The Sum Of 1 + 2 + 3 + 4 + 5 + ... Until Infinity Is Somehow -1/12

The sum of 1 + 2 + 3 + 4 + 5 + ... until infinity is somehow -1/12

Here's a fun little brain wrinkle pinch for all you non-maths people out there (that should be everyone in the world*): the sum of all natural numbers, from one to infinity, is not a ridiculously big number like you would expect but actually just -1/12. Yes, the sum of every number from one to infinity is some weird negative fraction. What the heck?

I'm not a maths guy but I'm pretty good at basic arithmetic. By that I mean I'm pretty good at adding really small numbers. By that I mean I would totally figure that if you add all the numbers until infinity, you're going to get a really big number. Not in this case. It seems like some mathematic hocus pocus at work here but it's real.

What's fascinating is that this idea that the sum of all natural numbers is -1/12 actually popped up way back in 1735 (as pointed out by Kottke). But seeing all those numbers actually come out to -1/12 is a whole 'nother story. So watch physicists Tony Padilla and Ed Copeland from the University of Nottingham walk through the process of getting -1/12. The video by Numberphile is fascinating because it's mind-blowing but also because you can see the real joy from the wonderful people proving this. I love it. I love them. I like maths a little more.

*Genius humans, please forgive us normal civilians for not taking advanced maths classes and forgetting how to calculiing the calculus. We know maths people exist somewhere.


Comments

    I call bullshit. You cannot come to the end of an infinite string of numbers so you cannot perform any arithmetic equations on it without stopping the aforementioned infinite string, at which point it no longer is infinite ad therefore nothing to do with the whole argument in the first place. It's like stopping Pi at 3000 decimal places. It's not infinite any more!!! Any infinite number equation could not ever have a finite answer. So if someone says to you, "What's the answer to 1+2+3+4+5 etc", what are you going to do? Sit there forever and wait until the end before jumping in with an answer? If the string of numbers doesn't stop, you'd never be able to do anything with it!! Just because these guys know some big words doesn't mean they are always right.

      Did you watch the video? He addresses that exact point right near the end.

      @slidin_sidewayz Um, yeah, actually it does mean they are doing it right. This is a mathematical proof. They clearly state that if you stop the sequence at a finite point, the result would be a large number. But if you treat the sequence as truly infinite, then this mathematical proof demonstrates that the result is -1/12. There are many fairly basic pure mathematics principles that allow us to deal with truly infinite sequences. A lot of them I learned in high school maths, and certainly at university undergrad level. You can call bullshit all you want, but, like most things in science, I'd like you know your qualifications.

      Your blanket statement that you "cannot perform any arithmetic equations on it without stopping the aforementioned infinite string..." is just mathematical ignorance. Here's a link (warning PDF) that might help you -

      http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-convergence-2009-1.pdf

      Finding the limits of a series, including an infinite one, is pretty standard stuff, so if you know how to search the internet (hint: Google) then please, get back to us. Otherwise, keep your bullshit calls to yourself...

        He can prove himself, with a proof by contradiction. He is off adding all the numbers up now and will get back to us...

      just coz you cant literally count to infinity & add all the numbers together doesn't mean it cant be done in a more abstract way.

      i mean if you could create a formula that describes 1+2+3+4+5=15, then apply that, just with no upper limit, then you'd get a result.

      it'd be a scenario sorta like
      y = 1/x
      where as x approaches infinity, y approaches 0

      man, i havent done limits & stuff since high school.

      Edit: alternatively, it could be like the 0.999999... = 1 'proof'

      x = 0.9999...
      10x = 9.9999...
      10x - x = 9
      9x = 9
      therefore, x = 1
      0.999... = 1

      it's not true of course, 0.999... =/= 1, but that 'proof' up there seems to suggest it is.

      Last edited 18/01/14 7:58 pm

        0.9 recurring IS equal to 1.
        http://us.metamath.org/mpegif/0.999....html

        Not sure if trolling, 0.99999... is equal to 1. I'm not going to try and convince you of the fact, there's many website dedicated to explaining it.

