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1+2+3+4... = -1/12
Posting a thread on this simply because it kinda blew my mind, something math hasn't done for me in a while.
The limit obviously converges to infinity, but the actual total sum of all of the positive integers comes out to be -1/12. Not only can this be derived in multiple ways, but is a statement that is applied and used in string theory. All I can say is: ...what |
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The actual sum does not give -1/12 using normal mathematics. There a lot of videos going on around showing how to get that result using simple algebra on infinite divergent series. That's something you can't do. It's like those proofs of 1 = 2 that divide by 0 to get to their result.
Anyway, there is a branch of mathematics called analytic continuation. Without going into detail, it deals with how analytic functions behave outside of their domain. And there is this well-known function called the Riemann zeta function defined as the sum of 1/n^s for s from 1 to infinity. This function is only defined for s>1. But if we were to take s = -1, we would get 1 + 2 + 3 + .... And using those rules of analytic continuation, you can get a value outside of the range of functions and in this case, the value is -1/12. So what is the value of the infinite sum? We made a mistake by just asking this question. Divergent sums don't have value until we give them one. The question is not what is the value but what we should define it as? There a couple of ways to go about it and each have their strengths and weaknesses. The usual way involving limits would not give any answer. The analytic continuation (and a couple others) gives -1/12. Since something is more useful that nothing, we decided to give the infinite sum the value -1/12. But neither is more correct than the other. PS: If you want to read more about it: http://terrytao.wordpress.com/2010/0...-continuation/ |
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For me, many concepts regarding infinity are not that difficult to accept or comprehend, but this is just something that is totally absurd and ridiculous no matter how I visualize it in my head. |
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Non-Euclidean Geometry is pretty neat too you know ;)
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it's like saying that just because you can get a maclaurin expansion of, for example: (1-x)^-2 = 1 + 2x + 3x^2 + 4x^3 + 5x^4 + ... and the expansion is valid for |x|<1 so you could put x=0.1 and get 1 + 0.2 + 0.03 + 0.004 + ... = 100/81, which is valid then you put x=10 and get 1 + 20 + 300 + 4000 + ... = 1/81 it's the same idea here, just replace "riemann zeta function" in whatever he said with "(1-x)^-2" and you'll see how ridiculous and nonsensical it is to talk of a divergent series converging to some value just because it has a non series representation outside where it converges so there'd never be a situation where you'd have to add 1+2+3+4... directly and get -1/12, if you wanted to work out zeta(-1) you'd work out zeta(-1), not try to add the divergent series, which won't make any sense edit: also non euclidean geometry sucks donkey balls, i did a course on that last year, and it was horrible and made me want to pull out all my hairs and superglue them to my chin. i think it was just badly taught, but maybe i just hate the subject |
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Non-euclidean geometry is pretty cool. I remember telling some guy that with spherical geometry I could make a triangle with 3 right angles and he just couldn't accept it XD
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I didn't get a chance to take the Non-Euclidean Geometry course at my university (because I was in Russia oops), but my best math friend explained it all to me in this way. |
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I haven't studied non-Euclidean Geometry like spherical Geometry, but I did study finite projective Geometry and I find that pretty....interesting lol Parallel lines don't meet? Well make them. Done. haha
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the concept of an infinite series is sick
if I ever make enough to get to college, I might take some advanced math classes |
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They could have used any symbol to describe the answer to this; the principles are the same. Insert the symbol and continue your mathy ways, or insert "-1/12" and continue your mathy ways. The solution is still based on the assumption that there is a solution to the convergent series. Whether it's -1/12 or a symbol, it doesn't matter.
