4 ms·
The Ramanujan sum of all positive numbers up to infinity is -1/12
- matmann2001 13y agoAccording to some people... The sum of alternating +1 and -1 equals 1 if you stop on an odd, and 0 if you stop on an even. Infinity is treated differently in different fields. I suppose in physics, it's probably treated as a superposition of even and odd, so statistically, you could interpret it as 0.5. And I'm sure that works great for physicists and it fits a number of scientific models. But there are probably other fields involving mathematics that would find this interpretation to be silly and not very useful.
- d4mi3n 13y agoYou're referring to Grandi's Number. Numberphile covered this pretty well on their youtube channel: http://www.youtube.com/watch?v=PCu_BNNI5x4 http://www.youtube.com/watch?v=PCu_BNNI5x4
- dj-wonk 13y agoLose 20 pounds -- and your sanity -- using this one weird trick! I suspect this kind of reasoning is useful in some fields for some particular kinds of infinite sums, but for the particular summation problem as stated, as most people understand it, this is rubbish.
- vinceguidry 13y agoYou should watch the linked video. The logic isn't that hard to follow, and finds real-world application all over the place.
- j2kun 13y agoExcept that all the math presented in it is, strictly speaking, completely false.
- dj-wonk 13y agoI was using humor there. Yes, I am open to this kind of thinking being useful. I did watch the video. I followed the logic. (That logic is contradictory from other perspectives -- see http://www.quora.com/Mathematics/Theoretically-speaking-how-can-the-sum-of-all-positive-integers-be-1-12 http://www.quora.com/Mathematics/Theoretically-speaking-how-...). Summary: as most people understand it, the sum of all natural numbers is (a) positive (b) very large (c) not finite (d) certainly not -1/12.
- gregw134 13y agoWikipedia says the sum (1 - 1 + 1 -1 ...) diverges. http://en.wikipedia.org/wiki/Grandi's_series http://en.wikipedia.org/wiki/Grandi's_series
- snnn 13y agoThat's right! 1 - 1 + 1 -1 ... = nothing!!!
- deevus 13y agoWell yeah it doesn't converge on any particular number so it must diverge. Like what has already been said I think this is helpful in certain fields, but bollocks in others.
- nemo1618 13y agoSee also Thomson's lamp: http://en.wikipedia.org/wiki/Thomson%27s_lamp http://en.wikipedia.org/wiki/Thomson%27s_lamp
- Zikes 13y agoAccording to the Wikipedia article [1], it equals -1/12 (R), except the R is written a rather fancy script. Now, I'm no mathematician, but I'm betting that (R) is important, and affects the true value of the -1/12, and probably should not be omitted. [1] http://en.wikipedia.org/wiki/Ramanujan_summation http://en.wikipedia.org/wiki/Ramanujan_summation
- valtron 13y agoThere's only one way to add a finite list of numbers, but several ways to "add" infinite lists. Ramanujan summation is just one way. There are also other ways to define "addition" on infinite lists: Abel, Borel, Cesaro, Euler, etc. summation. They're different from each other because they disagree on certain infinite sums, but they're still considered "summations" because, for finite lists, they give the same answer as normal addition.
- Zikes 13y agoSo the title should instead read "The sum of all positive numbers up to infinity (for a certain definition of 'sum') is -1/12".
- dj-wonk 13y agoUnintended consequence: my confidence in string theory just plummeted.
- dkhenry 13y agoThe Ramanujan Summation is not a sum in the traditional sense. So the title is missing a script R after the -1/12.
- klochner 13y agoAnd a non-linkbait title for the post would be: The Ramanujan sum of all positive numbers up to infinity is -1/12 Which is not sensational, exciting, or even that interesting.
- the_cat_kittles 13y agokey point! from wikipedia: Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a sum to infinite divergent series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties which make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined. ...so the way we normally think of sums, of course 1 + 2 + 3 ...grows infinitely large. Makes me wonder- what was the impetus for creating the idea of Ramanujan sums?
