so guys lesson today is if someone offers to give you 1 dollar today 2 dollars tomorrow ect ect dont take the deal since he is obviously trying to steal you

Top 10 pranks that went too far.

My IQ increased by -1/12 after watching this

If you're from 11th-12th science, and you got some amazing Professor who sometimes taught you this type of curious and out of the syllabus problem , just to keep you hooked to the wonder of science and Mathematics, you're lucky.

Me before watching this video: liar Me after watching this video: cheater

A simple stack overflow bug. God will patch it in the next update.

The analytic continuation of the Riemann Zeta function does indeed map -1 to -1/12, however this does not mean that the sum of all positive integers is -1/12. The whole point of analytic continuation is to extend the function to the domain where the original function is divergent, and after doing that u CANNOT say that the original function maps the analytically continued domain to all these extended points

>This sum is actually used in physics so we know it's true! >*shows a book on string theory* bruh

i always multiply both sides by zero. Seems to fix things up pretty well.

After having watched this video for infinite times, I realized that my knowledge had increased by a -1/12 factor every time I watched it.

We were allowed to make an intuitive conclusion about 1-1+1-1…, but weren’t allowed to make a much more intuitive conclusion about 1+2+3…

I think the biggest assumption is that S1 is 1/2 which I think is the reason why we got all the natural numbers sum to -1/12

Tony: "The answer can be either 1 or 0, so we take the average 1/2 Me: "Ok, now that's where you screwed up"

Let me prove that 1 = 0, using this premise: S1 = 1 + 2 + 3 + 4 + 5 ... = - 1/12 S1 - S1 = 1 + 2 + 3 + 4 + 5 + 6 ... - 1 - 2 - 3 - 4 - 5 ... = 1 + 1 + 1 + 1 + 1 + 1 ... Since S1 - S1 = - 1/12 - (- 1/12) = 0 It follows that 1 + 1 + 1 + 1 + 1 .... = 0 Let's name this sequence S2: S2 = 1 + 1 + 1 + 1 + 1 ... = 0 Now let's subtract it from itself: S2 - S2 = 1 + 1 + 1 + 1 + 1 ... - 1 - 1 - 1 - 1 .... = 1 Given that S2 equals 0, we can also write this as: 0 - 0 = 1 Which implies that 1 = 0.

It seems like there's all kinds of tricks you can pull to get whatever result you want, once you throw rigor out the window. For example, he took the average of 1 + 1 - 1 + ... to get 1/2. You could also do this: 1 - 1 + 1 - 1 ... = (1 + 1 + 1 ...) + (-1 - 1 - 1 ...) = (1 + 1 + 1...) - (1 + 1 + 1...) = 0

Trolley Problem: A trolley is on a track headed towards one person, and after this one person is two people, and after that is 3 people, and so on. You can flip a lever to send the trolley onto an empty track. Do you flip the lever?

To quote a math teacher from my uni: "It's extremely unpleasant to approximate solutions that don't exist."

Watching this makes me think of the mathematician who, after watching two people go into a house and then later seeing three people come out, declares that if one more person goes into the house it will be empty.

The statement that the sum of all natural numbers equals -1/12 is correct within the specific mathematical context of zeta function regularization used in theoretical physics and certain areas of number theory. However, it's important to emphasize that this result should not be interpreted as the sum of natural numbers in the traditional sense, which is a divergent series. In everyday arithmetic, the sum of all natural numbers is not -1/12. This concept is a result of mathematical manipulation and regularization techniques used in specific mathematical and physics contexts.

Mathematician: **calculates something, result doesn't make any sense.** Mathematician: "I define this as correct."

But S1 and S2 are divergent series, they can't be assigned a value. This video just shows that if you try to assign a value to divergent series you can prove nonsense such as sum of all positive numbers equal -1/12

This video represents negative knowledge; if you watch it, you will know less about mathematics than when you started.

When you do 2S2, isn't it going against rules when you shift the bottom set of numbers by one place ?

I’ve watched this video almost -1/12 times, and it never gets old.

A reminder of the golden rules to be adhered to when dealing with divergent series: 1) Do not use brackets. 2) Do not remove any zero (unless you have proven that the divergent series is stable). 3) Do not shuffle around more than a finite number of terms. Not adhering to these rules yields incorrect sums.