It's a weird choice to use an actual number, though. As emerald said: it's not normal mathematics and as frederic said: it's not the actual solution of the sum. But it confuses the regular people when it's written as a number instead of a symbol. Give people pi, people know pi. Give people the infinite series that results to pi, people don't know pi. Same here. Accept it for what it is and continue calculating. |
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math for the sake of math, how cool fun and useful
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I'm not going to harp on this topic for that long because math is confusing and people like you only make it more difficult to comprehend. More or less people will give me flack for writing this but I brought to you a logical conclusion as to why this is IMPOSSIBLE, no matter what combination of integers you input (Even if it isn't in this range here):
"1+2+3+4..." What is a sum? A combination of positive or negative numbers added together to produce a larger or smaller number. For this instance, all of the numbers in this range are both integer and positive, which means: • there's no possible way to produce a fraction as a result of this calculation. • there are no numbers in the calculation that are negative. So in logic, having simply "1" in the combination of numbers has already brought you above -1/12, and there is no such number that brings you below the number 1. The same can be said for the rest of the numbers we have here. There's probably some stupid math algorithm to justify your answer but the fact of the matter is that infinite numbers summed does not produce a negative rational fraction, unless that is you're trying to be completely incorrect... Wherever it is that you learned that OP, just ignore it. You're making math harder to justify than it needs to be. -.- "Non-euclidean geometry is pretty cool. I remember telling some guy that with spherical geometry I could make a triangle with 3 right angles and he just couldn't accept it XD" Well that isn't really hard to explain at all; just bend the lines. Some people really are dumb when it comes to geometry and I don't really blame them. PS: If your justification for the sum thing isn't smart please don't use it against me. I'm getting tired of people trying to say that they are correct just because others told them that they are. I get it way too often where I live and that's just an excuse to justify your incorrectness. -.-'... Mathematical Justification or Critical Thinking (Why this board exists): ƒ(n,s)=1/n^s where (s=-1), sub as ƒ(n,-1)=1/n^-1 ƒ(n,-1)=1/(1/n) ƒ(n,-1)=n Therefore, if (ƒ(n,-1)=-1/12) and ƒ(n,-1)=n, then n=-1/12. So your math is wrong anyways. ƒ(-1/12,-1)=-1/12 ƒ(ΣI,-1)≠-1/12 ∴ΣI≠-1/12 :P |
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I still don't know why it works, but Numberphile says the value -1/12 for that sum is used for practical quantum physics applications so I have no choice but to believe it.
My theory is black holes. |
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It got myth busted. 8-) YEEEEAAAAA lol |
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Obviously, for any finite number of terms, the sum of the integers starting from 1 can't possibly be fractional or negative. However, we're talking about an infinite series here, which characteristically makes the problem different. The reason why this result can even make sense is because we're talking about different contexts. Mathematical structures in different domains have different properties. When you have problems in which multiple answers can be justified, the conditions for the problem must be explained to obtain a particular solution. While in the sense of a divergent series this goes to infinity, through analytic continuation this statement can be justified. The result is actually used in quantum mechanics, and without it we wouldn't have the level of digital communication we have today. Just because you don't like the idea doesn't make it false. And whether or not pure mathematics is difficult to justify is irrelevant. Fermat's Last Theorem was notoriously hard to justify, but people spent hundreds of years to do it, and it's an important result. Quote:
making an assumption with no basis: "Therefore, if (ƒ(n,-1)=-1/12)" jumping to conclusions: "n=-1/12. So your math is wrong anyways." another assumption: "ƒ(ΣI,-1)≠-1/12" clearly math is not your strong suit |
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Wrecked
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i still maintain that it doesn't make sense to talk about a divergent series having a finite value as a sum.
