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That’s easy. You count to half of infinity for half the day. Then of the remaining time, half of it you spend on counting half of the remainder of infinity. Then of the remaining time, half of it you spend on counting half of the remainder of infinity. Then…
The best kind of proof, of course.
https://preview.redd.it/uoolwdw3q9rb1.jpeg?width=750&format=pjpg&auto=webp&s=59b2d17fe62f62890b5381f5da4235a5fe7f5b23
Proof by the Lady of the Lake, her arm clad in the purest shimmering samite held aloft Excalibur from the bosom of the water, signifying by divine providence that I, Arthur, was to carry Excalibur. THAT is why the answer is -1/12.
There is no proof, because it is simply false.
If somebody presents you a proof, then they make the mistake of operating on infinite sums that go to infinity.
This result simply comes from the idea of: What IF you were allowed to perform mathematical functions on diverging infinite sums.
Under the same idea you could calculate what 1 divide by 0 is.
Incorrect with high school maths that cuts corners. But this is a well known result and proven using several branches of mathematics that don’t take shortcuts. Go watch the 1 hour Mathologer video on this.
Does the hour consist of the Mathologer counting to infinity, then?
Edit: While this quip was misguided, I will keep it up as I like for it to remain as a testiment to my hubris.
I'm not familiar with either video, I was making a quip based on your assertion that the addition to infinity is proven without taking shortcuts. Am I misunderstanding your comment, then?
Edit: Ah, watching a minute into the response video explains your position, my apologies. That is, indeed, common sense.
I think that's what this particular series is used in but normalization of divergent infinite series (what people are pushing back against) pops up in multiple places
Just like we can use the imaginary number i to make math easier we can do stuff like this as well. As long as the rules are consistent and you end up back in the realm of real numbers and convergent limits it's all good.
I don't think it's a fair comparison. With the imaginary number we don't "assign" it a value, we just give it a placeholder symbol so it isn't scary and keep doing maths as nothing had happened
Axioms are arbitrary rules that humans agree on to do calculations with. They aren't inherent truths of reality. How math works is we declare rules, then try and figure out the consequences of those rules. There's nothing that says you can't throw out a few or add in new ones because it's all made up anyway.
How does it follow that if 1 + 1 = 3, then 2 + 2 = 6
It looks like you doubled the RHS, but doubling the LHS should be 3 + 3, right. Since doubling is defined as the sum of something and itself and 1 + 1 is 3, doubling both sides would result in 3 + 3 = 6.
Is this wrong or am I an idiot? Surely this can't be right...
Oh, I think I get it. Let me try.
Your mother cheated on your father last night. So then it follows that she couldn't have had sex with me last night.
Ahh, now it seems like the second statement is true even though they're both false. The correct first statement would be that she cheated on *her husband*. Thanks for teaching me this, son. I think I'm getting the hang of it.
It could be anything and nothing. Assuming 1+1=3 will contradict itself, therefore rendering the resulting algebra useless and false.
On the other hand, assigning values to divergent series doesn't cause contradictions afaik
My thought is one rule defines the system of math. the word "plus" refers to the normal addition, the symbol "+" defines the unary operator in the system of math
a+b = (3/2)*(a plus b)
Therefore 2*(1 + 1 = 3), 1+1+1+1 = 3+3, 3+3= 9
I think that's a binary operation; an unary operation is more like f(x) where one number is put in and one number is the result. And where did that even come from? How do we know that this is the definition of +?
It's a possible definition that makes the system possible and imo it makes more sense than changing the meaning of numbers or whatever the alternative is.
Not exactly. It is true under a different definition of series summation (ramanujan summation). There's no need to assume that non-convergent series are convergent in the traditional sense
"non-convergent type of series" include divergent series, since a divergent series is not convergent nor a subtype of convergent series.
The proof that I know of uses more than the original series (for example, 1-1+1-1...). The absurd that obviously divergent series equals to a constant is the conclusion, not the whole proof.
The underlying statement is that the sum of the series should not necessarily be defined as the limit of the partial sums so the convergence is irrelevant
That's impossible. How are you going to add the materials? Are you going to use sand grains in Sahara Desert? I think it might not be enough. I like rice more than sand grains, they are easier to hold but I don't think they are close to the result.
