Numberphile v. Math: the truth about 1+2+3+…=-1/12
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Numberphile v. Math: the truth about 1+2+3+…=-1/12


Welcome to the first Mathologer video of
the year. Today it’s about something very serious and so I’m wearing a totally
black t-shirt. You all like Numberphile right? Me too, except for this one video
over there in which they prove the infamous identity 1+2+3+…=-1/12 using some simple algebra that
even kids in primary school should be able to follow. Since this video was
published in 2014 over six million people have watched it and more than
65,000 have liked it. Unfortunately, pretty much every single statement made
in this video is wrong. And by wrong I mean wrong in capital letters. In
particular, as anybody who knows any mathematics will confirm 1+2+3+… sums to exactly what common sense suggests it should namely plus infinity.
And this video was not published on the 1st of April. Also, as we all know, the
Numberphile videos are presented by smart guys, in this case university
physics professors, who do know their maths and who are definitely not out to
mislead us. So how did they get it so horribly wrong and what did they really
want to say. Well, they started out with some genuinely deep an amazing
connection between 1 + 2 + 3 etc and the number -1/12 but in the effort to
explain this connection in really, really simple terms they just went overboard
and ended up with an explanation that is not just really simple but also really
wrong. Well 6 million views later and the comment sections of all maths YouTubers
are being inundated by confused one plus two plus three comments that are a
direct consequence of this video. For mathematical public relations it’s a
disaster. (Marty) It’s THE disaster. (Mathologer) Yeah, it’s THE disaster. And so I think it’s a good idea to have another really
close look at the Numberphile calculation step by step, state clearly
what’s wrong with it, how to fix it, and how to reconnect it to the genuine maths
that the Numberphile professors had in mind originally. Lots of amazing
maths look forward: non-standard summation methods for divergent series,
the eta function, a very well-behaved sister of the Zeta function, the gist of
analytic continuation in simple words, some more of Euler’s mathemagical tricks etc. Now, I’ve tried to make this whole thing self-contained. So you don’t have to have
watched my very different other video on one plus two plus three from over a year
ago or anything else to understand this one. Okay, let’s get going. So that we are
all on the same page, here real quick is the whole Numberphile calculation.
They call the unknown value of the infinite series 1 + 2 + 3+… S. As
stepping-stones for the calculation they first calculate the sums of these other
two infinite series. So 1-1+1-1+… and 1-2+3-4+… Adding up the terms of the first series, we get the partial sums
1. Ok 1 minus 1 is 0, 1 minus 1 plus 1 is 1, 1 minus 1 plus 1 minus 1 is 0 and so
on. These partial sums alternate between 0 and 1 and so the Numberphile guys
declare that the sum of this infinite series is CLEARLY the average of 1 and 0
which is 1/2 (Marty) That’s not all that clear to me. (Mathologer) Alright we’ll get to that. They also mention that there
are other ways to justify this. We’ll also get to that. Ok, second sum. Here they
start by considering what happens when you double this sum. So 2 times S2 is
equal to the infinite series added to itself but now before adding the two
infinite series they shift the bottom series one term to the right. Now 1 plus
nothing is 1, minus 2 plus 1 is minus 1, 3 minus 2 is 1, minus 4 plus 3 is minus 1,
etc. But that bottom series is the one we already looked at which, remember, is
equal to 1/2 and so …. Second sum done, great. Now for last sum, that’s the one we’re really after. Here the Numberphile guys start by subtracting S2, the
sum they just figured out, from S. Now 1 minus 1 is 0, 2 minus minus 2 is plus 4, 3
minus 3 is 0, 4 minus minus 4 is 8, etc. The zeros don’t matter so let’s get rid
of them. Take out the common factor 4. Ah the yellow that’s our 1+2+3+…
sum S again. Now solve for S, and my usual magic here, and we get -1/12.
And here the Numberphile guys take a bow. But, not so fast! !ll this is really
nonsense the way it was presented. In particular these three identities are