“So now do you believe me?” Me: *No*

Combining two positive numbers will always be a positive number, no matter how far you go. It’s like a function thats trending towards infinity. It just cannot be negative

obvious error, the sum of +1-1+1-1+1-1 ..is NOT 1/2 .. this is a DISCONTINUED function that alternate between 1 and 0 ... YOU DO NOT AVERAGE IT !

After watching this I have some idea why string theory went off the rails.

I think this video perfectly illustrates Proof by Contradiction: Start with nonsense, end with nonsense.

Shame you did not mention the great Indian (largely self taught) mathematician Srinivasa Ramanujan who first postulated this idea back in the early 20th Century. He died in 1920 aged 32. Even today, the work he left behind is still proving both challenging and useful.

The sum of 1 to infinity is given by Indian Mathematician 'Srinivasa Ramanujan'

Sum of first series is NOT DEFINED

One of the angriest KZfaq comment sections since the incident with the forest in Japan

I don't understand how can you shift a set of numbers to the right just like that ??

I saw this for this first time today when my son drew my attention to it, and I immediately knew that S1 wasn’t convergent and that the entire argument fell apart from that point. It gave me a headache to sit through. I felt like the person who yells at the television telling the teenagers not to go into the barn alone and unarmed in the middle of the night while that masked serial killer is still on the loose. Except I don’t watch those kinds of shows. One huge benefit, though, was in being able to have a nice discussion with my son about math and physics, mathematicians, theorists, and experimental physicists. And yes, I’m a physicist, too.

The problem/flaw of all this begins at the assumption that the "average" of 1-1+1-1+1.... equals 1/2

This is an excellent proof of the fact that if you attempt to sum a divergent series, you get a garbage result.

Pretty sure shifting the second set of S2 over one space is why this shouldn't work. While each sum of numbers is infinite, by adding them all together in such a way you technically leave out the very last number of the second sum which was added to the first sum. So instead of 2(S2) = 1/2, I think 2(S2) = 1/2 + (last number added in sum), which thus would make everything else inaccurate due to the last number being undefined and now making the set undefined. Just my guess though

Mathematics when KZfaq removes the dislike button:

I prefer to see this as a demonstration that 1-1+1-1+1-1... does NOT equal 1/2

Yesterday I solved an equation and got 2 solutions: 0 and 1. However, I wanted to save time and only wrote that there was only one solution and that was the average 1/2 . Dunno why, I got a bad mark

The message is loud and clear -- DO NOT MESS WITH A NON-CONVERGENT SERIES, YOU WILL NOT GET ANYTHING MEANINGFUL. Let me just illustrate what I mean. Consider the infinite sum mentioned also in the video: S=1-2+3-4+5-6+... One could write this as: S=1+(-2+3)+(-4+5)+... which tends to +infinity or as: S=(1-2)+(3-4)+(5-6)+... which tends to -infinity And please, please, please do not say we should sum the two to get S=0.

Only few people here would know that this was actually theorem by Srinivasa Ramanujan the greatest Indian mathematician of all time.

Just an astounding leap of logic. How can you say that a sum is an average? Average is a sum divided by the size of the data pool. A sum is a sum. Your sum 1+1-1+1 ... is divergent and cannot be solved. The case is closed.

You made two blatant mathematical fallacies in your video. 1. The sum of the first series you showed is absolutely not 1/2. It will never be 1/2. This is a divergent and discrete, oscillating series. You calculated the arithmetic mean of the series for every finite truncation, which will never be equal to the sum, because the sum doesn't exist. 2. When you add two series, you can't simply shift all of the terms to the right or to the left for the convenience of whatever result you're trying to attain. I can easily disprove that. Consider two series. The first is (1+2+3+4+5+6+7+...). The second is (-1 -2 -3 -4 -5 -6 -7 -...). The second series is simply the negation of the first. Obviously their sum is the convergent constant series, (0+0+0+0+0+0+...). However, if we inexplicably decide to shift all of the negative number in the second series to the RIGHT as you did in the video, and then add the terms of the two series vertically, we'll now get the series (1+1+1+1+1+1+1+...) which is a divergent series, and not a remotely accurate result. If string theory is based on this illogic, then theoretical physicists should refine their arithmetic abilities.