          Try and actually read someone's post before you comment.

          Even if you didn't catch how he had proof in quotation marks to show sarcasm, he clearly says "It's not true of course."

            Maybe you read someone's post before you comment.

            I said that 0.999... does equal 1, shoggoth said it doesn't

        That proof falls apart when you use 2x,3x etc. 10x is just a party trick to move a decimal place.

          Just because his arbitrary number is carefully chosen doesn't make it wrong..... (hint, you can pick 9 as well and skip the "-x" step, it's just not as simple to do the maths along the way)

          0.999... is definitely equal to 1, and 10 is important because this proof is operating in base10. There are identical proofs in each number base that do the same thing.

          Last edited 31/01/14 5:51 pm

        It is true actually! It's called the limiting value as the 9's go forever.
        You, in fact, have assumed that in the line 10x = 9.99999... the fractional 9's are the same value as before you multiplied by 10. If there were not an 'infinite' number of 9's then, yes, you would be correct.

      I don't think you understand the concept of infinity. Watch the other numberphile videos dealing with infinity. All the points you mentioned have been 'solved' by Cantor.

      I'm going to second your call of bullshit.

      Yes, I understand that it's accepted string theory, but I'm not going to go with the "believe me because they use this in physics" bullshit. There's a big difference between the strength of the theory of "gravity" or the theory of "relativity" than string theory.

      Sure, they've used some fancy maths to prove their point, the big "but!" is their stated assumption at the start:

      Here's why I call bullshit: The whole thing is based on the assumption that:
      1+1-1+1-1+1-1 ... = 1/2.

      The justification: "We all know the answer is "either 1 or 0" but because we don't know which, we just average it to 1/2. "

      Wait. No, F*cking stop for just one second. Let me just plug that into my little calculator:

      if( [0,1] == 0.5 )
      print "String Theory Bitchas!";
      else
      print "Of course one or zero doesn't equal half, are you retarded?";
      end

      While we're at it, I've solved the long standing question of Schrödinger's cat. It's 1/2 alive.

      My Prize?, Well, I've got either $2,000,000 or $1 in my bank account. You don't know which. Is anyone willing to give me a $1,000,000 cash advance?

      While I'm at it, I'm going to set this 1-bit register (for non CS readers, they traditionally store a 0 or a 1) to 1/2.

        This.

        You can't "average" out the two possabilities, that automatically invalidates everything.

        he was using the "average" argument because it's easy to understand intuitively. 1/2 is the Cesaro sum of Grandi's series (1 - 1 + 1 - 1 ...). Further reading:
        http://en.wikipedia.org/wiki/Grandi's_series
        http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation

      There is another one similar that will throw you:

      S=1+2+4+8+16+32...
      = 1+2 (1+2+4+8+16...)=1+2S
      so S=-1; it's true. Not one of those 1=2 nonsense proofs

      Go figure!

      Last edited 17/06/15 4:11 pm

    That sum has to greater than 1. Adding positive integers to 1 must yield a greater sum.

    -1/12 is less than 1, therefore it cannot be the correct sum.

    Not saying what it really is, just that it ain't -1/12.

    I guess a lot of physics is screwed now... :)

    Yeah but he fucks with the formula's to make his own result.

    If you don't put the "shift" (push the 1 along to align with the 2 from the first iteration of the equation) into 2S2 it obviously boils back down to the same equation...

    What if you did this? ( a double shift)

    2S2 = 1 - 2 + 3 - 4 + 5 - 6
    + 1 - 2 + 3 - 4
    =1 - 2 + 4 - 6 + 8 - 10

    Carry the 2 through

    S2 = 1/2 - 1 + 2 - 3 + 4 - 5

    Now what? I can push the equation along to suit my needs as well.

    S1 = 0 or 1 depending on where you stop it, since when can you just average out an infinite equation?