false. This means that if on any maths exam at any university on Earth you’re
asked to evaluate the sums of these infinite series and you give the
Numberphile identities as your answer you will receive exactly 0 marks for
your answers. It’s critical to realise that in mathematics we have a precise
definition which underpins the sum of an infinite series. Wherever you see
infinite series this definition and only this definition applies unless there are
some huge disclaimers to the contrary in flashing neon lights. Alright, now the
Numberphile guys did not include any such disclaimers and so they too should
get 0 marks for their effort. (Marty) Or maybe give them -1/12 marks. (Mathologer) Yeah I think I can agree with that. OK,
what’s this definition and what are the answers that will get you full marks on
your maths exam. To evaluate the sum of an infinite series you calculate the
sequence of partial sums just like the Numberphile guys did at
the very beginning. Now if the sequence of partial sums levels off to a finite
number, that is, if the sequence converges, or if it explodes to plus infinity, or if
it explodes to minus infinity, then this limit is the sum
of the infinite series. If no such limit exists, then the infinite series does not
have a sum. That’s it, that’s the definition. So for the first Numberphile
series the sequence of partial sums alternates between 0 and 1 and therefore
does not have a limit. This means that this infinite series does not have a sum,
neither 1/2 nor anything else. This is the correct answer for your maths exam.
Alright, what about the other two infinite series? Hmm, well, in the case of
1 plus 2 plus 3 the partial sums explode to plus infinity and so the sum
of the series is infinity. For the infinite series in
the middle the partial sums explode in size, but neither just to infinity or
just to minus infinity, and so this series also does not have a sum. So these
are the answers that get you full marks. In many ways the most important
infinite series are those with a finite sum which have not featured here yet. So,
to give some perspective, here’s a standard example, an infinite geometric
series: 1/2+1/4+1/8 and so on. Now here the partial sums exhibit a
really nice pattern and clearly they converge to 1. (Marty) Yeah, I think this one is clear. (Mathologer) This one is clear and so the sum of this infinite series is 1. Oh, before I forget, those
finite sum series are usually called convergent series and all the other
infinite series are called divergent series. Keeping this in mind
let’s have another look at the Numberphile calculation. Here’s the whole
thing at a glance. It’s just a transcript of the writing on the brown paper in the
Numberphile video. Again, as it was presented by Numberphile all this is
nonsense and worth 0 marks. (Marty) Or less! (Mathologer) Or less 🙂 THIS. IS. NOT. MATHEMATICS. Don’t use it,
otherwise you’ll burn in mathematical hell. Having said that there should be some method to this madness, right?
Those guys are smart! But if there is, then it’s clear that the sums you see
here cannot possibly represent the usual sums, as about six million people have
been misled to believe by this video. Ok let’s start by doing something that may
also seem a little bit crazy. At first glance, just for fun, and in denial of
reality, let’s assume for a second that those three Numberphile series were
actually convergent, that is, all had a finite sum. Then all, ALL highlighted
arguments would be valid. This includes the termwise adding and subtracting of
series that was performed here, … and here, … and even the shifting to the right
before the addition that a lot of people view with suspicion. Why would all these
operations be ok if we were dealing with convergent series? Because summation of
convergent series is consistent with termwise addition and subtraction, and
shifting. Let me explain that too. There are differences between finite and
infinite sums. For example, infinite series sometimes don’t have a sum
whereas finite sums always exist, rearrangement of the terms can change
the sum of convergent infinite series, etc. On the other hand, the sums of
convergent infinite series do share a lot of the properties of finite sums and
it’s exactly these properties that make them so useful. Here the most important
three such properties. Let’s say you have two convergent infinite series, okay

