum of all the natural numbers is -1/12. It's no more "nonsense" than assigning values to any other infinite series.
@Håkon You can not simply divide ∞/∞ because that is an indeterminate form. We need more information to know what the answer is, nor is there any reason to divide by ∞ anyway. ∞/∞ ≠ 1, because ∞ is not a number, but a concept. It does not cancel out, unless it can be shown that both the numerator and the denominator are the same variable which can be treated like a number. Look at the formula to find the slope of a line via 2 points on the line. The line already has the slope that it has. But let's lose some of the information, so that we have 2 points that are actually the same point. Then we have (y2-y1)/(x2-x1) = (y1-y1)/(x1-x1) = 0/0. The slope through just one point on the line, can be any slope that you would like for it to be. 0 slope, positive slope, negative slope, infinite slope. But only one of those answers is the correct one. The line can only have but the one slope that it has. It is not the fault of the line, that you were careless and did not gather enough information about it. But the one point does not give us enough information to determine what the slope of the line is.
it needs more symbols then, not just an equals. I even get how infinities cannot equal other infinities, there are different values to them....(but that's different notation, right?) and someone to describe why -1/12, how you could apply it in some function, why is squared 0 and cubed 1/120? what relation do they have? what is one of these examples where you can put this in as a replacement for infinity and come out correct, and how? If its some kind of descriptor why is it minus, after 'removing infinity'? how is infinity removed? obviously not by subtration or division! nor by turning it into a function or subsection or an average. -1/12 is outside, not even inbetween, not a single subtraction sign in there.....what does it represent then? then i might accept the answer!
The best explanation of this -1/12 business I've ever seen. There IS a lot more going on here than some of the other professors did a poor job of articulating.
The best explanation I’ve seen personally on how the -1/12 figure is obtained from the zeta function and the analysis of Complex Functions Mr.Riemann and the rest were involved in is the video on Analytic Continuation by 3blue1brown. It’s worth a try.
I think the most enlightening way is to take the integral of the closed form of the finite sum of positive integers, (n+1)(n)/2 between its zeroes, 0 and 1. This value ends up needing to be the linear coefficient of the closed form of the sum of squares, and its integral gives us the linear coefficient for the next power, etc. This makes sure that the closed form "lines up" with the finite sum. Because they form a crucial part of the coefficients of Faulhaber polynomials (the fancy name for the closed forms of sums of natural powers of positive integers), they're related to the Bernoulli numbers, which are related to the zeta function. Basically, if you're going to be summing a power of consecutive integers, the zeta function is gonna sneak in there somewhere.
"Euler was a kind of mathematical outlaw... a kind of a mathematical gangster..." Euler: don't worry dear Riemann, i`ll make 'em a proof they can't refuse... LOL
I love Professor Frenkel's accent and the way he explains complex mathematical concepts like this so amazingly well, so that even things that seem to make no sense suddenly make so much sense.
I think this is my favorite numberphile video because it explains so well the process of mathematical development and gives a glimpse into mathematical thought processes that most people do not see.
I feel so cheated by the public school systems for not teasing my mind with maths more as a child. When I revisited my education in my mid 20s (went back to school for my own enrichment) I was introduced to calculus I saw the horizon that is the magic of maths pushed well beyond by perspective at that time (and today). I do so wish I had seen that much earlier in life and now struggle to make up lost time only wishing I had more of it to do so.
You have taken the integral of your life and found out there is an upper limit. You only find out what it is at a moment before your death. But how long is that moment? Ahhhhh. That's the question!
I'm 18 and I barely understand the true meaning of calculus also I'm like the worst in geometry, but no way I'm gonna stop studying, there is no late time.
Cantor is another example of the prof.'s outlaws. The thing most non-mathematicians don't know is that the math that gets presented (such as in published papers, lectures, classrooms, etc.) is the "finished product." It seems clean and neat and perfect and irreducible - precisely because some mathematician (or more than one) toiled away at it (sometimes for years - even decades) to figure it out; it's clearest, simplest form, it's implications, etc. I've heard the process of doing math likened to a restaurant. Most of us sit in the dining area; we only get the finished meals. We never see the chaos that goes on in the kitchen! Can you imagine what a chef goes through? Experimenting with different ingredients, failing time and time again - until, finally, AHA! The perfect dish! That's what doing math 'feels' like. We aren't simply given the answers! We have to work at it - often fumbling around, until we get it right. Of course we're only going to present the world (that is, other mathematicians) with the completed, perfect recipe! Anything would be an embarrassment! Doesn't mean it's easy! And, as often as not, we STILL don't understand ALL the implications of our own discoveries; we may never do so in our own lifetime. That's the way it goes, kids. tavi.
Marshall Harrison - Guitarist - Sadly, 98% of English speakers don't even recognize the utility, or beauty, of wielding the language. They kinda notice when they encounter someone twice as articulate as themselves, but hardly discern when someone's command is twenty times that.
Hearing this guy come up with analogy after analogy to justify -1/12 is like listening to a person with sight trying to describe color to blind people. I really feel like he gets it. Maybe it'll make sense for me one day with enough analogies and experience.
On Facebook I received some backlash when I posted my amazement that the zeta function can assign -1/12 to the sum of all natural numbers (zeta(-1)). I am just as surprised at the backlash as I am about the -1/12 value. I can better-see how the discoverer of irrational numbers must have felt when they were ostracised long ago. I wonder if Reimann ever copped-it over his -1/12 value? Edit: Thank you, Brady. I love your videos. Keep it up, mate!
no sum of positive numbers is smaller than any number included. the sum of all numbers is not less than zero, unless you ask a faulty function. maybe that's why they didn't agree with you, it's not correct.
i is legitimate in every context. it stands in for sqrt (-1) so that operations can be performed on it to give it a real value. this value was obtained through error.
-1/12 reminds me of matrix determinant… It is a non-intuitive number that isn't the regular way to say "this equals that," but is still an essential "identity". I don't think the equal sign (=) is the proper way to represent this, since we defined *equal* as "considered to be the same as another in status or quality". In this case, the "1 + 2 + 3 + 4 + 5 + …" does not equal "-1/12", but is in a way the equivalent of a determinant (as with matrix). It is a property, but not an equivalent. The equal sign (=) isn't the proper sign… *It is only my opinion… I am not a mathematician.* Am I wrong? What do you think?
The notation is a surface complaint, and I think it shifts the problem elsewhere. The root problem is the difficulty with reconciling the philosophical nature of infinities. We have an intuitive understanding of sqrt(-1) as a number on a 2D plane, which is why we accept complex numbers without much problem. We know what the determinant of a matrix is, as it is a finite value and has useful properties in linear algebra. Once you get into infinities, there is a huge barrier with what infinity actually "is" and how this is communicated. For example, the idea of "potential" vs. "actual" infinities. An actual infinity is that which is realized as a mathematical object in some independent reality which can be reasoned with and manipulated. A potential infinity is something which can be constructed in the human mind, that is, a series going on forever without end, and cutting it off at some point lands you back into the land of finite numbers (partial sums). What we found with actual infinity is, by accepting their existence, it yields a bunch of mathematical results that have applicability everywhere from number theory, to actual, experimentally measurable phenomena in physics.
+Ruben Honestly, with my limited understanding of this, that could be correct. The Universe may not actually have a perfect mathematical system, and this is one of the "bugs". That's not the way I prefer to think about it, but it''s consistent with what we know about the universe.
Awwwww, I was hoping we would get see this thing in action. Like some problem you used -1/12 to solve, then showed an alternate path to prove that the answer was correct.
Wow. He really makes this clear in a way I haven't heard before. I have heard the explanation that this sum is actually -1/12 mod infinity before, but I never understood what that meant. When he gave the example of slicing off the infinity of dirt and just keeping the nugget, something just clicked. We already know from Cantor there are levels of infinities, so why couldn't this value be -1/12 in some finite field analog from a higher infinity where the lower infinity just happens to be the modulus? I hope to hear a lot more from Professor Frenkel. Really like his style.