    I understand how he gets to the result i just think this is retarded and if this has any practical and useful applications then i would like to see them because otherwise it is just a load of garbage

      He "pushes it forward" to show the terms lining up so you can see more clearly how he derives the answer. Just writing it offset from the other equation.

      There's an alternative proof here as well:
      http://youtu.be/E-d9mgo8FGk

        Nope, he pushes it forward, then adds (1 + nothing), then (-2 + 1). So at any point in S2, it is not the double of S2, because the second copy has been shifted.
        Let's take a computer example:
        ary1 = [+1, -2, +3, -4, +5]
        ary2 = [0, +1, -2, +3, -4]

        ary2 is shifted, there is a zero in the first index.

        Both array has 5 elements, but if you add the same index from both array (that is: ary1[i] + ary2[i]) the result will NOT be equal to ary1[i] *2.

          Yes, but it's not a *list* of numbers, it's the *sum* of those numbers.

      Actually, your double shift works just fine.
      When you add the double shifted, left shift I assume, you get

      2S2 = 1-2+4-6+8-10+12.....

      This is the same as

      2S2 = 1 - 2 (1-2+3-4+5-6...) = 1-2S2
      so S2 = 1/4

      Last edited 17/06/15 1:41 pm

      You almost had it!
      S2 = 1/2 - 1 +2 - 3 + 4 - 5......
      S2 = 1/2 - (1 - 2 + 3 - 4 + 5....)
      S2 = 1/2 - S2
      So S2 = 1/4

    You can't go to infinity, therefore, this question has no answer.

    You can do all the unjustifiable science you wish, such as "averaging" out 1 - 1 + 1 .... (which by the way doesn't have an answer also, unless you stop it at a point) and then applying that to other sums, but in the end, this is just some way for some mathematicians to create some weird excitement for themselves.

      This would seem to be sensible, except that physicists have encountered situations where this sort of sum drops out of the equations, and they have evidently used the result successfully (as indicated in the video - the guy making the video is a physicist.)

      Messing with infinities is enough to give anyone a headache. It's not an intuitive result, but mathematical results are not always intuitive.

    Um. Doesn't 2-2 equal 0 and not 4 and 4-4 equal 0 and not 8. Also, if you are going to add all the numbers up to infinity, why would you then add other numbers to the mix. Doesn't make sense. Also infinity means forever so you cannot possibly get an answer.

      If you look at the equation, it actually states:
      2-(-2), which is 2+2 (basic mathematics). Also infinity is a *number* of sorts, doesn't necessarily mean 'forever' or 'never-ending'. Tough to explain but watch there other videos on that one :P

        Infinity is not a "Number" of any sort. It is a concept. Something can tend towards infinity, a limiting value can converge to infinity... But infinity itself is not a number.

      I agree it sounds stupid, and if you watch through again you'll notice its not 2 minus 2, its minus -2, and when you minus a negative number, you actually add it. Hope that clears it up a bit :)

      Yeah, I don't think you get the concept of infinity.

      Let's say we have a running race and that you are much much faster than me. Since you are generous, you give me a little head start. You will never win the race!

      Because by the time you reach where I was when you gave me the head start, I would have run a little further on. And by the time you reach the spot where I was before, I would have moved a little further on. And so forth until infinity.

      But you know that eventually you will overtake me and win the race, but how can that be?

        Just Zeno's paradox. It takes your faster opponent an infinite number of steps to overtake you, but he does so in a finite time.

        I'll never win the race ever because the race doesn't end. At first you would be leading the race until i caught up and then i would be leading the race, but no one would ever win or lose because their is never going to be a finish line. I admit maths is not my strong point and all this doesn't make sense to me but the maths i know says if you add some numbers, the total has to be greater than any of the numbers you are adding. Anyway, this story is fascinating but i don't think any of my friends will understand it so I'll have to save the page to pocket as proof.

