# Why does the sum of all natural numbers equal -1/12? **URL:** \ **Category:** Factual Questions\ **Created:** [January 10, 2014, 5:58pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452 "2014-01-10T17:58:53Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![standingwave](https://avatars.discourse-cdn.com/v4/letter/s/9de0a6/32.png) [@standingwave](https://boards.straightdope.com/u/standingwave)\ **Post date:** [January 10, 2014, 5:58pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/1 "2014-01-10T17:58:53Z")
OK, Numberphile just blew my mind. The sum of all natural numbers (from 1 to infinity) produces an “astounding” result. [![](https://img.youtube.com/vi/w-I6XTVZXww/maxresdefault.jpg "ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12") ](https://www.youtube.com/watch?v=w-I6XTVZXww) I can’t help feeling that it’s a bit of a parlor trick like “proving” 1 = 2 via mathematical trickery. All that sliding infinite series one spot to the right or left to make things cancel. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 10, 2014, 6:26pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/2 "2014-01-10T18:26:50Z")
I haven’t watched the video yet, but I do remember we had a thread a few years ago on this: [1+2+3+4+… = -1/12. In what sense?](http://boards.straightdope.com/sdmb/showthread.php?t=420045) ---
**Author:** ![standingwave](https://avatars.discourse-cdn.com/v4/letter/s/9de0a6/32.png) [@standingwave](https://boards.straightdope.com/u/standingwave)\ **Post date:** [January 10, 2014, 6:52pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/3 "2014-01-10T18:52:29Z")
> [@Thudlow\_Boink](#): > > I haven’t watched the video yet, but I do remember we had a thread a few years ago on this: > [1+2+3+4+… = -1/12. In what sense?](http://boards.straightdope.com/sdmb/showthread.php?t=420045) :smack: Thanks. Should have checked the archives. ---
**Author:** ![kaltkalt](https://avatars.discourse-cdn.com/v4/letter/k/bb73d2/32.png) [@kaltkalt](https://boards.straightdope.com/u/kaltkalt)\ **Post date:** [January 10, 2014, 6:53pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/4 "2014-01-10T18:53:27Z")
i turned the video off when he decided he’d add two series together by spacing them apart. Like adding 100 + 100 and just deciding i’ll do it like this: ## 100 +100 1100 So 100 + 100 = 1100? I don’t think so. If doing that is a critical step to proving all the numbers to infinity add up to -1/12 I’m not interested. The sum of the first 100 numbers is 5050. No way the sum up to infinity (which includes the first 100) is less than that, let alone a negative fraction. The sum of any amount of positive numbers cannot be negative. The sum of any amount of whole numbers can’t be a fraction. This must be based on some sort of hidden mistake shorthand trickery crap. I’m just going to use common sense. ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [January 10, 2014, 6:58pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/5 "2014-01-10T18:58:00Z")
> [@kaltkalt](#): > > So 100 + 100 = 1100? I don’t think so. If doing that is a critical step to proving all the numbers to infinity add up to -1/12 I’m not interested. That’s not really how it goes. > [@kaltkalt](#): > > I’m just going to use common sense. Almost always a mistake. There’s a reason why mathematical brilliance also requires years of training. Human beings are generally pretty bad at logical reasoning. As the saying goes, “common sense” is neither. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 10, 2014, 7:07pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/6 "2014-01-10T19:07:34Z")
> [@kaltkalt](#): > > i turned the video off when he decided he’d add two series together by spacing them apart. Like adding 100 + 100 and just deciding i’ll do it like this: > > ## 100 +100 > > 1100 > > So 100 + 100 = 1100? I don’t think so. If doing that is a critical step to proving all the numbers to infinity add up to -1/12 I’m not interested. What you did above is _not_ the same thing that he did in the video. He wasn’t adding together two invidiual numbers, but two whole series of numbers, and the sliding/spacing just changed the order of the addition. As for your other objection, you’re _sort of_ right; but I think that’s addressed in the other thread I linked to above. If it makes you feel any better, think of it as a sort of mathematical equivalent of science fiction, starting with “What if we _could_ do this” and seeing where it takes you. ---