I really hope this whole affair didn't discourage Brady from covering mind-blowing topics because there are so many of them left. Maybe the take-home message from this is that it's ok to blow people's mind but it's a bad idea to leave them out in the rain with the crazy fact you threw at them just because you thought that would make the whole thing seem even more impressive.
Look up Derivation of Casimir effect assuming Zeta regularization. You will need to sum n^(3-x), let x=0 and rewrite as 1/n^{-3) Now you can input -3 into the Riemann Zeta Function and get 1/120
***** You are wrong, the Casimir effect has been demonstrated. The sum of natural numbers being infinity is intuitive but it is not useful. On the other hand -1/12 is counter-intuitive but is used in Quantum Electrodynamics and Quantum Field Theory with the derivation of the Casimir Effect
This video is the perfect explanation on why math is a tool, first and foremost, in the sense that "if something works we'll use it". Before taking Calculus at my university I always saw math as something mystical and arcane, but thanks to explanations like this one I finally overcame my fear and started taking real interest in this beautiful subject
This explanation is way more acceptable than the ones in the other videos! There is actually something unknown about the subject. It's not just "Hey, this divergent sum equals -1/12, period"
Actually you can see why. The teachers/professors/the people who tries to explain it to whom did not understand it, does not explain it like that, they are explaining it like it is literally -1/12. I look at the topic like there are several realms of mathematics, like these two. In one of them it is equal to infinity and in one of them it is -1/12.
This professor speaks so well and looks at the camera just right. I really was able to absorb the whole topic. Thanks so much for this. I hope he can speak about another interesting topic soon.
This man must give fascinating lectures. He seems to have an exceptional ability in explaining mathematical topics. I love his answer to Brady's question about "breaking the rules" when calculating divergent series. I'm definitely buying his book soon.
This is what I love about maths. It’s so logical yet so bizarre. By the way he did a great job explaining such unordinary topic in a way, that even I can follow the ideas.
Really enjoying Professor Frenkel's explanatory methods. If almost any other professor tried to explain this concept to me I'd be more lost than Malaysian Airlines. Too soon?
The easiest way of picturing it is an infinite line as opppsed to an infinite plain defined by two infinite lines. Thesre are of course many more, and these two would fall into the same category under xertain classification, but it is easy to see how one differs from the other fundamentally.
Thanks. The analogy drawn between the square root of a negative number and renormalization of the naively infinite sums by the construction of a more inclusive mathematical setting was just beautiful in its phrasing. Bravo! I'm nostalgic for my course of complex analysis many decades ago. Talk about a nugget of gold in the infinite dirt of the internet.
Watched the whole video. Thank you so much, Dr. Frenkel (as well as Mathologer) explained this issue the best I have ever heard. Too bad the Ultra-finists mis-quote him. :(
I agree. The way mathematics (and physics) relate the ordered sum of all natural numbers to the value -1/12 isn't saying that they are equal to each other, but that the process of summing has some characteristics that can be represented by -1/12. This characteristics is unique even if you change your approach to finding it, as many (if not all) methods of assigning finite values to that sum yields -1/12. And this result is useful in physics as it can help us describe and predict the events in the universe.
@@jackren295 - Absolutely no explanation was given to how the value of -1/12 was arrived at, which was the whole reason I wasted 15 minutes watching the video.
Now that imaginary numbers are "out the bag", Numberphile's surely got to explain that old chesnut: e^(i*pi) = -1 or as my maths teacher used to prefer: e^(i*pi) + 1 = 0
The core of that identity, and the more interesting part really, is the formula Euler (or Cotes, perhaps) presented: e^(iθ) = cosθ + isinθ As can be seen, when θ=π then you have e^(iπ) = cosπ + isinπ = -1 + i(0) = -1. This also results in e^(i2π) = cos2π + isin2π = 1, and e^(iπ/2) = i, and so on.
Well, the specific case for pi comes from Euler's formula e^(i*x) = cos(x) + i*sin(x). Substituting pi for x gives -1, because the sin term vanishes and the cos term evaluates to -1. As for where the more general Euler's formula comes from, I don't know for sure, but it can be proven by comparing the Taylor series expansions of either side of the equation. It comes from calculus though, and I'm not sure whether Brady typically includes calculus topics in the Numberphile videos.
These has been most intresting videos in mathematics I have ever seen. It's disturbing, counter-intuitive and facinating. Also it ties quantum physics and mathematics together in ways that I did not expect.
The way I see this is many mathematical processes seem to have more than one valid, but not necessarily "correct" answer. For example, the square root of 4 is both 2 & -2, but if an actual square has an area of 4 square metres the length of the sides is 2 metres, not -2 metres, as only positive distances exist in reality (you can define it as negative, but only by cheating & counting a positive distance backwards from a reference point at the other end of the side). In this case it seems like this sum has two values again, infinity & -1/12, but this time we discard the "correct" value of infinity, and use the incorrect value of -1/12, because this time the "wrong" answer is the one that exists in real life. It's like sometimes we can only access one of the two values for any mathematical function, and usually it is the "correct" one that fits into reality, but very occasionally it's that one which falls outside the scope of what can really exist & we have to make do with the other one.
+David Harrison Well I wouldn't say that you could view any of a number of valid answers as being the universally correct answer. For example with the square root of 4, in terms of pure mathematics 2 and -2 are both equally valid. When applying that to area and distance, negative numbers aren't allowed so -2 no longer is valid in that context. But if I were to ask you the following question: "You owed a man a sum of money; however, that man decided that he was going to be generous and forgive the debt that you owed him and give you twice the money you owed him. He gave you 4 bucks, how much money did you have (assuming that you would pay off any debt you had if you had the money)?" The answer would be -2 because if you had money then according to the question you would've paid off the debt. While the area and distance context appears much more often IRL than the example I gave, there are times when both the positive and negative answers of a square root have meaning. For example with quadratic equations, the quadratic formula has a square root and the two answers found are when the parabola crosses the x axis, which when applied to the context of projectiles, are the two times the projectile is on a flat surface. The same's true for the -1/12. When considering the context that most people assume, that if I were to get $1 + $2 + $3 and so on, this really doesn't make sense. Of the two valid answers (infinity and -1/12) infinity makes much more sense. But apparently in many different contexts in mathematics and physics, -1/12 is much more useful and makes more sense. By itself there is no universally correct answer, even just assuming infinity is probably derived from us putting it in the context that we normally put numbers in, representing physical objects. But when taken to infinite sums, such a context really doesn't make too much sense as there so far isn't anything that we're certain is infinite. (except maybe stupidity, sorry Einstein joke, not directed at you). In a pure mathematical context, where the answer isn't being used for anything, any answer that can be derived is equally valid.
It seems to me that the mistake is to think that the usual rules of mathematics are "real". Addition, multiplication etc. are also abstract concepts that were invented by people. 2x5=10 in the sense that you get that answer if you follow the rules. It's also "real" in the sense that those concepts can be applied to real things, which is what makes mathematics so useful. It sounds like 1+2+3...=-1/12 is real in the same sense, if a little harder to picture.
He has awesome command over English. What I liked most that he called imaginry numbers 'imaginry' without any worry, unlike many other videos. Though, imaginry numbers are as real as other numbers, but I have seen so many videos where I see that it would be explained that we should not think of these numbers as imaginary. He didn't emphasize on that. People like me who learn from these videos will certainly say something about as we are influenced by these videos.
Thank you for mentioning Ramanujan, as I vaguely remember him sending this solution to Hardy, and Hardy and Littlewood both laughed. But, this should have been well known to both of them by 1913 ?
Crazy how Euler could come up with something so abstract with seemingly no possible applications, and then however many, many years later it is actually used in a branch of physics euler would have never been able to comprehend
This has always intrigued me and I've done a bit of research here and there... but hearing Professor Frenkel explain the topic is a piece of gold in of itself!
A huge like. Excellent explanations. I would have added that this regularization helps make, say, the zeta function a whole rather than discarding half of it and clamming up like this: we cannot sum over infinity a divergent series - let's stop doing math in that area. Let's call divergent series to be a taboo. They are not taboo, i.e, they need to be studied too just as complex numbers were studied and developed. It's surprising to see how the Casimir effect seems to be in line with the zeta function and summations like this.
