The Ramanujan sum of all positive numbers up to infinity is -1/12(kottke.org)
kottke.org
The Ramanujan sum of all positive numbers up to infinity is -1/12
http://kottke.org/14/01/the-sum-of-all-positive-integers
31 comments
There'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.
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.
So the title should instead read "The sum of all positive numbers up to infinity (for a certain definition of 'sum') is -1/12".
According 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.
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.
You're referring to Grandi's Number. Numberphile covered this pretty well on their youtube channel: http://www.youtube.com/watch?v=PCu_BNNI5x4
The Ramanujan Summation is not a sum in the traditional sense. So the title is missing a script R after the -1/12.
And 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 Ramanujan sum of all positive numbers up to infinity is -1/12
Which is not sensational, exciting, or even that interesting.
key 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?
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?
I asked this question on the mathematics stack exchange: 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.
It is only true if you redefine what the "sum" of an infinite series is.
Achilles 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!
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!
I 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...
This 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.
They 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.
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.
Unintended consequence: my confidence in string theory just plummeted.
Wikipedia says the sum (1 - 1 + 1 -1 ...) diverges.
http://en.wikipedia.org/wiki/Grandi's_series
http://en.wikipedia.org/wiki/Grandi's_series
Well 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.
Like what has already been said I think this is helpful in certain fields, but bollocks in others.
See also Thomson's lamp: http://en.wikipedia.org/wiki/Thomson%27s_lamp
That's right! 1 - 1 + 1 -1 ... = nothing!!!
He 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.
there 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_n...
http://www.reddit.com/r/math/comments/1usu93/1_2_3_4_5_112_n...
I 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.
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.
If 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...?
Note that 1+1-1+1 doesn't fluctuate between positive and negative, it fluctuates between 0,1,0,1,0,1.
What is presented here is actually a reductio ad absurdum proof that 1 - 1 + 1 -1 ... does not equal 1/2 (or any other number).
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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.
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.
You should watch the linked video. The logic isn't that hard to follow, and finds real-world application all over the place.
I 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-...).
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.
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.
Except that all the math presented in it is, strictly speaking, completely false.
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Did Euler overflow his abacus or something?
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