100 Comments

  • Matt Gregory

    Would +1-1+1-1… converge if you added up all the +1's first and then added up all -1's? What if you make two groups of an infinite number of +1's and just one group of an infinite number of -1's and summed them together? Would that make the sum 1 rather than 1/2? Infinite sums are ridiculous.

  • Scott Thomson

    The irony is numberphile even said yes it goes to infinity but we are looking for a number that gives the series some meaning and like they said the values they presented are used in real applications and work properly how can you say they are wrong

    Math is still evolving please don't get stuck behind

  • wikichris

    The main problem is that it doesn’t pass any test of human value. Mathematics has delivered huge value to humans via physics and engineering and computing and finance. Using these kind of results have never given humans any value. Quantum physics has given humans a lot of value. String theory never has.

  • Techy Savage

    Earlier I posted a comment explaining how he was wrong. Later realizing that I was wrong. I didn't belive that it was but that's what most porffsesers had said. They were not saying that it was indeed the answer but to actually point out that even mathimations make mistakes like svintria ramanujan

  • Lashy

    22:20

    So what is the supersum of "1+0-1-0+1+0-1-0…"? I'm getting partial sums of 1,1,0,0,1,1,0,0,… and psmean of (1),(1),(2/3),(1/2),(3/5),(2/3),… tending to 1/2.

  • Cactus Galactus

    Eh, I wasn't impressed with this video. My take away is that you weren't happy with numberphile generalizing the issue for the lay viewer. So you claim the video is completely wrong but wind up coming to the same conclusions anyway. The difference being this video says the real answer is more general than specific. Anyone with sense understands numberphile is just trying to get people interested in math, not give them answers that will help them on their math exams.

    Idk just has that feel of trying to ride the coat tails of numberphile that a lot of response videos have. I also didn't care for the guy off camera, his jokes didn't land well to me and seemed more spiteful than humorous. Normally I've enjoyed your videos but this one didn't work for me, I just don't see the need to go "actually" for over 40 minutes.

  • TheFarmanimalfriend

    This is totally off topic, but I thought you would find it interesting:
    Something is 'wrong' with Nicolas Oresme's proof. 1/2 is .5. .5 summed to infinity will blow up, however as you get farther from the origin, 1/2 decreases in magnitude until it is really tiny. The harmonic series is divergent, but not because the sum of its elements blows up. The harmonic series is divergent because it never approaches a single number.

  • M Siemons

    So I think you have been a little bit too harsh on the numberphile video. If I understand correctly, and please correct me if I'm wrong, the – 1/12 is not the sum, but if you want to assign a meaningful number to the series, then it should be -1/12. While numberphile used completely wrong math to get there, it might inspired a lot of non-mathematicians to learn more about math.

  • rrr00bb

    it isn't wrong. it's the difference between a recursive and an iterative definition. recursively, it works. and you can't do an infinite number of iterations.

  • Jay B

    they said that its found in physics and that the complex numbers were also thought to be a crazy idea. I still don't get how they got 1-1+1-1+….=1/2. Using the big "if," we could also not shift the numbers of s1 and get 0. so s1 =1/2 and s1=0

  • Rob Fielding

    consider this writeup of the general difference between a recursively defined sequence vs iteration. S = Limit[S] – Tail[S] …. and it makes perfect sense for a simpler example case such as powers of 2. S can be finite because Limit and Tail can grow without bound. https://www.overleaf.com/read/tzsfpnyykgpq

  • Eric Lager

    Please let me know that people have 11 fingers. Count five one hand + 6 on the other is 11! Mathematically you can't add divergent summations! Did you teach mathematics to Republicans?

  • François Cauneau

    OK, Numberphile'demo' was false… the S1 and S2 were reported centuries ago by Grandi as 'strange numbers' (not calculable)

    but you must know that us, Physicists, are allowed to play with wrong Maths… Anyway, I'm sure (as a Physicist, so) I could show using strange numbers that the sum of integers is equal to your birthday date 🙂

  • Mike-Tysons Lisp

    Everybody who can answer 3+3
    Put your mothafukn hands up and follow me
    Everybody who can answer 3+3
    Put your mothafukn hands up… look, look

    Numberphile stands tough
    But notice that this man did not have his hands up
    This math world has got you gassed up
    Now who's afraid of mathologer?
    1, 2, 3 and to the 4
    1 plus 2 plus 3 plus 4
    4 plus 3 plus 2 plus 1
    Infinite? -1/12? None.
    This guy aint no mothafukn physicist
    I know everything he's got to say against us
    I am white, my head is bald like a babies bum
    You can play bongos on my head like its a drum
    Compared to the greats my math may be less than some
    I do got dumb friends named Numberphile who shoots themselves in the leg with their own sums
    Tho me by myself out sums all six of you chumps
    This equation did fuck with minds
    Like you stood outside screaming "fuck the minds of mandkind"

    But don't ever try to judge me dudes
    You don't know what the fuck my logic has been through
    But I know something about you
    You went to Cranbrook, that's a low IQ school
    What's the matter dawg, you feeling bad?
    This guys a math wiz? His real name's Chad
    And Chad lives at home with both dads
    And Chad's dads would both beat his ass
    This guy don't want battle, he's shook
    Probably never even looked inside of a math book
    He's scared to death, he's scared to look because his argument is futile
    FUCK Numberphile

    Fuck a beat, and fuck a Ramanujan summation
    Numberphile bring lesser multiples of this heat like permutations
    Fuck everybody, fuck yall if you doubt me
    That sum cannot equal -1/12 and I'll say it proudly!
    And fuck this battle, I quit I'm sick of this static
    Here, tell these people something they don't know about mathematics.