        I just looked up infinity maths wise. It gave me a headache, i yelled at the phone and i gave up. I think i have one of those brains that use the other hemisphere. Found school easy, maths hard. Fractions don't make sense and my brain refuses to understand why there is letters in something that is all about numbers.
        When i was reading about infinity just then, it said infinity minus infinity does not equal zero. WTF. If infinity was an Apple and you take away an Apple, then you have zero, but I'm wrong for some reason and I'm starting to get a headache again. I hate maths witha passion. Its voodoo

      I think the sums he mentions are 2 - - 2 and 4 - - 4 and so on. This gives 4, 8 etc

      Yes 2-2 =0 however they were dealing with 2 - (-2) and 4 - (-4). Obviously subtracting a negative gives a positive therefore the answer is 4 and 8. I'd also recommending googling what infinity actually means.

    It's actually -(1/12), not -1/12.

      Same thing dude!

        Whoops. I just derped hard. You're completely right.

    The answer is the same as ∞-1+∞-2+∞-3+∞-4 repeated until infinity

      No, everyone knows that answer to that summation is 42, as proved by the famous mathematician Douglas Adams.

    They're not talking about summing in the usual sense. The sum of the natural numbers is infinity, there are several ways to do this. (Take the formula for the sum of the first n natural numbers, then take n to infinity.)

    They're talking about Zeta function regularisation, see link.

    Their derivation using infinite series is flawed because the series in question are not absolutely convergent, so they cannot switch terms without changing the value of the series.

    Link: http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

    I typed the summation to infinity formula into my trusty HP42C programmable calculator and ran it. After a few minutes 13/12th's of the mass of the calculator disappeared in a puff of quantum string energy, thereby proving this assertion. Take that doubters.

    This isn't true. The comes from their opening proposition, that

    1 - 1 + 1 - 1 + 1 ..... = 1/2

    This is not true. If we look at the values of this series, as we continue adding to it, we have 1,0,1,0,1,..... it never converges to a value of a 1/2, and as such, we cannot assume that 1 - 1 + 1 - 1 + 1 - ...... = 1/2. Since this point is wrong, the rest of the argument must in turn be wrong.

    To prove that the sum of all positive numbers must be positive, we can look at the series of numbers

    1,3,6,10,15,.....

    If the n-th number in this series is positive, then to get the (n+1)th number we simply add (n+1). As (n+1) will always be positive, therefore if the n-th number is positive, then so must the (n+1)th. From inspection of the first values of the sequence, when n=1, then every time we add another number to it, the series must stay positive. Ergo, the quoted proof is simply untrue.

    Another thing we can do is look at the value of this sum when we add the first n numbers together. When we add 1 + 2 + 3 + 4 + ..... + n = n*(n+1)/2. For example, if n = 5, then 1 + 2 + 3 + 4 + 5 = 15 = 5*(5+1)/2. To then examine what happens when we add all the numbers up to infinity, we can take the limit as n goes to infinity. When you have infinity * (infinity + 1)/2, well, that's infinity.

    For a little bit of a complicated explanation, the sum 1 + 2 + 3 + 4 + 5 + ..... can be subjected to what is known as a convergence test, to see if it converges upon a single value as the number of terms added together goes to infinity. For simplicities sake I'll skip the maths here, but if you add up the sequence of all positive numbers, then the series doesn't converge.

    For discussions on this video from another source, see http://scientopia.org/blogs/goodmath/2014/01/17/bad-math-from-the-bad-astronomer/

      Numberphile actually proves that 1 - 1 + 1 - 1 + 1... = 1/2 in an earlier video.

        It's pseudo-science.

        They basically say "It's either one or zero, but because we don't know which, it's okay to just average it".

        Just like you can average winning the lottery, becoming pregnant or programming a bit register... oh, and by the way, Schrödinger's cat is 1/2 alive.

        Just because it is used to calculate string theory, doesn't mean it is mathematically correct in the traditional sense.

          Really!

          S = 1-1+1-1+1-1....
          This is the same as
          S = 1 - (1-1+1-1+1-1+1....
          So
          S = 1 - S
          Therefore S = 1/2
          No pseudo science there

            But you have asserted that: 1-1+1-1+1-1.... = 1 - (1-1+1-1+1-1+1....
            Which leads you to a logical conclusion that the product of an infinitely long discrete osculation between zero and one is 0.5.