Using the methods used in this video I was able to show that the summation from n=2 to inf of {S(Pn)*[Pn+G(Pn)]+n} = 1 where S(Pn) is the summation of all of the numbers with non trivial factors strictly greater than Pn. Pn is the nth prime if you include 1 as being prime P1. Thank you all for the inspiring video.

Just curious are there any videos explaining where this divergent series sum is being used in physics.

This whole "astounding" fact sums from the fact that people are mistaking Grandi's series for the ACTUAL sum. The sum of alternating ones is not a half, it SHOULD be a half. A half is an approximation, not the actual answer. The actual answer is that there is no defined sum. There's a big difference..

The US government is trying this with the national debt.

Wow! No visible dislikes, I'm sure the community really loved this one!

What pains me is in that textbook, they used Ramanujan's result. But did not mention or gave him any credit. He lived his whole life full of struggle and poverty, still contributed to the archives of knowledge. Give the man the most deserved respect, even thou he left us.

This is not an astounding result, it is simply a false one. Whatever result this leads to cannot be the sum in the sense of the result you get when you add all the positive integers together. if you start with 1, the sum of all the integers must be greater than 1 because there are other integers to add. If you take the first 2 integers, the sum of all of them must be greater than three, because there are other integers left to add. And so on, potentially forever.. Since I'm not a mathematician, I can't deny that there may be a relation between the positive integer set and -1/12. You could call it the Riemann zeta function sum or the Rajamujan sum or some such (pun intended), but it clearly cannot be the sum in the first-grade sense of that term. To claim it is only tends undermine the integrity of mathematics. The men's demonstration is not at all convincing. They simply changed the subject, and never even addressed how these other series have anything to do with the original problem.

Dear God, I'd like to file a bug report (see attached video) Amen.

The trick is S2 is divergent so you cannot just shift it along to evaluate a rational number out of it.

I just have a simple question- to deduce 2S2, why did he move S2 along by a digit when he was adding ? Is there any significance of that ?

I'm no professional mathematician but I figured out why it is wrong. At least the method used here. You cannot do a shift addition or subtraction with a divergent infinite series. Remember how they get Grandi's Series to be 1/2? They manipulated the second row so it's shifted by one place, and assume the second row to be the same as the first row. In fact it's not: Grandi's series: 1-1+1-1+1-1+1...... "Shifted" Grandi's series: 0+1-1+1-1+1-1...... But you will say "well anything plus zero is itself isn't it?". No, it's not in this case. The Grandi's series follows the pattern 1,0,1,0,1,0; the "Shifted" series is 0,1,0,1,0,1, now every term in the "Shifted" series is different from the original. Therefore by adding a zero to the beginning you get a different series. So now you cannot use 2S1=1. Try this at home: dilute the Grandi's series with 0 after each negative one, and do shift addition with three rows, you will get a number. Then dilute it by placing the zero after each positive one and do shift addition again. Compare the results (I will not spoil your fun of doing this). This happened at the end of the proof where he assumed 4+8+12+16......=4*(1+2+3+4......). Well it's right, but this is not the series appeared here. It's in fact 0+4+0+8+0+12+......You will figure out why they are different if you do the dilute Grandi's series experiment. In conclusion the method of shift addition to sum a divergent infinite series is inherently flawed. I cannot comment on the average partial sum method though. But I guess partial sum method is not enough to prove 1+2+3+4+......=-1/12.

If A=B, then AB=B^2 AB-A^2=B^2-A^2 A(B-A)=(B-A)(B+A) A=B+A A=2A 1=2 Math makes so much sense now!

S1 goes 1, 0, 1, 0, 1, 0 and the average is 0,5. Therefore S2 should be 0 as it goes 1, -1, 1, -1, 1, -1 where the average is zero. The final equation should state S - 0 = 4S => S = 4S => 3S = 0 => S = 0

i have a question regarding the addition of S2 to itself @ ~ 3:12 . by shifting the value one number to right, even though both are infinity, aren't you adding two "different values" of infinity ? and not strictly 2*S2

The issue started when you assumed S1 = 1/2 when you divided (1+0)/2. All the points after that make sense but they are built on a questionable foundation. S1 does not end, simple as that.