    That took way too fucking long to make what am I doing with my life

  • Charlez PlaysMC

    Iam not a math genius like I’m just a teenager. but what i think is “quantum math” or vice versa is gonna be a thing in quantum computing. 1, 1/2, 2 more like, it is there or is it? It must be there. Like a quantum particle does.

    Me: just a teenager thought about this video

  • Gladstone Moises Arantes Junior

    Thank you for saving the math for me! I was going crazy with those other videos of Numberphille… I though it could be inconsistent, not just incomplete!

  • singamar

    I am not a mathematician. As a layman, I read a comment like this on a Numberphile video on this topic:

    "If 1+2+3+4 …….infinity = – 1/12, then if A arranges $1 to be given to B everyday till a very large number of days, B should end up owing $ 1/12 to A in the end!" The practical value of this comment struck me immediately.

    But, afterwards, I came to learn that 'Experts' also assert that the result 1+2+3+4+ …….infinity has been found to be useful in Quantum Mechanics and String theory! Should a person like me conclude that the concept of 'right' and 'wrong' is no more there in Mathematics/Physics? I look forward to knowledgeable people like you to please clarify. Thank you!

  • singamar

    I am not a mathematician. As a layman, I read a comment like this on a Numberphile video on this topic:

    "If 1+2+3+4 …….infinity = – 1/12, then if A arranges $1 to be given to B everyday till a very large number of days, B should end up owing $ 1/12 to A in the end!" The practical value of this comment struck me immediately.

    But, afterwards, I came to learn that 'Experts' also assert that the result 1+2+3+4+ …….infinity has been found to be useful in Quantum Mechanics and String theory! Should a person like me conclude that the concept of 'right' and 'wrong' is no more there in Mathematics/Physics? I look forward to knowledgeable people like you to please clarify. Thank you!

  • Charles Kusniec

    https://oeis.org/wiki/User:Charles_Kusniec/Offset_Applications#Misuse_of_offset_in_1.2B2.2B3.2B4.2B5.2B6.2B7.2B….3D-1.2F12

  • free thinker

    You likely want the female male thing for another video ( 0 or1) cause saying zero isn’t nothing sometime would confuse mathematician or student

  • Pierre Denis Trichet

    well well well! I know, it's bad. It can't be true. BUT!!!! Here comes Casimir. Hendrik Casimir was a Dutch physicist. He makes research on attraction between 2 plates in a vacuum. He has found an equation showing the sequence (1 + 2 + 3 + 4 + 5…) In physics, infinity is very bad. It leads to black hole, explosions… Then he simply replaced the sequence (1 + 2 + 3 + 4 +5…) by -1/12. Measurements of the Casimir effect have shown that Casimir's formula was accurate. Quite disturbing, isn't it ?

  • Radagast The Brown

    22:25 still getting 1/2, what am I doing wrong?
    1 0 -1 0 1 0 -1 0 1 0 …
    1 1 0 0 1 1 0 0 1 1 …
    1 1 2/3 1/2 3/5 2/3 4/7 1/2 …
    so it's { (2k+1)/(4k+1), (2k+2)/(4k+2), (2k+3)/(4k+3), 1/2 } and everything converges to 1/2 with k -> inf

  • free thinker

    This page is gosted ! Please guys , if you have posted link in the comment , remove it ! Otherwise this page can’t be seen by others !