**Author:** ![kaltkalt](https://avatars.discourse-cdn.com/v4/letter/k/bb73d2/32.png) [@kaltkalt](https://boards.straightdope.com/u/kaltkalt)\ **Post date:** [January 10, 2014, 7:08pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/7 "2014-01-10T19:08:25Z")
Ok fair enough, forget “common sense” and replace it with basic mathematical conjectures. The sum of two whole numbers is a whole number. The sum of two positive numbers is also a positive number. Basic pre-algebra principles of math. In the video, that’s how he added two (equivalent) series together - by spacing them apart one digit and adding straight down. I call bullshit on that. It was exactly like my 100+100=1100 example. He did it like this: S1=1+2+3+4… 2(S1) = 1+2+3+4… - 1+2+3+4… * * * 1+3+5+7… No. It would be: ## 1+2+3+4… 1+2+3+4… 2+4+6+8… So if that’s the type of mistake this is premised on, I’m not interested in seeing any more. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 10, 2014, 7:08pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/8 "2014-01-10T19:08:59Z")
> [@standingwave](#): > > :smack: > > Thanks. Should have checked the archives. In your defense, it’s pretty hard to search for 1+2+3+ … = -1/12. And we wouldn’t have been discussing that particular video before, since it just got posted. ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [January 10, 2014, 7:12pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/9 "2014-01-10T19:12:51Z")
> [@kaltkalt](#): > > S1=1+2+3+4… > 2(S1) = > > 1+2+3+4… > > - 1+2+3+4… > > * * * > > 1+3+5+7… > > No. It would be: > > ## 1+2+3+4… 1+2+3+4… > > 2+4+6+8… > > So if that’s the type of mistake this is premised on, I’m not interested in seeing any more. Explain how either is a mistake. Neither really is. The mistake here is supposing this is a “basic” mathematical operation. Yes, it involves only addition. But it involves addition of two divergent infinite series, which is not well defined. While that’s bad enough, your 100+100 example is actually worse, since it involves incorrect addition on top of everything else. At no point in either addition of infinite series is the value of any individual term changed. But in your addition, you increase the value of one of your terms by a factor of 10 for some reason. That makes no sense and does not occur in the addition of the infinite series. If you want the equivalent, it’s more like this: ``` 1234 + 1234 = (1000 + 200 + 30 + 4) + (1000 + 200 + 30 + 4) ``` = 1000 + 200 + 30 + 4 - 0000+ 1000 + 200 + 30 + 4 = 1000 + 1200 + 230 + 34 + 4 = 2468 VS = 1000 + 200 + 30 + 4 +1000 + 200 + 30 + 4 = 2000 + 400 + 60 + 8 = 2468 Both are equivalent ways of writing this addition. In neither case do we change the value of any of the terms, as you do in your 100+100 example. ---
**Author:** ![kaltkalt](https://avatars.discourse-cdn.com/v4/letter/k/bb73d2/32.png) [@kaltkalt](https://boards.straightdope.com/u/kaltkalt)\ **Post date:** [January 10, 2014, 7:13pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/10 "2014-01-10T19:13:22Z")
The sliding changed the answer. Just like how it changes 100+100 from 200 to 1100. He wanted to multiply the series by 2, and adding it to itself is an easy way to do that. But you don’t get to slide the numbers around. That changes the answer. I understand it’s a series as opposed to a single number, but moving it around to add down in columns changes the answer. You have to add the first number in the first series to the first number in the second series, and so on. You can’t add the first number to the second number, 3rd to the 4th, and so on. That’s arbitrary. Why not add the 1st number to the 3rd number? Let’s move it apart two spaces instead of one. ---