This is bollox. For S1, you can't just stop on odd or even. Infinity is infinity. It is a concept. It isn't a number. You have to keep going.

Consider that sequence as the partial sum n(n+1)/2 and plotting that you'll see that -1/12 is the integral between [-1, 0] - this is a hint that this result is not a sum but something that makes sense in another ruleset. So this result needs to be considered _contextually_ in the sense that it's not measuring a sum, but it's the result of the Riemann Zeta function on domain points that only make sense by analytic continuation. The same way that sqrt(-1) is nonsense in |R but makes sense and gives valid results when we change the context to the |C complex domain.

Why are these guys so poorly funded that they’re writing their maths on old envelopes instead of fresh paper?

The key thing to note is that one should never add or minus with infinite on each side of equation. For example, 5 + oo = oo and 10 + oo = oo. Therefore, 5 = 10. That is how the mathmaticians trick our ordinary folks.

I am amazed at the number of people who claim they know 100% that this is rubbish. The nice thing about some parts of physics is that you can think about it without having learnt the maths behind it, but this is not one of those things. I have doubts and issues with what was shown here but I not so stupid to claim that huge branches of physics are just completely wrong, I just don't know enough about this...

sloppy use of the = sign The String Theory book used -> for a reason.

I just cannot comprehend how adding positive numbers can ever lead to a result that's smaller than you started with. How does it ever go down?

Damn, i was hoping the answer would be 42.

You can reject the claim that 1-1+1-1... = 0.5 and instead say that it has no solution, or an indeterminate value. If you do that, the entire system falls apart. The thing is, this uses a different summation method than what most people are used to.

The great Indian mathematician gave this sum called THE RAMANUJAM PARADOX.

Nonsense is still nonsense, no matter how you explain it.

"The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes …" - Niels Henrik Abel 1820

chatgpt: the series of natural numbers diverges, and we represent this divergence by saying the sum is infinite (∞). The idea of the sum being -1/12, as mentioned in certain contexts such as in complex analysis or theoretical physics, is a regularization technique and should not be interpreted as the actual sum in the traditional sense of arithmetic summation. In standard mathematical conventions, the sum of natural numbers is not a finite value.

I looked into this a bit, and I believe I've found the flaw (but when I say "flaw" I'm merely attempting to explain why the intuitive result differs from the mathematical proof shown here) Also since I am unable to use the fancy formula symbols, I'm just going to paste the raw data that will format itself in desmos . The important thing to note here is that you are choosing an arbitrary stopping point, so I've rewritten the formulas to account for this point that may or may not be infinity. For example, s1(3) = 1-1+1, and s1(6) = 1-1+1-1+1-1 The formula for s1 would be s_{1}\left(x ight)=\sum_{n=1}^{x}\left(-1 ight)^{n-1} The formula for s2 would be s_{2}\left(x ight)=\sum_{n=1}^{x}n\left(-1 ight)^{n-1} . Now, you say that s2+s2=s1, but that's not entirely true. You're offsetting the stopping point by one, so the true formula must reflect this: s_{1}\left(x ight)=s_{2}\left(x ight)+s_{2}\left(x-1 ight) (this one doesn't have a summation so I could write it normally, but I'm just being consistent for the sake of copy/pasting into desmos) . This altered formula does still hold true, but I feel like it breaks the reality of adding the "same" summation to itself due to the differing stopping point. There will always be that one missing term from the shortened version, and that singular term will become infinitely large as the stopping point extends toward infinity. To account for this last term, I've added it onto the formula as shown here: s_{2}\left(x ight)+s_{2}\left(x-1 ight)+x\left(-1 ight)^{x-1} or alternatively s_{1}\left(x ight)+x\left(-1 ight)^{x-1} This extra term should be considered a part of s2+s2 now that they have the same stopping point, and this means that the sum is not equal to s1 anymore. Therefore any of the following conclusions drawn from this claim can no longer be considered valid either because 2 * s2 ≠ s1

How can you just average the stopping point and call that a solution? You should call that an average?

If 1+2+3+4+5...=-1/12, does that mean that 1>1+2+3...?

I've come up with a counterproof: For 1+2+3+4+...+n If 1+2=3, then 1+2+3+4+...+n>3 (where n>n-1) -1/12 is not greater than 3 Therefore 1+2+3+4+...+n=-1/12 is disproven.