  • free thinker

    You might have to remove comment with link of any kind ! Otherwise your video can’t be seen ! (Alphabet is very jumpy about link in comment ! (I found this out recently ) yep this is why some page aren’t visible ! Twisted ? Yep ! An awesome page like this being gosted Huet mathematic futur fan a lot and discredit even more mathematic since you did an honest review of mathematic ! Wich was very much needed to add credibility to the awesomeness at mathematic under the right context (assuming a context is even supply ! As you mentioned ! Keep up the good work

  • free thinker

    Adding or removing a zero doesn’t change anything ?
    Ok let’s apply logic to that notion
    So if you add Full disk and suddenly you have the hole of a disk you can ignore it ? I don’t think this mathematic is valid either

  • free thinker

    Want real world exemple of zero ? Take tnt and had a spherical hole with a perfect vacuum in its center (about a 1 mm spherical hole should do ! And try the tnt again ! The yield will be greater

  • free thinker

    Yes my exemple is only a short duration but it’s a cheat mathematic shouldn’t ignore even if it last only until the momentum has filled the vacuum with something then reversal of fortune occur (or at least try to but this is another thing

  • free thinker

    Could you change the title ! When we try to save it to favorite or signet in iOS YouTube think you stole numberphile video and basicly redirect the favorite and the signet in iOS to numberphile page instead ! I don’t know invert and say mathologer vs numberphile instead or whatever ! It’s like someone doesn’t like this video and make sure it is impossible to view ! You would likely have way more viewer if it was not so well obfuscated for some obvious bad reason !

  • Andrea Parma

    I'm not getting why the supersum 1+0-1+0+1+0+… should give a different result!

    Partial sums: 1 1 0 0 1 1 0 0 1 1 0 0 …
    Average of the first n terms: 1/2 if 4|n ; 1/2+1/(2n) if n is odd ; 1/2+1/n otherwise
    This sequence converges to 1/2… what am I missing?

  • Justin Rose

    It must so sad to have such a rigid and unimaginative view of the world like this guy. The inability to read into context and tone is astounding.

  • Charles Reynolds

    You mention that you will do a video on the Euler-Maclaurin formula. Did you ever get round to doing this? Would very much like to see it if so, but can't seem to find it!

  • Shane Pearson

    This equation is used in freaking string theory which isn’t usin any wrong equations be cause it’s a stepping stone to a theory of every thing

  • T Oadaly

    Thank you, and to those wondering why 1 – 1 + 1 – 1… has no sum, rather than saying "split the difference and call it 0.5", why not do the following?
    1 – 1 + 1 – 1 … =

    (1 + 1 + 1 …)
    -(1 + 1 + 1…), which we can rearrange as
    (1 + 1 + 1 + …)

    -( 1 + 1 + 1…) = 1, or
    (1 + 1 + 1 + 1…)
    -( 1 + 1…) = 2, etc

    You can make it equal any integer number you like from -inf to +inf, including -242, which causes the final answer in the numberphile approach to equal 42.

  • R Marinov

    2! = 2*1 = 2 AND 3! = 3*2*1= 6 AND 2 – 3 = – 1
    1 IS NOT PRIME NUMBER, AND 5<5! AND
    5 < (5!) – 5 ; 2!=2 AND 3! – 3 = 3 AND П>3>2
    INTEGER(11<4П<13) = 12 ; (0 – 0!)/12 = 1+2+3+…..
    (0 – 0^0)/INTEGER(4П) = – 1/12 = (2-3)/2!/3! =?

  • Poważny Mikey

    I'm not a pro but it seems like when for few years you are tought that you cannot divide by 0 and then after all this time you meet limits. I meen I watched only half of the video and in some way you have right but maybe there is something more than basics or something. 😉

  • Nirvana Supermind

    Your “supersum” already sort of exists as Cesaro summation, but that slightly effects some infinite series. 1+2+3+4+5+6+7+8… might be inf, -1/12, -inf, 100 or anything else depending on how you define the sum, if it’s Cesaro sum then it’s -1/12 though.

  • David Morris

    I dont understand. Whats the difference between what they did and using complex numbers?

    Wouldnt old mathematicians also claim that complex numbers "are not math"?

  • Trevor Perkins

    26:14 – "now let's play a game."
    Me: sweet I love games
    Shows a graph
    Me: is this some kind of German game that I'm not structured/organized enough to understand?

  • Bullfrogz100

    I could not comprehend that people believed that the result was -1/12, all was a trickery. I suspect this was promoted to get people used to negative interest rates: here is the money, now its gone, that's mathematics bro!

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