let's go back to my earlier example of Quote:
you're saying: if i plug in x=10, then i get a value of the sum 1+20+300+4000+ ... = 1/81 which is justified by analytic continuation, since there is a unique analytic continuation of the sum 1+2x+3x^2+4x^3... outside its radius of convergence, and this analytic continuation just happens to be equal to (1-x)^-2. here's the thing. if you wanted to do that, you'd need to tell me what the function you're trying to take out of its radius of convergence is. and because the series is divergent, you're not going to get the same answers whichever way you do it. so you're trying to say that 1+2+3+4... represents the analytic continuation of 1^-s + 2^-s + 3^-s + 4^-s + ... | s=-1 at s=-1, which is the zeta function at -1, which is -1/12. there is no way to determine that when you wrote 1+2+3+4... you're starting from the zeta function and working things out from that. you could be trying to get it from the analytic continuation of another function. here's another way to write it. how about this? 1+2+3+4+... = 1 + 2x + 3x^2 + 4x^3 + ... | x=1 = (1-x)^-2 | x=1 = 0^-2 = "infinity" you're still not convinced? 1+2+3+4... = 1 + 2x + 3x^2 + 4x^3 + ... | x=1 = (1-x)^-2 | x=1 = ( (1-x)^-1)^2 | x=1 = (1 + x + x^2 + x^3 + ...)^2 | x=1 = (1 + 1 + 1 + 1 + ...)^2 = (zeta(0))^2 = 1/4 yeah. none of this would be happening if you used a valid x inside the radius of convergence, since everything would agree on a certain value. this is the point. there may be occasions when you're trying to work out what the analytic continuation of 1^-s + 2^-s + 3^-s + ... is at s=-1. in these cases you'd write it as zeta(-1), because the zeta function is defined to be this analytic continuation. you'd never write it out as 1+2+3+4+... because nobody knows what you mean when you're writing down the sum of a divergent series. it's complete trash. just like your face (oohh burn) |
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straight up gangst thread |
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check out dirichlet series and grandi series
this also reminds me: check out thompson's paradox. edit: http://en.wikipedia.org/wiki/Thomson's_lamp It's really easy to abuse the concept of infinity |
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Zapmeister...you can't use (1-x)^(-2) at x=1 as an analytic continuation of 1+2x+3x^2+... because (1-x)^(-2) itself is not analytic at x=1. You have to remember that analytic continuations are unique (in some sense). I still don't see any contradictions or inconsistencies when we use the notion of analytic continuation to "redefine" things.
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I think the best way to explain this is by looking at an analog of a much more simpler example. When it comes to mathematics, we like to define a lot of things. However, sometimes we're stuck by the definition we have which restricts us. So we look for a way to "extend" our definitions to:
1) make everything still consistent with the old definition (i.e. not break maths) 2) have "nice" features in this new definition that is desirable Let's take an example with exponentiation. When first introduced to exponentiation, we learn this as repeated multiplication: if we see something like 2^3, it's simple: 2^3 = 2*2*2 = 8. However, then you see something like 4^(3/2) and you're like "da fuq? how am I supposed to do one and a half multiplications?" With our original definition, we are powerless, we can not do anything, we are dumbfounded with the notion of trying to 1.5 multiplications. However, we can redefine exponentiation for fractional powers in a way that is still consistent with the previous rules and we find that 4^(3/2) = sqrt(4)^3 = 2^3 = 2*2*2 = 8. We had to redefine what exponentiation meant for fractional powers to solve the problem. Did we redefine in any ol' way? No, we had to redefine it in a way that was still consistent with the previous rules. Think of this as the "fractional continuation" of the exponentiation function. We could have left it "as is" and just say it means nothing when our power isn't a whole number, but we wanted to work beyond the whole numbers. Mathematicians have extended this definition all the way up to the real numbers. What about complex numbers? Well, looking only at the definition for what we have for real numbers, we can't do anything. We can redefine exponentiation for complex numbers quite nicely in a way that makes the function f(z) = z^c analytic, where z and c are complex numbers. What it means for a function to be analytic is beyond the scope of this post, but it's basically a "nice" property that is desirable for a function with complex variables. Although technically, this isn't an analytic continuation of f from the reals to the complex, the idea is nearly the same: we can extend the domain of many complex functions using analytic continuation (and since it can be shown that analytic continuation is unique, we have less to worry about in terms of inconsistencies!) So although 1+2+3+4... doesn't really equal anything (unless you count infinity) with our original definition (i.e. the value of the limit of the partial sums), we can extend our definition by means of analytic continuation and result in -1/12. So is 1+2+3+4...=-1/12? Both yes and no. Sticking with the usual definition which most of us are familiar with, this is bogus nonsense. Using this other definition which some mathematicians (particularly complex analysts and it seems quantum physicists use): yes. In this re-definition correct? Definitions are neither correct or incorrect, they just are so whether this re-definition using analytic continuation is a good one...that's your call. |
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This thread has got to be the flagship of the critical thinking thread.