- jezell 13y agoNo, the sum of all positive numbers up to infinity does not equal -1/12. Bad math is bad. Amazing people actually buy this dude's "math."
- smegel 13y agoDid Euler overflow his abacus or something?
- jimmaswell 13y agoHe shifted the series over and added it to itself, which I've seen be called out as an invalid thing to do on another sum 'proof' with a weird result.
- chacham15 13y agoI asked this question on the mathematics stack exchange: http://math.stackexchange.com/questions/633285/is-the-sum-of-all-natural-numbers-frac112 http://math.stackexchange.com/questions/633285/is-the-sum-of... It is only true if you redefine what the "sum" of an infinite series is.
- zem 13y agothere was a good discussion of this on /r/math a week ago: http://www.reddit.com/r/math/comments/1usu93/1_2_3_4_5_112_numberphile/ http://www.reddit.com/r/math/comments/1usu93/1_2_3_4_5_112_n...
- dmunoz 13y agoI was visiting this comment thread with the intention to post that link if nobody else had already. For all the hate reddit gets, I love it for content such as the discussion contained in the above. Reddit has major issues due to its user base, but one thing I do is browse certain subreddits once a week (month) by filtering by top posts in the past week (month). The day by day drone in most subreddits is tiring, but quality threads hit the top posts.
- deleted 13y ago[deleted]
- valtron 13y agoAchilles runs at 1 unit/s and Tortoise at x < 1 unit/s. Tortoise gets a head start of 1 unit. How long until Achilles catches up? Well first, Achilles has to travel 1 unit because Tortoise got a head start. That takes 1s. By then, Tortoise managed to run x units, so Achilles has to catch-up, which takes another x seconds, and by then... etc: The time it takes Achilles to catch up is 1 + x + x^2 + ... Since x < 1, this converges and the answer is 1/(1-x). It takes 1/(1-x) seconds for Achilles to catch up. Plot the lines: Achilles runs along A(t) = t, the tortoise T(t) = 1 + x t. Solving for t: t = 1 + x t = t (1 - x) = 1; t = 1/(1-x). Ok, now what if x > 1? Let's use x = 2. 1/(1-2) = -1. It takes -1 seconds for Achilles to catch up. Don't believe me? Plot it: A(t) = t, T(t) = 1 + 2t; Achilles catches up with Tortoise 1 second before the race starts! There you have it, a physical interpretation of why 1 + 2 + 4 ... = -1. (I have no example right now for 1 + 2 + 3 ..., sorry.) So you see, this isn't nonsense, though I admit it DOES depend on definitions!
- keerthiko 13y agoI am fairly certain sum=1/(1-x) holds true only for x<1. when x>1, it is a divergent series... and I can't tell if you're joking or not, or I would just laugh...
- j2kun 13y agoThis video has been showing up all over the internet, and the math in it is completely imprecise. Numberphile has lost a lot of respect in my eyes due to how much misinformation this video has spread.
- jgg 13y agoThey also created a pretentious video where they explained that the NSA hacked people's e-mails through a rigged random number generator, complete with an attempt at explaining elliptic curves and modular arithmetic, that ignores the fact that barely anyone used that PRNG and most e-mail isn't encrypted properly in the first place. Either I'm a really grumpy person, or their videos are actually terrible. I think there are enough interesting things in math that can be explained without falsely embellishing uninteresting results and ideas.
- pazimzadeh 13y agoIf 1+1-1+1-1+1... can be approximated to 0.5, then 1000+1000-1000+1000-1000+1000... must be approximately 500, but shouldn't 1+1-1+1-1+1...be equivalent to 1000+1000-1000+1000-1000+1000...?
- kibibu 13y agoNote that 1+1-1+1 doesn't fluctuate between positive and negative, it fluctuates between 0,1,0,1,0,1.
- PopePompus 13y agoWhat is presented here is actually a reductio ad absurdum proof that 1 - 1 + 1 -1 ... does not equal 1/2 (or any other number).