This helped a lot. I've seen way longer videos on this that only confused me more. This at least gave me a basic sense of what's going on here through the use of analogy and discussion of context and explaining that even by mathamaticians and physicists may not have this fully understood.
I was one of the many people who commented on the first video, crying out heresy at Tony and Ed when they claimed to be able to calculate the sum of an infinite series. This video really helped me better understand the value of these otherwise illegitimate calculations.
sk8rdman Calculating the sum of an infinite series is simple, it is finding the sum of an infinite divergent series which can be a bit more complicated.
You should do a video showing an actual calculation with one of these infinite sums. Use something where you get the right answer using the -1/12 substitution and then solve it using a different method to show that you did indeed get the correct answer.
It would probably require integration, derivative calculations and lots of other stuff that in movies is called "scientific mumbo-jumbo". It would be understandable for very few subscribers.
Ha, I was going to say the same thing but didn't want to be too controversial :D The chance of understanding everything (or anything) up to the point in the book that uses this is very remote.
MaC S3th the way I learned it involved using the Gamma function and finding patterns which let you define the extend the domain of the Zeta function one vertical strip of width 1 at a time. It took a few two-hour lectures.
Yes, Euler's discoveries in mathematics are monumental in the history of knowledge. His work on headdresses, on the other hand, was unable to garner enduring appreciation.
The more I think about it, the more the -1/12 answer makes sense to me, at least as a possibility. I know the common wisdom is that if a number keeps getting bigger, the final answer has to be bigger, but that's only true until you break infinity. I recall seeing in high school and college, a lot, when talking about limits and asymptotes, that a number will approach infinity as it approaches the asymptote from the one side and approach negative infinity as it approaches the asymptote from the other side and then at the asymptote it seems to simuletaneously take on many and no values at the same time, sometimes a range, usually all real values. At any rate, it is definitely not one defined value. But lately I've been watching math vidoes that talk about the fascination with infinity. Many people erroneously assume there is a real number called infinity, when, really, infinity desribes the set of all real numbers. So, all real numbers are less than infinity. If you add a finite number to a finite number, it too will be less than infinity. And infinity is not itself a real number. The video I saw talked about magnitudes of infinity, of how there are things bigger than infinity, they are just not real numbers, and more so are used to describe sets of numbers. Again this take me back to college to two things. One is the concept of a delta function if engineering, which is the deritive of a step function. It is infinity big at one value and 0 elsewhere and it's directive over the discrete value is 1, essentially it is a rectangle of 0 width and infinity height whose area is 1, and you can multiply constants by it to make it bigger and the area likewise will just be a multiple bigger. I recall it being uses to simulate signal spikes. I assume multiplying 0 by infinity and getting a constant is something pure mathemeticians don't like. But also I recall, I think it was working with the stability of systems, a value could go to infinity at an asymptote, wrap around the entire coordinate system multiple times, and then come back on the same side or opposite side from the other side of the asymptote. I wish I remember more about what it exactly meant and how you determine how many times a value could "wrap around infinity". Still, now this -1/12 is making physical sense in my head (and trust me, I have trouble reconciling things in my head, so I'm always asking "why did this happen" until I understand things). So, if you add a finite number to a finite number, you are still less than infinity, but if you add numbers beyond infinity, things wrap around back to negative infinity, and then if they keep going it could happen more than one time. Well, when you keep adding increasing bigger numbers together an infinite amount of times, you get something that's close to half of infinity times infinity. In fact, for finite numbers, the formula is n*(n+1)/2. It is a number beyond infinity, so I can absolute see it wrapping back around to a negative number or even to a finite positive number in our single finite set of real numbers. It's hard to imagine a number being bigger than infinity sinse we experience life in a seemly limited scrope. I cannot imagine what it exactly means in a physical sense. But I can see rules fit together. However, I've notice when numbers hit asymptotes, they seem to become all real numbers (or even just a range) at once and seeming no numbers at once, but there do seem to be times when a specific number does seem to represent the special circumstances, so I like the talk in this video about removing the cruft to get to the gold nugget, removing the infinite values that make up the answer until you have the one that best represents what is going on. So, yeah, information from multiple sources, 20 years ago, 6 months ago, and just now just helped me make sense of something that on the surface looks weird, but seems to actually make sense. (Now as to the proof, I've never attempted it before but I do like the simplicity of Euclid's solution despite the divergent function problem-- which I've seen some comments complain about and then other explain it does follow special rules that can be used.) I certainly miss mathematics, though.
Infinity does not "wrap around". We are not talking of modulus arithmetic here. limit (n • (n + 1) / 2) as n approaches ∞ simply equals ∞ and not "beyond infinity". When I asked my graphing calculator what 1/0 is, it said undef. A graph of 1/x shows what the problem is. It diverges at x=0. It does not "wrap around", but rather the sign of x switches at 0 between negative and positive. My graphing calculator objects to the lack of convergence it seems. But when I asked it, what the answer is to limit (1/x) as x approaches 0 from the positive direction, it gave me the answer of ∞. What if there is no "gold nugget" and it just fools' gold, or there simply is no gold? Isn't it just make-believe nonsense? limit (n • (n + 1) / 2) blows up to ∞ as n approaches ∞. End of story. Why do we need a make-believe nonsense "assigned" value to it? You can remove all of the integers from the series that they do not like, and still you will not get -1/12.
I agree with the professor. If we only go by 'what is allowed' we'd never advance. This is especially true in the history of Mathematics from the early Greeks and perhaps earlier in pre-history; however, Pythagoras had "his own way" of seeing numbers, and his theorem is timeless still used up to our day. I say we dare see things our own way by what is NOT allowed or conventional. We may yet discover the next Pythagorean Theorem or someting as useful :)
The whole point is that the context defines what is allowed though. You still need a filter to distinguish correct statements from incorrect ones (i.e., what is allowed versus what isn't). Under one set of rules you're not allowed to sum divergent series. Under a different set of rules, you are. The problem I have with the Numberphile videos is that they use a non-conventional set of rules and imply that the result is valid for the conventional set of rules. That's an obvious error, due to lack of rigor.
I'm specifically addressing this sentiment that mathematics is somehow advanced by 'breaking the rules'. That's incorrect. Mathematics is advanced by making new rules and rigorously following them to arrive at useful results. Never rule breaking; logic always applies. Just like for complex numbers: they can lead to real number solutions, but only insofar as complex numbers can be converted back into real numbers. It's just like transforming between spatial and frequency domains. You still have to be mindful of the restrictions keeping you from mixing one framework with the other.
As a non mathematician, this actually makes a lot of sense. He's basically saying each infinite sum that has a defined pattern has a value that is not the sum, or an expected value, but almost a measuring tool, a comparator to be used to compare to other infinite sums. Almost like the sine and cosine functions find a very accurate measurement of something calculated using an irrational constant (pi)
The keyword here is "regularized sum." It goes through a transformation so writing '1+2+3+4+5+6.... = -1/12' is not accurate unless you include some notation to indicate that the sum of positive integers has been transformed like 'f(1+2+3+4+5+6....) = -1/12'.
No its more than that. If the "regularized sum" was just something like a weird function that assigns values to infinite sums, it would not explain why you can replace the naked sum 1+2+3+... with -1/12 in many situations and get the right answer. When you replace it with -1/12, there was no 'regularized-sum-function' there... it's just the sum of all integers. So in a sense -1/12 seems to actually be *equal* to the sum of all integers... of course the sum of all integers also seems to be infinity. Why and how both can be at the same time, is the mysterious part... it certainly seems like both results are actually true.
@@YEC999 If that is so, then why does it give the right answer in the Casimir effect? That is an experiment you can do in a lab, replacing the infinite sum with -1/12 gives the same result as the value that is measured? That -1/12 is not some obscure mathematical hocus pocus... its the real deal, its in the real world, it can be measured and tested.
![Revolutionary Math Proof No One Could Explain...Until Now [Part 1]](/img/n.gif)
A blog with more links and info - www.bradyharanblog.com/blog/2015/1/11/this-blog-probably-wont-help
"Does the square root of negative one exist? Come on, Brady." He taunts Brady with a glint in his eye and an evil grin.
i
To prove that this series sums to -1/12 is correct too.