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which is funny because there's really not much debate in this thread
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best i can say now is that if you try to define sums of divergent series using the formulae and stuff you derive from convergent series, you have to throw away a lot of favourable nice assumptions you make about how they behave. e.g. sum(n: n=1 to infinity) = x (here x is assumed to be -1/12 but that doesn't matter) => sum(n: n=0 to infinity) = x => sum(n-1: n=1 to infinity) = x => sum(n: n=0 to infinity) - sum(n-1: n=1 to infinity) = 0 => sum(1: n=1 to infinity) = 0 => sum(1: n=0 to infinity) = 0 => sum(1: n=0 to infinity) - sum(1: n=1 to infinity) = 0 => 1=0 however i'm pretty sure term-by-term addition/subtraction like this is not justified in the way stargroup100 is trying to treat these things, although he never explicitly states that in any of his posts. i'm still trying to find a contradiction that i'm satisfied with but for the moment it looks like you got me here >_< edit: i was hoping my 300th post would be less self-demeaning than this but whatever |
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I'd have to say, the most intriguing mathematical truth is:
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I'm (a bit) surprised so many people seem to have taken complex analysis.
I find it interesting that so many people get so heated over what they believe is the right answer regarding math. It's like that one problem floating around involving the order of operations...I'm pretty sure there have been like legit fist fights over that one...probably. |
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Funny we were just discussing this in my calculus class. In fact the infinite series of positive numbers is divergent (infinity or undefined). This instance is kind of like how my calc teacher told us that taking the integral of natural log yielded 1=0 . But then there came the +C that corrected the equation. I think something like that will come up but for now it can't be explained. I find it funny that the tool we created to better understand our universe, math, we can't even fully figure out ourselves.
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A quick google search https://www.youtube.com/watch?v=w-I6XTVZXww
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However, there is a lesson to be learned here. The lesson being that this way of defining how to given values to divergent series isn't "nice" and doesn't have the "nice" properties like the ones we see with convergent series (see this and this). We know with convergent series we can add term-by-term...but your thing shows that that is clearly not the case anymore (or else otherwise we get the absurdity that 0=1). To be honest...this way of defining how to assign values to divergent series doesn't sit too well me either. It works well for complex analysts since analytic functions are "nice". However, this definition doesn't have a lot of the "nice" properties one is looking so if you find a "nicer" one let me know :) |
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What I do care about however, is how people approach and understand math. There are way too many misconceptions about the nature of mathematics and how it's applied. Example of what I mean: Quote:
Actually this topic has been explored pretty well. It's nothing new for mathematicians, and while there are questions that are still unanswered, we generally have a pretty good idea of what goes on here. Math can be thought of as a tool we created to better understand our universe, this is fine. There should be absolutely nothing surprising about not being able to fully figure out mathematics as a whole. Just because you created something doesn't necessarily mean you understand it completely, and there are countless examples of this in the real world. Why is math special in that it needs to be completely understood? Mathematics in many ways has infinite complexity in many dimensions. Not only are there probably an infinite number of systems and objects, but different mathematical objects behave differently under different circumstances/domains/etc. There are no "concrete rules" in mathematics. You absolutely must follow the rules you set up within your own system, but none of these rules are universal to all systems. In fact, complete axiomization of all mathematics has already been proved to be impossible (See Gödel's incompleteness theorems). What is important is not whether or not we fully understand math, it's how much we understand and how we use it. Over time we build upon our existing knowledge and expand it, which then results in more applications to real life. There are always going to be things we don't know; it's literally impossible to know everything. Our objective is not to have perfect knowledge, but to increase our knowledge as much as possible. |
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I think this problem is too advanced and we should find something fascinating in math that's in the realm of understanding for everyone. Any ideas?