**Author:** ![DrCube](https://avatars.discourse-cdn.com/v4/letter/d/a3d4f5/32.png) [@DrCube](https://boards.straightdope.com/u/DrCube)\ **Post date:** [January 10, 2014, 7:13pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/11 "2014-01-10T19:13:29Z")
> [@standingwave](#): > > I can’t help feeling that it’s a bit of a parlor trick like “proving” 1 = 2 via mathematical trickery. All that sliding infinite series one spot to the right or left to make things cancel. It is a parlor trick. Most proofs of “1=2” divide by zero at some point, rendering the rest of the proof invalid. This one relies on treating a divergent series like one that converges, which is equally invalid. The sum of all positive integers does not equal negative one twelfth. It doesn’t equal anything, because the series grows without bound. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 10, 2014, 7:18pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/12 "2014-01-10T19:18:47Z")
> [@kaltkalt](#): > > In the video, that’s how he added two (equivalent) series together - by spacing them apart one digit and adding straight down. I call bullshit on that. It was exactly like my 100+100=1100 example. He did it like this: > > S1=1+2+3+4… > 2(S1) = > > 1+2+3+4… > > - 1+2+3+4… > > * * * > > 1+3+5+7… > > No. It would be: > > ## 1+2+3+4… 1+2+3+4… > > 2+4+6+8… > > So if that’s the type of mistake this is premised on, I’m not interested in seeing any more. He wasn’t shifting the _digits_; he was shifting the _numbers_. If the series were finite, this sort of thing would be perfectly legitimate. For example: ```auto 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 --------------------------------------------------- 2 + 4 + 6 + 8 +10 +12 +14 +16 +18 + 20 ``` = 110 If you shift, ```auto 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 +10 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 +10 --------------------------------------------------- 1 + 3 + 5 + 7 + 9 +11 +13 +15 +17 +19 +10 ``` = 110 ---
**Author:** ![mcgato](https://avatars.discourse-cdn.com/v4/letter/m/ac8455/32.png) [@mcgato](https://boards.straightdope.com/u/mcgato)\ **Post date:** [January 10, 2014, 7:18pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/13 "2014-01-10T19:18:57Z")
I didn’t have the sound on, but the result for S1 is incorrect. He puts: S1 = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + … = 1/2, which is incorrect. Everything after that pretty much falls apart. S1 does not converge to anything. ---
**Author:** ![kaltkalt](https://avatars.discourse-cdn.com/v4/letter/k/bb73d2/32.png) [@kaltkalt](https://boards.straightdope.com/u/kaltkalt)\ **Post date:** [January 10, 2014, 7:20pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/14 "2014-01-10T19:20:33Z")
I really hope this sort of thing isn’t actually used in physics to conclude there are 26 dimensions in string theory and whatnot. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 10, 2014, 7:21pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/15 "2014-01-10T19:21:29Z")
> [@kaltkalt](#): > > You have to add the first number in the first series to the first number in the second series, and so on. You can’t add the first number to the second number, 3rd to the 4th, and so on. That’s arbitrary. Why not add the 1st number to the 3rd number? Let’s move it apart two spaces instead of one. If my previous post didn’t make it clear: you seem to be confusing **numbers** with **digits** of a number. It’s basic math that when you have a finite set of numbers to add together, you can add them in any order whatsoever without changing the result. 1 + 2 + 3 + 4 is the same as 1 + 3 + 4 + 2 is the same as 4 + 1 + 2 + 3 and etc. ---
**Author:** ![kaltkalt](https://avatars.discourse-cdn.com/v4/letter/k/bb73d2/32.png) [@kaltkalt](https://boards.straightdope.com/u/kaltkalt)\ **Post date:** [January 10, 2014, 7:24pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/16 "2014-01-10T19:24:31Z")
> [@Thudlow\_Boink](#): > > He wasn’t shifting the _digits_; he was shifting the _numbers_. If the series were finite, this sort of thing would be perfectly legitimate. For example: > > ```auto > > 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 > + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 > --------------------------------------------------- > 2 + 4 + 6 + 8 +10 +12 +14 +16 +18 + 20 > > ``` > > = 110 > > If you shift, > > ```auto > > 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 +10 > + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 +10 > --------------------------------------------------- > 1 + 3 + 5 + 7 + 9 +11 +13 +15 +17 +19 +10 > > ``` > > = 110 Ah okay, that does help. He said he was trying to multiply the same series by 2. I was using “digits” interchangeably with “numbers” (or spaces, places, whatever). I guess it’s like adding all the even numbers and adding all the odd numbers, presumably the sums are the same. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 10, 2014, 7:24pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/17 "2014-01-10T19:24:38Z")