No it doesn’t
@@iPlayDotaReligiously I mean, what he says in the video is not that. Dude says not that it's The Answer, but that it is A Useful Answer, and that we could really use you do a lot more math to understand why exactly is a useful answer.
@@stretchyone more like more theoretical physics like quantum mechanic, but yeah.
"Euler was a mathematical gangster."
- Prof. E. Frenkel
+Pelle Olsson So you drink until 4 am and listen to gg allin and you are into advanced mathematics. You're neat.
thug life
life iz a struggle
420 likes
Future plans set. I hope Mathematical gangsters get payed a lot :/
Euler did some painstaking work and lost his vision.
After the loss of vision he said "Now I will have fewer distractions"
Now thats gangster lol
Russian Jaime Lannister makes some compelling points.
hahaha you are so right
Ha ha! I was thinking the same thing. But I thought his accent sounded Braavosian.
I was thinking that he was Russian since I first heard his accent, but now I'm sure.
Russians! (I'm also Russian. .w.)
Somewhat strange, but I understand J. Grime's accent better than this' obviously Russian speaker. (Although I'm Russian)
"There is magic, but we always want to explain it." What a perfect encapsulation of scientific endeavor.
Amazing words
Any significantly analyzed magic is indistinguishable from science
yeah humans.... always trying to find the answers
that's what they are paid for
And failure.
man this professor must be an amazing educator. he makes things so clear and easy to understand.
So did P.T.Barnum. This is the amazing Egress.
@@NormReitzel egress?
Indeed! I finally understood imaginary numbers
@@NormReitzel
Frenkel is no charlatan like PT Barnum.
He’s one of the greatest mathematicians alive.
He has an incredible mastery of a wide variety of subjects, and is a leading researcher in the Langlands program, with dozens of brilliant papers within it.
Keep in mind that the Langlands program is already one of the deepest, hardest subjects to study within mathematics, and Frenkel is at its forefront.
Not only this, but he has created many of the links between Langlands duality and mathematical physics.
I can’t speak of any specifics (I don’t study the subjects he does), but I know enough to have a tremendous respect for him.
His lectures are available online. He also has made a movie.
"One thing which is important in mathematics is that we just can never leave [...] loose ends. [...] Mathematics is rigorous, and at the end of the day we are looking for a rigorous justification of everything. In other words, we are not content with just saying that there is some magic over there. There is magic, but we always want to explain it."
Edward Frenkel, Professor of Mathematics
I just... this... wow...
This is not rigorous. You don't cut off a diverging series irrespective of weighting function.
This man is BRILLIANT. The way he clearly explains complex topics in a language that is not his mother tongue is astounding.
Imagine being in an infinite job where you're paid 1 more dollar every day then at the end get a paycheck with - 1/12 dollars.
Ah, but the end is at infinity.
and lots of dirt
Ehhh, that series doesn't "equal" that perse
It's all fiat money anyways
I prefer not to live till infinity.
his accent just make everything better
yeah... it's kinda soothing to listen the explanations through that piece of particular pitch and frequency
Makes.
I am a physicist and I deal with quantum-electro-dynamics (QED). I want to share with you (or show you ) the existance of a finite value for divergent sums.
In QED we calculate vertices (meaning something like electron interacts with another electron via one photon). And we can calculate the magnitude of this interaction. But if one includes fluctuations (which are known to be omnipresent) we always get infinity. The vertex "electron interacts with itself via a photon" is for instance always infinite and can be added to any interaction. And now the above video comes into the game. We can renormalize the infinte sums just the way described above. and the outcome of the theory is absolutly impressive. QED has given values of physical constants up to the 14 spot behind 0 correctly, compared with experiments (which is unreached in any other physical theory). Therefore the existance of finite values for divergent sums is not a mathmatical fantacy, it is the TRUE reality.
oops... I should ahave watched the video to the end before responding.., he mentions what I just described :D sorry..
And that's why Pauli-Villars and Hadamard Regularization exist.
@@michaelzimmermann9804 Don't know if it is okay to bother you with subject 4 years after your work, but i really wish you can add more precisions and details about your explanation
Wonderful
Michael Zimmermann doesn’t renormalization depend on experiments???
6:52 "Does the square root of -1 exist". I laughed so hard (in delight) when he said that because his point was made blindingly obvious by asking such a simple question. Elegance at it's best. I'd sit through one of his lectures any time any where.
i'd like, but it's at 169 haha
Of course root -1 exists; we can conceive it, describe it, and use it.
@@paulbennett7021 exactrly, we can concecve -1/12 describe it, and use it, its not the sum to the series, but its a special result. he called it regularized sum, although i dont like using the word sum. i would say it is a special result, and this is in fact the rieman zetta function. -1/12 is an important output of the series, kind of like a function, but its not the sum but still a very important number.
@@rohangeorge712 I don't really understand. He keeps referring to -1/12 as the infinite sum. But he says you get there by removing the infinite part. So -1/12 can't be the infinite sum, right? So what is it? Is it like a function, and you're just taking a slice of the series?
@@paulbennett7021 well you technically cannot take the square root of a negative number. The definition of i isn't √-1 but i²=-1 for a reason
Numberphile has managed to take the subject I disliked the most in school and turned it into one of my favourite and inspiring subjects on UA-cam! Bravo!
that's great
Numberphile for some reason that seemed like sarcasm without a ! at the end XD
that's great factorial
James Flyleaf y u no like math... Unlike all the language and humanity subjects, math and science have a definite answer, as in if you are right, you are right. If you are wrong, you are wrong. There is no such thing in math and science where your are half right or wrong, which is what I love.
lol y do u have such a problem with the subject of sums of infinite divergent series
He looks like a guy that would be taunting Bruce Willis over the phone in a diehard movie.
lol I totally can see it
+Yung Brizzy and a lovely accent too!
lol
Simon says what is the square root of negative 6
w
I respect this professor’s mastery of my language (English), his second language, in addition to his mastery of mathematics.
It's much easier to accept the answer on an intuitive level, when you think of the result as a number that describes the series, or describes a certain property of the series, rather than the actual sum. Whether or not it is the actual sum, is like asking whether or not the square root of -1 is a number. Mathematics is so crazy, I like that nature can spring up essentially patterns that exist, that blow our intuition out of the water.
Well, there is an = there, so by mathematic law it means that the mathematic operations on the other side of = equal the number on the left.
If we had a "describes as" symbol I would be fine with this, but what he is showing here is nonsense.
And yes, I get it, "a root cannot be negative" and such have held us back and its important to push boundaries.
But, if you are gonna push boundaries, then do it well.
The sum of any infinite series is a matter of definition. If we adopt the definition of the Ramanujan summation, then the sum of all the natural numbers is -1/12. It's no more "nonsense" than assigning values to any other infinite series.
@Håkon
You can not simply divide ∞/∞ because that is an indeterminate form. We need more information to know what the answer is, nor is there any reason to divide by ∞ anyway. ∞/∞ ≠ 1, because ∞ is not a number, but a concept. It does not cancel out, unless it can be shown that both the numerator and the denominator are the same variable which can be treated like a number.
Look at the formula to find the slope of a line via 2 points on the line. The line already has the slope that it has. But let's lose some of the information, so that we have 2 points that are actually the same point. Then we have (y2-y1)/(x2-x1) = (y1-y1)/(x1-x1) = 0/0. The slope through just one point on the line, can be any slope that you would like for it to be. 0 slope, positive slope, negative slope, infinite slope. But only one of those answers is the correct one. The line can only have but the one slope that it has. It is not the fault of the line, that you were careless and did not gather enough information about it. But the one point does not give us enough information to determine what the slope of the line is.
it needs more symbols then, not just an equals.
I even get how infinities cannot equal other infinities, there are different values to them....(but that's different notation, right?)
and someone to describe why -1/12, how you could apply it in some function, why is squared 0 and cubed 1/120? what relation do they have? what is one of these examples where you can put this in as a replacement for infinity and come out correct, and how?
If its some kind of descriptor why is it minus, after 'removing infinity'? how is infinity removed? obviously not by subtration or division! nor by turning it into a function or subsection or an average. -1/12 is outside, not even inbetween, not a single subtraction sign in there.....what does it represent then? then i might accept the answer!