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When you find something fascinating, it doesn't have to be something you can understand. There are lots of facts about the world, not just mathematics, that are amazing, and many of them even top academics don't understand yet. It's more important to first understand mathematics as a whole, the things I've mentioned in my last post. Once people understand these basic ideas, there will be far fewer misunderstandings about math, far more people that will appreciate it, and it will contribute greatly to people being better at math. |
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i'm not seeing it. why should/would i appreciate the result "1+2+3+4+stuff = -1/12", or find it "fascinating", if by analytic continuation any series that has a rule for the terms can be "summed" by taking a power series or (in your case here) a dirichlet series and extending it outside its radius of convergence. if it's about the applications of the actual value -1/12 then why bother with writing it out like that instead of writing it as zeta(-1) and using properties of the zeta function.
have at me. |
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-.-
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when I first learned about calculus, the idea that you can find the area under any polynomial curve in trivial time was mindblowing to me. now I take it for granted because I'm so familiar with basic integrals, but that didn't make it any less surprising when I first learned about it Quote:
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All i've seen from you is "some guy who is awesome said"
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Wait what?
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I haven't seen him actually attempt to prove his point/theory. stargroup that is
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The reason the exact proof is not provided is because the proof requires knowledge in a field that most people on this site would not be familiar with (primarily complex analysis). If you read the thread however, it has been said numerous times that it can be justified using something called analytic contiunation which I and some other people in the thread have given everyone a flavor of.
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what reuben said
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If you're talking about the notion that different objects in different domains can have different properties and therefore many different statements can be justified/proven under certain contexts/conditions/systems, here's an example: Polynomials of degree n with real (or even complex) coefficients have exactly n compelx roots, but not necessarily n real roots. Certain applications of polynomials, such as most geometry problems, disregard the complex solutions because many kinds of complex lengths/measurements cannot be physically realized (you can't draw a line i units long). However, the result is fundamentally important to understanding polynomials and is used extensively in other fields of math. Here's a directly related result: Exponentiation is an example of a function that has different behaviors in different domains. The exponent of e^n tells us to multiply e by itself n times. This makes sense for integers, and if we consider the inverse of this function, also makes sense for rational numbers. However, we can easily plot this relation and make a smooth line to extend it to real numbers. Once we find a real definition for the exponential function, we can observe its properties and extend it to complex numbers (by Taylor expansion for instance) and get results such as e^x = cos x + i sin x. However, this is where it gets a bit interesting. If x is a rational number, e^x has d solutions, where d is the denominator of x in simplest form. This result is directly related to the extension of polynomials to complex numbers we mentioned above. If x is irrational, then it "probably" (I'll explain in a second) only has one solution in the complex plane. Once we extended our definition, the behavior of rational numbers in this function has changed! Even more bizarre is when you take the relation e^(a*ln(x)) = x^a. Here, we have a case where even though the exponent could be irrational, and yet we can change the base to make the exponent rational again! And this is only scratching the surface of complex analysis, and complex numbers are just one type of mathematical object. Other mathematical objects include matrices, vectors, tensors, sets, groups, operators, rings, partitions, spaces, manifolds, functions, geometric figures, etc. Even today, we're still investigating all of the properties of these objects and new ones that we've conceptualized. I know I'm not directly replying to anyone's specific ideas, so I wrote this more for myself as practice. I figured I might as well post it if anyone is interested in reading it. EDIT: I just realized that I pretty much just did a layman's reiteration of one of reuben's earlier posts. I was wondering why something was bugging me when I posted this LOL |
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A few years ago, I was offered a free masters in math. As in, this prof liked me and wanted to teach me and eventually research with me. However, I was just back in school to get the necessary math credits so I could teach high school math. I was broke as it was and needed to get working, so I turned it down and started teaching.
This thread makes me regret that decision. I want to take all that non-euclidean nonsense and real analysis and modern algebra and topology and all that garble. I feel like I really missed something. |
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If you have both then yeah you probably missed something. You've probably "wasted" far more time than I have, and even I feel depressed about not spending more time working on this kind of stuff when I was younger. |
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