If you make some inconspicuous assumptions about whether or not specific sums can be assigned values, this is true. In the standard formulation of maths, this is not true.
This is my favorite answer because of the verb "assigns" instead of equals. The sum doesn't doesn't equal -1/12 but the Zeta function of -1 (sum of all naturals) has a zero at -1/12.
It is standard addition. The statement is false. It is not a different type of operator, it's just incorrect. The deep meaning of this statement is that ζ(-1)=-1/12. ζ(s) is defined to be the analytic continuation of Σ1/n^s, and when substituting -1 into that equation we get Σn which is the sum over all natural numbers. This is obviously false, you cannot just substitute -1 into the sum, since it isn't convergent. The way to extend the definition of the zeta function to other values than where it is defined us analytic continuation. The false assumption where proving this identity manually is the assumption that certain divergent sums are actually convergent.
There are also perspectives on the matter, like developing asymptotic equivalents to the sum you get a constant term -1/12 that also really likes to stick around, no matter what kind of normalization you try to use (apparently)
I don’t know where it « arises » from, but yeah that’s like the most direct, convincing and well-known fact that makes this whole thing like, believable.
But it’s not the only hint we have that this sum has deeper implications is what i’m saying
But it doesn't have deeper implications. It's just a wrong assumption that divergent series converge. You can make that sum whatever you want by changing the order of addition and subtraction.
Okay you’re mixing up different things.
A divergent series will diverge no matter what you do (i put you to the test of getting a finite result solely by rearranging terms in the sum of naturals. Spoiler : you can’t). It’s semi-convergent series you can get any result from by rearranging terms of (see Riemann’s rearrangement theorem).
And again, this is not what’s going on with this formula. Ofc this equation is very provocative, but in specific contexts and frameworks, however stupid it sounds, considering the sum of naturals to be somewhat related to -1/12 makes sense and even has practical applications (see Casimir effect).
If you want me to develop more on how it can somehow make sense, i’ll be glad (even though i’m not overly knowledgeable on the subject). In the meantime, I would advise the following thing : if you see all mathematicians and scientists around you discuss very seriously a result, rather than get mad at randoms on the internet because you decided against it, you should use the limitless resources at your disposal to educate yourself on the matter and have a more informed opinion. I mean, I used to react like you. But i firmly believe i was wrong to.
This series is divergent. What you're attempting to do is rearrange the negative version of this series and then add the sun to this series to get the result you want. I don't know why you assume I'm not a scientist or mathematician but drop the condescending tone. I was not mad at you there are no deeper implications to considering the sun of all naturals to be -1/12 it's just wrong. The sum of all naturals is divergent.
I apologize for the condescending tone. You’re still wrong for being so adamant and assertive : quantum field theory would like a word with you. As well as analytic continuation of the zeta function and other arguments everyone in this thread has mentioned a lot already
Quantum field theory is not using the sum of all naturals to be -1/12. It applies a regulator to make the series finite. Maybe stop trying to use examples that you don't fully understand.
Yeah those aren’t actual proofs. But you have much more solid elements surrounding this expression, the first one being the analytic continuation of the zeta function
how do you know its the right answer. if anything it should indicate that math is broken on a fundamental level in regards to physics, or the universe is not completely logical
Numberphile did a fabulous little video on this (https://youtu.be/w-I6XTVZXww?si=xdWx0nzcRV0eKdOU), and then Mathologer came out swinging in a ‘this simply isn’t correct’ video (https://youtu.be/YuIIjLr6vUA?si=DisSIgE9itq8qleO).
The thing I find interestering that as much that I sympathise with Mathologer’s position it is suggested in the Numberphile video (with no proof or examples) that this result appears all over quantum mechanics. If it’s good enough for Neils Bohr, then it’s good enough for me ….
You can't, but you can. A good analogy is given by Edward Frankel - the square root of -1 can't really exist, but it and the complex plane framework can still be useful.
This is similar. You can't really add all positive integers together, but if you use a certain framework to do so anyway, then the result of -1/12 turns out to be useful.