@georgesimpson1406 I think these are the perfect kinds of questions you bring to your professor's office hours XD
Man, mathematics is wild.
The best explanation of this -1/12 business I've ever seen. There IS a lot more going on here than some of the other professors did a poor job of articulating.
I still don't get exactly how the -1/12 number is obtained.
Yeah he didnt explain how they got it at all. Thats what I was curious about.
The best explanation I’ve seen personally on how the -1/12 figure is obtained from the zeta function and the analysis of Complex Functions Mr.Riemann and the rest were involved in is the video on Analytic Continuation by 3blue1brown. It’s worth a try.
xnalebb @RubalCava Eddie woo explains a way
I think the most enlightening way is to take the integral of the closed form of the finite sum of positive integers, (n+1)(n)/2 between its zeroes, 0 and 1. This value ends up needing to be the linear coefficient of the closed form of the sum of squares, and its integral gives us the linear coefficient for the next power, etc. This makes sure that the closed form "lines up" with the finite sum.
Because they form a crucial part of the coefficients of Faulhaber polynomials (the fancy name for the closed forms of sums of natural powers of positive integers), they're related to the Bernoulli numbers, which are related to the zeta function. Basically, if you're going to be summing a power of consecutive integers, the zeta function is gonna sneak in there somewhere.
I just watched Srinivasa Ramanujan movie, "The man who mew infinity"
"Euler was a kind of mathematical outlaw... a kind of a mathematical gangster..."
Euler: don't worry dear Riemann, i`ll make 'em a proof they can't refuse... LOL
*refute
Euler has a dream
@@elnico5623 errm.. no!
what do we do with a divergent series?
we just ignore them.
good advice.
_video ends_
How can you just ignore it ?? It bothers me horribly !!!!
The books or the movies? Or just both?
@@smritisivakumar3291 hey !
@@smritisivakumar3291 I'm also from India do you wanna discuss pure mathematics?
We used to “throw away” negatives, I wonder if someday looking back we’ll understand those series better
we do?
Heck a very long time ago the concept of zero was controversial. We just move concepts from controversial to accepted in certain contexts.
I love Professor Frenkel's accent and the way he explains complex mathematical concepts like this so amazingly well, so that even things that seem to make no sense suddenly make so much sense.
I think this is my favorite numberphile video because it explains so well the process of mathematical development and gives a glimpse into mathematical thought processes that most people do not see.
This guy is very fun to listen to. I get excited about the way he's explaining these concepts.
I feel so cheated by the public school systems for not teasing my mind with maths more as a child.
When I revisited my education in my mid 20s (went back to school for my own enrichment) I was introduced to calculus I saw the horizon that is the magic of maths pushed well beyond by perspective at that time (and today).
I do so wish I had seen that much earlier in life and now struggle to make up lost time only wishing I had more of it to do so.
+George Viaud Unfortunately, a love for math gets beaten out of children at an early age, and very few people find it again once they lose it.
+Jeffery Wells Yeah... I'm in love with numbers and mathematics, but the subject Maths is just agonising.
That's probably Dewey's education system you attended. Too bad it's used in the USA :(
You have taken the integral of your life and found out there is an upper limit. You only find out what it is at a moment before your death. But how long is that moment? Ahhhhh. That's the question!
I'm 18 and I barely understand the true meaning of calculus also I'm like the worst in geometry, but no way I'm gonna stop studying, there is no late time.
the most interesting video i've seen for a long time. fascinating stuff, thank you
glad you liked it
Numberphile what is the weird S notation thing on the board?...
No question. Mind blown.
GameOver That would be an integral for your knowledge.
+GameOver I think that's a zeta.. Looks like a description of the Riemann Hypothesis with the Re z = 1/2 at the bottom
Cantor is another example of the prof.'s outlaws. The thing most non-mathematicians don't know is that the math that gets presented (such as in published papers, lectures, classrooms, etc.) is the "finished product." It seems clean and neat and perfect and irreducible - precisely because some mathematician (or more than one) toiled away at it (sometimes for years - even decades) to figure it out; it's clearest, simplest form, it's implications, etc. I've heard the process of doing math likened to a restaurant. Most of us sit in the dining area; we only get the finished meals. We never see the chaos that goes on in the kitchen! Can you imagine what a chef goes through? Experimenting with different ingredients, failing time and time again - until, finally, AHA! The perfect dish! That's what doing math 'feels' like. We aren't simply given the answers! We have to work at it - often fumbling around, until we get it right. Of course we're only going to present the world (that is, other mathematicians) with the completed, perfect recipe! Anything would be an embarrassment! Doesn't mean it's easy! And, as often as not, we STILL don't understand ALL the implications of our own discoveries; we may never do so in our own lifetime. That's the way it goes, kids. tavi.
Wow, probably one of my favourite Numberphile videos of all time. Very eye-opening. Thank you.
thanks
I would wrestle a bear to sit in on a lecture by this guy. I love his presentations.
Try Fozzie Bear.
I'm really enjoying his voice too.
What accent is that?
@@phasepanther4423 He's from Russia. And then he moved to the US to work on his PhD.
@hey, folks! That can be arranged. It's UC Berkeley after all.
STRONK MATH! KOMRADES, RUSSIAN MATH SUPERIOR TO DECADENT CAPITALISTIC RUBBISH.
The Prof. wields the English language like most native speakers could only wish.
Marshall Harrison - Guitarist - Sadly, 98% of English speakers don't even recognize the utility, or beauty, of wielding the language. They kinda notice when they encounter someone twice as articulate as themselves, but hardly discern when someone's command is twenty times that.
@@samuelluria4744 source for that statistic?
Eric Williamson - It's my own lifelong attention to the empirical evidence.
@@samuelluria4744 you can't cite yourself
Eric Williamson - It's not a citation. It's an assertion.
This is far & away the best Numberphile video out there & I've seen every single one multiple times.
Once you've seen them -1/12 times, you'll know you've finished.
Hearing this guy come up with analogy after analogy to justify -1/12 is like listening to a person with sight trying to describe color to blind people. I really feel like he gets it. Maybe it'll make sense for me one day with enough analogies and experience.
Best discussion I've seen yet on this topic. Also, Frenkel's book (Love and Math: The Heart of Hidden Reality) is excellent!
I did not know he wrote books, thank you for this information!
On Facebook I received some backlash when I posted my amazement that the zeta function can assign -1/12 to the sum of all natural numbers (zeta(-1)). I am just as surprised at the backlash as I am about the -1/12 value.
I can better-see how the discoverer of irrational numbers must have felt when they were ostracised long ago.
I wonder if Reimann ever copped-it over his -1/12 value?
Edit: Thank you, Brady. I love your videos. Keep it up, mate!
you can get backlash on facebook for saying 1+1=2
+MMorgattto You numberist oddophobe
no sum of positive numbers is smaller than any number included. the sum of all numbers is not less than zero, unless you ask a faulty function. maybe that's why they didn't agree with you, it's not correct.
+chill dude woosh
i is legitimate in every context. it stands in for sqrt (-1) so that operations can be performed on it to give it a real value. this value was obtained through error.
-1/12 reminds me of matrix determinant… It is a non-intuitive number that isn't the regular way to say "this equals that," but is still an essential "identity".
I don't think the equal sign (=) is the proper way to represent this, since we defined *equal* as "considered to be the same as another in status or quality". In this case, the "1 + 2 + 3 + 4 + 5 + …" does not equal "-1/12", but is in a way the equivalent of a determinant (as with matrix). It is a property, but not an equivalent. The equal sign (=) isn't the proper sign…
*It is only my opinion… I am not a mathematician.* Am I wrong? What do you think?
The notation is a surface complaint, and I think it shifts the problem elsewhere. The root problem is the difficulty with reconciling the philosophical nature of infinities.
We have an intuitive understanding of sqrt(-1) as a number on a 2D plane, which is why we accept complex numbers without much problem. We know what the determinant of a matrix is, as it is a finite value and has useful properties in linear algebra. Once you get into infinities, there is a huge barrier with what infinity actually "is" and how this is communicated.