What bothers me about this is that to get 1/2 and to get 1/4, you have to end the infinite sum, yet you can’t end the infinite sum of 1+2+3+4+…. Sounds like they’re using different rules for the different infinite sum…

I love how he says it's not a bunch of mathematical hocus pocus one second then says you have to do the mathematical hocus pocus in order to reach such a result.

According to riemann's rearrangement theorem; Infinity - infinity = (*any number*) Its just the way you rearrange the series...

This is what happens when you let physicists try to do maths

“This is not mathematical hocus pocus” “You have to do the mathematical hocus pocus to believe it”

There is no logical way you can get a NEGATIVE fraction from only adding positive numbers.

I have now had multiple friends ask me to explain to them why this video is wrong. I don't care much that you want to keep things informal and allow for casual fun maths. What bothers me about the video is that you're claiming this is unconditionally true (by the fallacy of authority), and that there's nothing deeper going on for people to read about when in fact there is and the particular proof given in this video is flat out wrong. It doesn't matter that the "result" is used in physics (physicists are well known to abuse mathematics because the "results" are interesting), or that there is a second video explaining things in more detail (though I don't think it goes far enough to make it clear where the line between truth and falsity was blurred in this video). What matters is that this video, standing by itself, is spreading massive amounts of misinformation. This is numberphile's blessing and its curse: it's so popular now, and has gained so much influence, that the majority of ignorant viewers (which is the vast majority of all viewers) take what is presented as gospel. You might say that's their problem for being ignorant and not questioning things, but I think it's also seriously dishonest to knowingly do such a thing. To think that mathematicians, who so rarely get as wide an audience as numberphile has, would knowingly lie about mathematics! I can hope it was more of a misunderstanding on the editor's part, but until I see evidence of that, this video has made me lose a lot of respect for numberphile.

A while back I learned what p-adic numbers are, and I’m starting to think this absolutely makes sense in a mathematical context.

can't believe 4 years has gone by and this is still ridiculous.

Is this how the financial crisis happened? Add together ever-stacking credit risk to get no credit risk? Note to investment bank CEOs, do not hire physicists.

The golden rules to be adhered to when dealing with divergent series are: 1) Do not use brackets 2) Do not remove any zero 3) Do not shuffle around more than a finite number of terms

S2=1/4 can be done directly as a 4 term partial sum average on even term sums and odd term sums giving 1/4[ ( n-n)+(-m+m+1) ] =1/4 as a variety average of types.

The law of whole numbers says that whole numbers are closed for addition and multiplication. Which means that the sum or product of 2 or more whole numbers is always a whole number. There is also a rule that states that the sum of 2 or more positive integers is always a positive integer.

i'm just astonished how a infinite sum of positive numbers is a negative number, and people act like it is physicaly possible... i bet it happens in "theoretical physics", that is still unproven and higly theorectical, it might explain everything, or it might just be wrong as everything... i have a really low education in maths, but an infinite sum of a alternating series, doesnt converge to a number, is diverging, and certainly not 1/2

I don't think everyone in these comments realizes that they're using averages, since those sequences don't have an actual sum.

I wouldn't think that the first sum equals 1/2. I would just think it was undefined.

If i understood this correctly, 3 Blue 1 brown provided a better explanation of this. The sum is not -1/12 except when you modify it to fit the context of providing symmetry for the Riemann hypothesis; it is a modification.

This video is like that episode of fairly odd parents where timmy got a mathematician to prove to his teacher that 2+2=5

So by only adding positive numbers you get a negative number?

first mistake he made here was assuming any arrangement of natural numbers summed up gives you unnatural numbers

problem 1: there was no test for convergence. one of the first things taught when you learn to evaluate infinite sums is that you have to test for convergence. this sum does not converge, it just goes to infinity. so, even if you find tricks like this to solve it, it will not converge 1.1: even if this sum converged, this all hinges on the fact that 1+1-1… converges. which it doesn’t. it oscillates, which means it does not converge 2. you can’t displace an infinite amount of terms when you add together 2 series like this. when you “stretch out” one series and add it to the other, it will not have the same value as before

2,818 people think they're smarter than the majority of mathematicians and scientists.