> [@mcgato](#): > > I didn’t have the sound on, but the result for S1 is incorrect. > He puts: > S1 = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + … = 1/2, > which is incorrect. Everything after that pretty much falls apart. > > S1 does not converge to anything. If you’d had the sound on, you’d have heard that he refers you to a previous video for the justification of that. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [January 10, 2014, 7:29pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/18 "2014-01-10T19:29:46Z")
Under the standard definitions, that sum is undefined. If, however, you extend the definitions in any of a variety of obvious ways, the sum becomes equal to -1/12. By analogy: What’s 3 divided by 2? It doesn’t make sense: 3 isn’t a multiple of 2, so there’s no integer that you can multiply by 2 to get 3. But if we assume that it does make sense, and extend the definition of “number” to compensate, we can get a sensible answer. Or another one: What’s 4^2.5? Well, 4^2 = 4_4, with two 4s, and 4^3 = 4_4\*4, with three 4s… But what the heck does it mean to multiply two and a half 4s together? That’s clearly nonsense! Except that we can come up with a new definition of exponentiation to accommodate that, and find that it makes perfect sense to say that 4^2.5 = 32. It’s the same thing here. Using the simple set of definitions, it looks like (1+2+3+4+…) is nonsense. But if we extend the definitions, we can get an answer for it. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 10, 2014, 7:38pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/19 "2014-01-10T19:38:00Z")
> [@DrCube](#): > > This one relies on treating a divergent series like one that converges, which is equally invalid. The sum of all positive integers does not equal negative one twelfth. It doesn’t equal anything, because the series grows without bound. Pretty much, yeah. If you take a class in which infinite series are covered (often second-semester Calculus), they’ll tell you you can only perform manipulations like these on series that are absolutely convergent (or they’ll be careful to work with partial sums and worry about when and how those sums converge). But centuries ago, mathematicians like Euler played fast and loose with infinite series like this and got some pretty interesting results that way. Here we do get a “nonsensical” result; but is it the kind of nonsensical result like “1 = 2”, that just plain isn’t true, or is it the kind of nonsensical result like “sqrt(-1) is a number,” that leads somewhere interesting and useful if you allow yourself to accept it? ---
**Author:** ![kaltkalt](https://avatars.discourse-cdn.com/v4/letter/k/bb73d2/32.png) [@kaltkalt](https://boards.straightdope.com/u/kaltkalt)\ **Post date:** [January 10, 2014, 8:01pm UTC](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452/20 "2014-01-10T20:01:23Z")
> [@Thudlow\_Boink](#): > > Here we do get a “nonsensical” result; but is it the kind of nonsensical result like “1 = 2”, that just plain isn’t true, or is it the kind of nonsensical result like “sqrt(-1) is a number,” that leads somewhere interesting and useful if you allow yourself to accept it? I think it’s like the former, not the latter. It’s just as nonsensical as 1=2. Imaginary numbers are not clearly wrong. If you want to say sqrt (-1) = i so be it. If you can get useful results from that, wonderful. IF you want to say sqrt (-1) = 1 or 0 or 2 or 50 or -1/12 then no, that’s not gonna work. Worries me to think any theoretical physics might use this sort of thing. I have no doubt such physics will remain theoretical. [Next page](https://boards.straightdope.com/t/why-does-the-sum-of-all-natural-numbers-equal-1-12/678452.md?page=2)