For example, the idea of "potential" vs. "actual" infinities. An actual infinity is that which is realized as a mathematical object in some independent reality which can be reasoned with and manipulated. A potential infinity is something which can be constructed in the human mind, that is, a series going on forever without end, and cutting it off at some point lands you back into the land of finite numbers (partial sums).
What we found with actual infinity is, by accepting their existence, it yields a bunch of mathematical results that have applicability everywhere from number theory, to actual, experimentally measurable phenomena in physics.
Euler was a mathematical gangster. He was the father of mafiamatics.. :)
Maybe the sum of all the natural is -1/12 just because the Universe's calculator overflows.
+Ruben That may have been a joke, but so far it is the best explainaition of why replacing infinite series with -1/12 works!
+titubakom It really is.
+Ruben Honestly, with my limited understanding of this, that could be correct. The Universe may not actually have a perfect mathematical system, and this is one of the "bugs". That's not the way I prefer to think about it, but it''s consistent with what we know about the universe.
this implies that we live in the matrix
physical world is illusion which appears in eternal universal consciousness..#pseudo_logic
Awwwww, I was hoping we would get see this thing in action. Like some problem you used -1/12 to solve, then showed an alternate path to prove that the answer was correct.
excatly, ur not an expert
Wow. He really makes this clear in a way I haven't heard before. I have heard the explanation that this sum is actually -1/12 mod infinity before, but I never understood what that meant. When he gave the example of slicing off the infinity of dirt and just keeping the nugget, something just clicked. We already know from Cantor there are levels of infinities, so why couldn't this value be -1/12 in some finite field analog from a higher infinity where the lower infinity just happens to be the modulus? I hope to hear a lot more from Professor Frenkel. Really like his style.
I really hope this whole affair didn't discourage Brady from covering mind-blowing topics because there are so many of them left.
Maybe the take-home message from this is that it's ok to blow people's mind but it's a bad idea to leave them out in the rain with the crazy fact you threw at them just because you thought that would make the whole thing seem even more impressive.
Sounds like the solution to the national debt!
@Håkon r/whoooosh
Lol...
Imagine presenting national debt as the sum of prime numbers
This video single-handedly gave me all the closure I needed about -1/12. The analogy with the square root of -1 was perfect. Thank you!
Watched the whole video? Seen the links? Watched the other videos?
Then why not leave a comment! :)
***** Except -1/12 is meaningful in physics.
The Riemann zeta function is used in the Derivation of Casimir effect with s=-3 and that gives 1/120
Look up Derivation of Casimir effect assuming Zeta regularization. You will need to sum n^(3-x), let x=0 and rewrite as 1/n^{-3)
Now you can input -3 into the Riemann Zeta Function and get 1/120
*****
You are wrong, the Casimir effect has been demonstrated.
The sum of natural numbers being infinity is intuitive but it is not useful.
On the other hand -1/12 is counter-intuitive but is used in Quantum Electrodynamics and Quantum Field Theory with the derivation of the Casimir Effect
***** Fine, derive it here without ever involving -1/12, go ahead.
***** Your entire objection is "it's not that way, cause I said so." Why would I take you seriously?
Great video. I could never reconcile this result in my mind but the context explanation and comparison to complex numbers really helps!
This video is the perfect explanation on why math is a tool, first and foremost, in the sense that "if something works we'll use it". Before taking Calculus at my university I always saw math as something mystical and arcane, but thanks to explanations like this one I finally overcame my fear and started taking real interest in this beautiful subject
This is proof that a base-12 system is the most natural.
Not really, but I like the idea.
This explanation is way more acceptable than the ones in the other videos!
There is actually something unknown about the subject. It's not just "Hey, this divergent sum equals -1/12, period"
+Marco Curvello Ok, maybe the previous videos weren't so simplistic as I implied, I exaggerated. But still, my point remains the same.
What it proves is that the math is wrong and that these guys are idiots for thinking any differently.
3:05 yes make a distinction.
-1/12 is a regularized sum
Infinity is the naive sum...
This is what folks had difficulty with.
Actually you can see why. The teachers/professors/the people who tries to explain it to whom did not understand it, does not explain it like that, they are explaining it like it is literally -1/12. I look at the topic like there are several realms of mathematics, like these two. In one of them it is equal to infinity and in one of them it is -1/12.
This professor speaks so well and looks at the camera just right. I really was able to absorb the whole topic. Thanks so much for this. I hope he can speak about another interesting topic soon.
5:10 I always forget what comes after 2, bro. Don't feel bad.
When you are so used to being a mathematical gangster, it's hard to count to ten
😆 I know the best mathematicians are bad at arithmetic, but that was painful.
Professor Frenkel is the best person I have ever seen at explaining things. I could watch him speak all day man
This man must give fascinating lectures. He seems to have an exceptional ability in explaining mathematical topics. I love his answer to Brady's question about "breaking the rules" when calculating divergent series. I'm definitely buying his book soon.
Love that you gave plenty of time in the edit for Prof Frenkel to talk. Great stuff!
The analogies of prof Frenkel are brilliant. They just clear out everything. He is very clear in explaining!
This is what I love about maths. It’s so logical yet so bizarre. By the way he did a great job explaining such unordinary topic in a way, that even I can follow the ideas.
Really enjoying Professor Frenkel's explanatory methods. If almost any other professor tried to explain this concept to me I'd be more lost than Malaysian Airlines.
Too soon?
too soon... too soon...
Nah, it was only like a week ago....
You just made my day! haha
The fact that there are many levels of infinity I can actually almost believe it, not understand it, but believe it.
The easiest way of picturing it is an infinite line as opppsed to an infinite plain defined by two infinite lines.
Thesre are of course many more, and these two would fall into the same category under xertain classification, but it is easy to see how one differs from the other fundamentally.
Thanks. The analogy drawn between the square root of a negative number and renormalization of the naively infinite sums by the construction of a more inclusive mathematical setting was just beautiful in its phrasing. Bravo! I'm nostalgic for my course of complex analysis many decades ago. Talk about a nugget of gold in the infinite dirt of the internet.
Watched the whole video. Thank you so much, Dr. Frenkel (as well as Mathologer) explained this issue the best I have ever heard.
Too bad the Ultra-finists mis-quote him. :(
I agree. The way mathematics (and physics) relate the ordered sum of all natural numbers to the value -1/12 isn't saying that they are equal to each other, but that the process of summing has some characteristics that can be represented by -1/12. This characteristics is unique even if you change your approach to finding it, as many (if not all) methods of assigning finite values to that sum yields -1/12. And this result is useful in physics as it can help us describe and predict the events in the universe.
@@jackren295 - Absolutely no explanation was given to how the value of -1/12 was arrived at, which was the whole reason I wasted 15 minutes watching the video.
@@jowbloe3673 I mean, it's a very deep topic, that majority of the viewer wouldn't understand. I don't think you'd understand either.
@@jowbloe3673 they have a bunch of videos to show it. This is more philosophical
@@jowbloe3673 numberphile already has 2 other videos on the topic that this video mentions that the beginning.
Best answer I've seen so far: (paraphrasing) "We don't really understand it." We should not be afraid to say we don't understand something.
Now that imaginary numbers are "out the bag", Numberphile's surely got to explain that old chesnut:
e^(i*pi) = -1
or as my maths teacher used to prefer:
e^(i*pi) + 1 = 0
I think every mathematician prefers the e^(i*pi) + 1 = 0
The core of that identity, and the more interesting part really, is the formula Euler (or Cotes, perhaps) presented:
e^(iθ) = cosθ + isinθ
As can be seen, when θ=π then you have e^(iπ) = cosπ + isinπ = -1 + i(0) = -1. This also results in e^(i2π) = cos2π + isin2π = 1, and e^(iπ/2) = i, and so on.
Well, the specific case for pi comes from Euler's formula
e^(i*x) = cos(x) + i*sin(x).
Substituting pi for x gives -1, because the sin term vanishes and the cos term evaluates to -1.
As for where the more general Euler's formula comes from, I don't know for sure, but it can be proven by comparing the Taylor series expansions of either side of the equation. It comes from calculus though, and I'm not sure whether Brady typically includes calculus topics in the Numberphile videos.
