Dear researchers:
As counterintuitive as it can be, the infinite sum of all the integers : 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 ... to infinity can be understood as = -1/12
i.e. Sum(all positive integers) = -1/12
Despite of its weirdness, this sum is used in the math of several fields in theoretical physics, and some of these results were tested with experiments,
What do you think about this result? Can we get to understand it ? Or, Is it something we'll never get to understand ?, and Something that we have to take as a sort of axiom, a mathematical trick of convenience ?
Regards, !
Dear Franklin
Please see the following link in RG in which the same question is asked. I have also attached a file to the question. The file scrutinizes this question in some details.
https://www.researchgate.net/post/Why_do_we_need_tricks_in_mathematics
Ziaedin Shafiei :
Thanks for sharing ! I am checking it out now ,
Best Regards !
I found the following interesting explainations, I hope will be only way to understand what is beyend this unbeleveble quetion:
There you are living your life, content in your grasp on how the world works: up is up, down is down, the Sun rises in the east and sets in the west. Then, out of nowhere, a bunch of mathematicians try to tell you that the sum of all positive integers, that is, 1 + 2 + 3 + 4 + 5 + 6 + . . . and so on to infinity is equal to . . . −1/12.
Well, that's clearly ridiculous, right? How can increasingly big numbers, when added together, make a small number? How can whole numbers make a fraction? How can positive numbers make a negative?
Tony Padilla and Ed Copeland, physicists at the University of Nottingham in the UK, appear in and narrate the eight-minute video. Minute by minute, what they explain requires from the viewer no mathematical grounding beyond simple addition and a smattering of the most basic algebra. But in the aggregate, what they argue goes well beyond that simplicity.
Phil Plait, science blogger at Slate, did a posting about the video, corrected it, and then—in response to complaints—had to post a revised version.
The American Physical Society’s PhysicsCentral website offers the Physics Buzz Blog posting “Correction: Does 1 + 2 + 3 + 4 + . . . = −1/12? Absolutely Not!” It begins by pointing to a graph: “Brief Summary: 1 + 2 + 3 + 4 + . . . is not equal to −1/12, but both the infinite series and the negative number are associated with each other in a way that can be seen in this graph.”
At the Times, Overbye quips, “After watching the video myself, I checked to make sure I still had my wallet and my watch.” But he also reports something that Padilla and Copeland adduce in the video. They open the textbook String Theory, by Joseph Polchinski of the University of California, Santa Barbara, and point to a page showing his use of the mathematics in question.
On the web, Padilla has posted a write-up for physicists. He begins with this explanation:
It’s by no means obvious, but this is the only sensible value one can attach to this divergent sum. Infinity is not a sensible value. In my opinion, as a physicist, infinity has no place in physical observables, and therefore no place in Nature. David Hilbert, one of the founding fathers of quantum mechanics, described infinity as “a mathematical abstraction that does not have a physical content.” I think most physicists would firmly agree with this sentiment. The trouble is that divergent sums like the one we discuss in the video do appear in calculations of physical observables, such as the Casimir energy, or in the dimensionality of the Universe in bosonic string theory. Therefore, only a very brave individual would dream of attaching the value infinity to sums like this. Minus a twelfth is far less crazy a value when you start talking about physics.
Padilla notes that the result is “utterly counterintuitive,” but he explains it at some length and in some mathematical detail, and he offers links for further reading.
He also defends the license he and Copeland have taken to reach out to the public. Here’s his write-up’s final paragraph:
There is an enduring debate about how far we should deviate from the rigorous academic approach in order to engage the wider public. From what I can tell, our video has engaged huge numbers of people, with and without mathematical backgrounds, and got them debating divergent sums in internet forums and in the office. That cannot be a bad thing and I’m sure the simplicity of the presentation contributed enormously to that. In fact, if I may return to the original question, “what do we get if we sum the natural numbers?”, I think another answer might be the following: we get people talking about mathematics.
Dear Franklin Uriel Parás Hernández
There are many similar relations to that
1 + 2+ 3+.....= -1/12 and many questions were raised to discuss
such relations.
Let me tell you that all of them are wrong and absurd in mathematics.
An introductory course in mathematical analysis removes all ambiguity about such relations.
In brief: We can rearrange the terms of any divergent infinite series to reach any answer. We can't rely upon divergent series, so we need to test the convergence of the given infinite series.
Let me show you a simple example:
What is 0+0+0+.....+0+0+.... it is divergent, and the answer
0+0+0+.....+0+0+....= 0 is the wrong answer.
Why?
Assume that 0+0 +0 +.....+ 0 + 0 + ....= 0,
then (r-r) +(r-r) +(r-r) +.......+(r-r)+.........=0
implies r -r +r -r +r - r + r -r.....+r - r .... =0
rearrange the terms as
r + (-r+r) + (-r+r)+..........(-r+r)+ (-r+r)......=0
we obtain r +0+0+0+0+........0 .....= 0, that is r + 0 = 0
hence r = 0 which is a contradiction !!
So the fake results are due to the divergence of the infinite series.
Best regards
Dear Issam Kaddoura :
Yes you are right !, Now I can see the whole point,
This discussion for me was very useful
Thank you all !
Hi Franklin,
Let me add a couple of comments. First, as mentioned by others, the result stated like that without further comments is, indeed, incorrect. What happens for instance in the calculation related to Casimir forces, is that a sort of convenient resumation can be made. A regularized version of this sum can be shown to converge to some large quantity minus, precisely, 1/12. At least in the calculation one does in physics, this divergent part exactly cancels another diverging term coming from somewhere else. So, in a certain sense, there is a formal asymptotic representation of that sum as something infinitely large minus a 1/12 and, again, in the physics problem it is only this "remainder" what is left.
I just want to add that in pure mathematics, there is a way in which one can prove convergence of that sum to -1/12. See, the "=" sign when talking about infinite sums does not have exactly the same meaning as with finite sums because a limit is implied. So, there are notions of convergence, and correspondingly, of metrics (or distance) intrinsicaly linked to the "=" sign in these cases. One can show that there is some metric (called 2-adics) in which this series indeed converges to -1/12. But in the usual metric in the reals, i.e. the absolute value, this series is divergent.
Best,
Reinaldo.
Reinaldo, Thank for this informative and clear explanation
Regards !
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