            The absurdity of this maths can be explained with a real world example: If you were to apply this in the real world, you could try to charge your 5 volt iPhone 6 with a power supply that provides 10 volts for 1 second then negative 10 volts (or zero volts, depending on your interpretation) for one second. Maybe, if you never unplug it, it won't blow up your iPhone, because 10-10+10-10... = 5.

            I disagree with the assertion 1-1+1-1+1-1.... = 1 - (1-1+1-1+1-1+1.... because I think it violates the laws of performing operations with infinity.

            Since we're dealing with an infinitely long string of 1+1-1, we could express 1-1+1-1+1-1.... as infinity * (1+1-1). Since you are asserting that (1+1-1...) = 1 - (1-1+1), could also express it as infinity * (1+1-1+1). Therefore, infinity * (1+1-1) = infinity * (1+1-1+1). For simplicity, lets re-write this equation with infinity expressed as "i". Therefore, i(0) = i(1) or the mathematically sound conclusion that 0=1.

            Which is like saying, since 20 * 0 = 0, and 19 * 0 = 0, then 240 * 0 = 120 * 0 = 0 * 0, or 240 = 120 = 0.

            Again, if you trust this, feel free to use 120 volt rated stuff in 240 volt power points, or hell, just stick a knife into the power point, because 240 volts is really just as harmless as zero volts.

            This is why mathematicians have arbitrary rules about dividing by zero, or working with infinity.

            Last edited 17/06/15 7:39 pm

        Please note I'm saying this without having watched the earlier video. However, I can guarantee that they're incorrect - yes, it is possible to assign a value to 1 - 1 + 1 - 1 + 1 .... = 1/2. There are several techniques, such as Cesaro summation, that allow a numerical value to be assigned to an infinite series that diverges or doesn't converge.

        However, and here is the point where Numberphile went wrong, these summations are not the actual value of the sum, they are only useful for studying the sums under certain mathematical frameworks. They cannot be used to directly substitute a value for an infinite divergent sum, so you cannot just say that 1 - 1 + 1 - 1 + 1 - ..... = 1/2, and then use it in the type of process that they attempt to use in the video. It simply doesn't work. What they've done is a mathematical sleight of hand, trying to conflate several concepts which cannot be mixed together, and then using it to derive what seems like a magical result.

          Yes numberphile have really confused the issue by making you think you can simply add these terms forever and somehow get a result. Euler himself commented that the meaning of the word 'sum' should be extended when referring to counter intuitive results like these.

      http://en.wikipedia.org/wiki/Summation_of_Grandi%27s_series

    ITT: Laymen trying to prove mathematicians wrong

      The trick is in the "shifting" of 2S2. From then on he doesn't add 2x the same value (which he states), as the second series has been shifted. He adds 1 to 0, -2 to 1, +3 to -2 but if it is a double, he should really add 1 to 1, -2 to -2, +3 to +3, etc.

    Nearly everyone posting here should read up on Cantor and Hilbert, then come back and delete all their dumb 'intuitive' alternative answers.

      Alan Jones told me that infinity is bullshit made up by leftie greenie latte sippers in their ivory towers.

    I can't even think of a witty comment to weigh in with...

      You need an infinite number of monkeys with typewriters working on your rejoinders, something snappy will pop out eventually. BTW, if I had 1 monkey and added 2 monkeys, followed by 3 more monkeys and so on ad-infinitum, will I end up with -1/12 monkeys? I think not, therefore my belief in the truth of the original proposition is now shattered. Take that Hilbert and Cantor.

    Hey, just realized that this means the sum of an infinite 1s is zero,

    basically 1+2+3+4+... =-1/12
    (1+1+1+1+...)+(1+2+3+4+...)=-1/12
    (1+1+1+1+...)-1/12=-1/12
    (1+1+1+1+...)=0

    am I wrong in this, can someone point me towards somewhere i might get more info on this?

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