Its tied with one of Euler's equations
Or the /real/ identity, e^(i*tau) = 1
its like nikolaj coster waldau and martin freeman had a baby and raised it in russia
Raised *it*?
@Fremen theyre both popular actors. Nikolai played Jaime Lannister in Game Of Thrones and Martin Freeman played Bilbo Baggins in The Hobbit
Amazed by how proficiently Professor summed such infinitely deep concepts in nice finite words.
These has been most intresting videos in mathematics I have ever seen. It's disturbing, counter-intuitive and facinating. Also it ties quantum physics and mathematics together in ways that I did not expect.
This is the best explanation I've seen of this weird idea. I like his take on the matter.
After spending an infinite amount of time at the office, I received a salary of -1/12. I was forced to pay it in gold instead of euros.
Oof
This is excellent.
The way I see this is many mathematical processes seem to have more than one valid, but not necessarily "correct" answer. For example, the square root of 4 is both 2 & -2, but if an actual square has an area of 4 square metres the length of the sides is 2 metres, not -2 metres, as only positive distances exist in reality (you can define it as negative, but only by cheating & counting a positive distance backwards from a reference point at the other end of the side). In this case it seems like this sum has two values again, infinity & -1/12, but this time we discard the "correct" value of infinity, and use the incorrect value of -1/12, because this time the "wrong" answer is the one that exists in real life. It's like sometimes we can only access one of the two values for any mathematical function, and usually it is the "correct" one that fits into reality, but very occasionally it's that one which falls outside the scope of what can really exist & we have to make do with the other one.
+David Harrison Well I wouldn't say that you could view any of a number of valid answers as being the universally correct answer. For example with the square root of 4, in terms of pure mathematics 2 and -2 are both equally valid. When applying that to area and distance, negative numbers aren't allowed so -2 no longer is valid in that context. But if I were to ask you the following question: "You owed a man a sum of money; however, that man decided that he was going to be generous and forgive the debt that you owed him and give you twice the money you owed him. He gave you 4 bucks, how much money did you have (assuming that you would pay off any debt you had if you had the money)?" The answer would be -2 because if you had money then according to the question you would've paid off the debt. While the area and distance context appears much more often IRL than the example I gave, there are times when both the positive and negative answers of a square root have meaning. For example with quadratic equations, the quadratic formula has a square root and the two answers found are when the parabola crosses the x axis, which when applied to the context of projectiles, are the two times the projectile is on a flat surface.
The same's true for the -1/12. When considering the context that most people assume, that if I were to get $1 + $2 + $3 and so on, this really doesn't make sense. Of the two valid answers (infinity and -1/12) infinity makes much more sense. But apparently in many different contexts in mathematics and physics, -1/12 is much more useful and makes more sense. By itself there is no universally correct answer, even just assuming infinity is probably derived from us putting it in the context that we normally put numbers in, representing physical objects. But when taken to infinite sums, such a context really doesn't make too much sense as there so far isn't anything that we're certain is infinite. (except maybe stupidity, sorry Einstein joke, not directed at you). In a pure mathematical context, where the answer isn't being used for anything, any answer that can be derived is equally valid.
It seems to me that the mistake is to think that the usual rules of mathematics are "real". Addition, multiplication etc. are also abstract concepts that were invented by people.
2x5=10 in the sense that you get that answer if you follow the rules. It's also "real" in the sense that those concepts can be applied to real things, which is what makes mathematics so useful. It sounds like 1+2+3...=-1/12 is real in the same sense, if a little harder to picture.
He has awesome command over English. What I liked most that he called imaginry numbers 'imaginry' without any worry, unlike many other videos. Though, imaginry numbers are as real as other numbers, but I have seen so many videos where I see that it would be explained that we should not think of these numbers as imaginary. He didn't emphasize on that. People like me who learn from these videos will certainly say something about as we are influenced by these videos.
-1/12 is a gold nugget?
(puts -1/12 into calculator)
(gold nugget appears)
+Christopher Gudgeon Almost everyone tried this now.
dont use a ti85 or you are gonna lose your money
Bilal Baig .....Mathematician are funny😂😂😂😂😂😂😂😂😂😂
Mathematician discovers one weird trick to generate endless gold (economists hate him)
Petty lies just to get some comment likes.
Mathematical gangsters like me got thrown in jail for spilling the beans on -1/12
Ramanujan calls this "regularized sum" the "constant" of a series. It seems to be capturing some sort of defining characteristic of the series.
Thank you for mentioning Ramanujan, as I vaguely remember him sending this solution to Hardy, and Hardy and Littlewood both laughed.
But, this should have been well known to both of them by 1913 ?
Crazy how Euler could come up with something so abstract with seemingly no possible applications, and then however many, many years later it is actually used in a branch of physics euler would have never been able to comprehend
+kvnd7331 Euler could have comprehended it eventually, he was a smart cookie.
interesting politics you have
Welcome to the world of mathematics! The problems may seem trivial and useless now, but they will be of infinite use to those in the future.
What branch of physics is this being used in? String theory?
What he said.
His accent ИЗ СТРОНГ )))
+1 😊
Not really. Look up the interview with the Tetris creator, Alexey Pajitnov.
Zrc 😂
What's strange, considering he's been living in USA since 1991. Maybe earlier, Wikipedia doesn't show precise year.
Nathan Drake his accent is strong*
I love that the rigorous mathematical framework that he talks about is up on the board behind Prof. Frenkel.
I love this channel. awesome stuff.
Are ya a bot?
@Fremen no
I lost it after he said 'mathematical gangster'
ThisIsRTSThree999 due to my brain having been put in a blender when I was 8, I am confused by your response
This has always intrigued me and I've done a bit of research here and there... but hearing Professor Frenkel explain the topic is a piece of gold in of itself!
A huge like. Excellent explanations. I would have added that this regularization helps make, say, the zeta function a whole rather than discarding half of it and clamming up like this: we cannot sum over infinity a divergent series - let's stop doing math in that area. Let's call divergent series to be a taboo. They are not taboo, i.e, they need to be studied too just as complex numbers were studied and developed. It's surprising to see how the Casimir effect seems to be in line with the zeta function and summations like this.
I believe my explanation is more up-to-date than a golden nugget analogy.
Am I crazy or is this guy ridiculously handsome? No homo bro, really.
+Alexis Pius i'm straight too, but there is something about his looks that's really captivatingly pleasant. what's the math behind this phenomenon?
+iranjackheelson his accent helps a lot too.
Perhaps it's his mod-ish hairdue. It rejuvenates him a bit.
+iranjackheelson I was thinking this too haha
+Alexis Pius You are certainly not crazy, he definitely is...
This helped a lot. I've seen way longer videos on this that only confused me more. This at least gave me a basic sense of what's going on here through the use of analogy and discussion of context and explaining that even by mathamaticians and physicists may not have this fully understood.
I was one of the many people who commented on the first video, crying out heresy at Tony and Ed when they claimed to be able to calculate the sum of an infinite series.
This video really helped me better understand the value of these otherwise illegitimate calculations.
sk8rdman Calculating the sum of an infinite series is simple, it is finding the sum of an infinite divergent series which can be a bit more complicated.
0:23 When somebody looks at you like that, you know there's something going on behind those eyes. I don't meet enough of those people unfortunately.
This guy speaks with passion, it's videos like this that inspired an interest in mathematics for me.
Ive watched this video three times and i still havent got a clue what its about.... Back to the fail vids for me
jeremy western so funny. 😭At like 5:00 in, I'm like wait, what did I miss?
Yes, that's because he did not explain anything. He just blabbered on and on senselessly about nothing.
Perhaps you can explain it better then. Go right ahead.
Daniel Slick It's been 6 months. I don't think they can :P
You should do a video showing an actual calculation with one of these infinite sums. Use something where you get the right answer using the -1/12 substitution and then solve it using a different method to show that you did indeed get the correct answer.
They showed a book in the first video (I believe) that does use this substitution. I'm sure you'll be able to find the book somewhere.
It would probably require integration, derivative calculations and lots of other stuff that in movies is called "scientific mumbo-jumbo". It would be understandable for very few subscribers.
Ha, I was going to say the same thing but didn't want to be too controversial :D
The chance of understanding everything (or anything) up to the point in the book that uses this is very remote.
MaC S3th the way I learned it involved using the Gamma function and finding patterns which let you define the extend the domain of the Zeta function one vertical strip of width 1 at a time. It took a few two-hour lectures.
Yes, Euler's discoveries in mathematics are monumental in the history of knowledge. His work on headdresses, on the other hand, was unable to garner enduring appreciation.
The more I think about it, the more the -1/12 answer makes sense to me, at least as a possibility.
I know the common wisdom is that if a number keeps getting bigger, the final answer has to be bigger, but that's only true until you break infinity. I recall seeing in high school and college, a lot, when talking about limits and asymptotes, that a number will approach infinity as it approaches the asymptote from the one side and approach negative infinity as it approaches the asymptote from the other side and then at the asymptote it seems to simuletaneously take on many and no values at the same time, sometimes a range, usually all real values. At any rate, it is definitely not one defined value.
But lately I've been watching math vidoes that talk about the fascination with infinity. Many people erroneously assume there is a real number called infinity, when, really, infinity desribes the set of all real numbers. So, all real numbers are less than infinity. If you add a finite number to a finite number, it too will be less than infinity. And infinity is not itself a real number. The video I saw talked about magnitudes of infinity, of how there are things bigger than infinity, they are just not real numbers, and more so are used to describe sets of numbers.
Again this take me back to college to two things. One is the concept of a delta function if engineering, which is the deritive of a step function. It is infinity big at one value and 0 elsewhere and it's directive over the discrete value is 1, essentially it is a rectangle of 0 width and infinity height whose area is 1, and you can multiply constants by it to make it bigger and the area likewise will just be a multiple bigger. I recall it being uses to simulate signal spikes. I assume multiplying 0 by infinity and getting a constant is something pure mathemeticians don't like. But also I recall, I think it was working with the stability of systems, a value could go to infinity at an asymptote, wrap around the entire coordinate system multiple times, and then come back on the same side or opposite side from the other side of the asymptote. I wish I remember more about what it exactly meant and how you determine how many times a value could "wrap around infinity".
Still, now this -1/12 is making physical sense in my head (and trust me, I have trouble reconciling things in my head, so I'm always asking "why did this happen" until I understand things). So, if you add a finite number to a finite number, you are still less than infinity, but if you add numbers beyond infinity, things wrap around back to negative infinity, and then if they keep going it could happen more than one time. Well, when you keep adding increasing bigger numbers together an infinite amount of times, you get something that's close to half of infinity times infinity. In fact, for finite numbers, the formula is n*(n+1)/2. It is a number beyond infinity, so I can absolute see it wrapping back around to a negative number or even to a finite positive number in our single finite set of real numbers. It's hard to imagine a number being bigger than infinity sinse we experience life in a seemly limited scrope. I cannot imagine what it exactly means in a physical sense. But I can see rules fit together.
However, I've notice when numbers hit asymptotes, they seem to become all real numbers (or even just a range) at once and seeming no numbers at once, but there do seem to be times when a specific number does seem to represent the special circumstances, so I like the talk in this video about removing the cruft to get to the gold nugget, removing the infinite values that make up the answer until you have the one that best represents what is going on.
So, yeah, information from multiple sources, 20 years ago, 6 months ago, and just now just helped me make sense of something that on the surface looks weird, but seems to actually make sense. (Now as to the proof, I've never attempted it before but I do like the simplicity of Euclid's solution despite the divergent function problem-- which I've seen some comments complain about and then other explain it does follow special rules that can be used.) I certainly miss mathematics, though.
Infinity does not "wrap around". We are not talking of modulus arithmetic here.
limit (n • (n + 1) / 2) as n approaches ∞ simply equals ∞ and not "beyond infinity".
When I asked my graphing calculator what 1/0 is, it said undef. A graph of 1/x shows what the problem is. It diverges at x=0. It does not "wrap around", but rather the sign of x switches at 0 between negative and positive. My graphing calculator objects to the lack of convergence it seems. But when I asked it, what the answer is to limit (1/x) as x approaches 0 from the positive direction, it gave me the answer of ∞.
What if there is no "gold nugget" and it just fools' gold, or there simply is no gold? Isn't it just make-believe nonsense? limit (n • (n + 1) / 2) blows up to ∞ as n approaches ∞. End of story. Why do we need a make-believe nonsense "assigned" value to it?
You can remove all of the integers from the series that they do not like, and still you will not get -1/12.
@@yosefmacgruber1920 Quantum electrodynamics
@@polygontower
How is that relevant to mathematics?
Man, I've never seen someone explaining this sum as well as you do, thank you!
More of this guy. Context is extremely important.
Also videos about Riemann, analytic continuation and so on.
I agree with the professor. If we only go by 'what is allowed' we'd never advance. This is especially true in the history of Mathematics from the early Greeks and perhaps earlier in pre-history; however, Pythagoras had "his own way" of seeing numbers, and his theorem is timeless still used up to our day. I say we dare see things our own way by what is NOT allowed or conventional. We may yet discover the next Pythagorean Theorem or someting as useful :)
The whole point is that the context defines what is allowed though. You still need a filter to distinguish correct statements from incorrect ones (i.e., what is allowed versus what isn't). Under one set of rules you're not allowed to sum divergent series. Under a different set of rules, you are. The problem I have with the Numberphile videos is that they use a non-conventional set of rules and imply that the result is valid for the conventional set of rules. That's an obvious error, due to lack of rigor.
I disagree. The example he mentions are imaginary numbers; he's right through imaginary numbers we arrive at real number solutions.
I'm specifically addressing this sentiment that mathematics is somehow advanced by 'breaking the rules'. That's incorrect. Mathematics is advanced by making new rules and rigorously following them to arrive at useful results. Never rule breaking; logic always applies.
Just like for complex numbers: they can lead to real number solutions, but only insofar as complex numbers can be converted back into real numbers. It's just like transforming between spatial and frequency domains. You still have to be mindful of the restrictions keeping you from mixing one framework with the other.
I see what you're saying, I still say dare.
No diea, sorry.
There, this is a fulfilling explanation! I loved the point of view and choice of words of Dr. Frenkel in this video. Very well done!
This explanation has made the most sense so far regarding why and how this sum can be said to be -1/12
lol "mathematical gangster"
Nishant Gogna thats me
How come this is less popular than reality tv?
As a non mathematician, this actually makes a lot of sense. He's basically saying each infinite sum that has a defined pattern has a value that is not the sum, or an expected value, but almost a measuring tool, a comparator to be used to compare to other infinite sums.
Almost like the sine and cosine functions find a very accurate measurement of something calculated using an irrational constant (pi)
The destination is infinity, how you get there (the journey) is - 1/12 :)
The keyword here is "regularized sum." It goes through a transformation so writing '1+2+3+4+5+6.... = -1/12' is not accurate unless you include some notation to indicate that the sum of positive integers has been transformed like 'f(1+2+3+4+5+6....) = -1/12'.
No its more than that. If the "regularized sum" was just something like a weird function that assigns values to infinite sums, it would not explain why you can replace the naked sum 1+2+3+... with -1/12 in many situations and get the right answer. When you replace it with -1/12, there was no 'regularized-sum-function' there... it's just the sum of all integers.
So in a sense -1/12 seems to actually be *equal* to the sum of all integers... of course the sum of all integers also seems to be infinity. Why and how both can be at the same time, is the mysterious part... it certainly seems like both results are actually true.
The problem that I have is that, if -1/12 is not the "real" sum, then WHAT IS the real sum?
@@zhadoomzx No not even the prof from California said that. It is an assigned value that has something to do with it in no way it is equal.
@@YEC999 If that is so, then why does it give the right answer in the Casimir effect? That is an experiment you can do in a lab, replacing the infinite sum with -1/12 gives the same result as the value that is measured?
That -1/12 is not some obscure mathematical hocus pocus... its the real deal, its in the real world, it can be measured and tested.
That's such a compelling demonstration from proffesor