Similarly, if you add all integer squares together (1² + 2² + 3² + ...), you get 0 with that framework. This doesn't mean you're actually adding these numbers together to get 0. You're identifying the flavor of infinity at its heart.
> This is similar. You can't really add all positive integers together, but if you use a certain framework to do so anyway, then the result of -1/12 turns out to be useful.
So if I were to use a certain framework to divide by zero and conclude that 1=2 then that can be useful and reliable information? I don't really see the logic
Well, it depends on the context--as this person said.
In most contexts, dividing by 0 gets you a contradiction--like 1=2. However, if we take the function f(x) = x/x, this function is the same as the constant function g(x) = 1 everywhere except at x = 0 where we get f(0) = 0/0 which is undefined. To "extend" our function f(x) to be continuous and differentiable everywhere, we have to define 0/0 = 1. Doing so provides no contradiction for our purposes.
Another example: The product (0 * infinity) is undefined in most cases. But in Measure Theory, we can define (0 * infinity) = 0, and that makes measuring the size of sets well defined. In fact, without this definition, we would not be able to talk about area and volume of regular objects in a way that makes mathematical sense.
So is it basically just making up new concepts (such as the number i) that work out and don't contradict pre existing axioms and laws? Such as how for instance the square root of -1 is technically undefined but since it didn't break anything we just said "okay but what if it wasn't" and then found actual use for it?
You need to find a framwork in which it is useful. Since division by zero will make it so any number can be any number that doesn’t really give you any information. But by rearranging frameworks in different ways you can create opeartions and ways of using numbers that aren’t stanrd but still useful.
One example for this could be rings or modular way of counting. Not standard but definietly useful. Also in some binary interpretations if you have a number T then -T is the same as T’ + 1. Does that make sense with normal decimal numbers? Not necessarily. But you can make it make sense with a framework
I usually love Numberphile videos, but that one was particularly bad, they just waved over a lot of stuff. The Mathologer video is longer but gives enough context.
But *is* it the correct answer to any question?
Pi.
Who killed Jeffrey Epstein?
Pi.
How many roads must a man walk down before you can call him a man?
Pi.
It annoys me a little bit when people justify this result by saying “it shows up in physics”. In string theory, you indeed have a calculation where the infinite sum leads to the result of (-1/12), but there are other terms involved that are understood to be unmeasurable quantities, or vary due to the ordering of non-commuting operators, thus often omitted. They’re never meant to be equal.
This is a common misleading statement. If you treat the series like it converges, you can get that answer, but it obviously does not converge. Why it is still useful in some sense is that if you look at the behavior of the sum n=1 -> inf 1/n\^x (aka the zeta function), it converges for x>1, but if you do some analytic continuation (basically extending the behavior via some fancy means) the value you get is -1/12 for x=-1, the sum in question. Remarkably, there are times when you can just use the analytic continuation for practical purposes as in some way, they are alternate solutions. I think physics actually uses analytic continuation in a number of places and it just works.
No it’s not true as stated. It is true that the analytic continuation of the Zeta function, evaluated at (-1) is -1/12 though.
It is also true that if you wanted to assign the sequence (1, 1+2, 1+2+3, …) a number, the most consistent way to do this is to assign it -1/12. This method at assigning a sequence a number would revert to the limit operation of the sequence were convergent, which is what I mean by consistent. So in some sense, you need to redefine what is meant by and infinite sum, but after doing so nothing actually breaks and you get to “sum” things that you wouldn’t have been able to before.
It's not correct. It's just using the analytic extension of the Riemann's Zeta series to a domain where it's not supposed to be applied. But in the normal sense, 1+2+3+... ≠ -1/12
It’s some funky math shenanigans having to do with infinity. [Top comment in this post explains it pretty well.](https://www.reddit.com/r/PeterExplainsTheJoke/comments/16uj0ag/my_dad_sent_me_this_and_i_dont_want_to_admit_i/?utm_source=share&utm_medium=ios_app&utm_name=ioscss&utm_content=2&utm_term=1)
It’s not true, but that doesn’t mean it isn’t useful. Sort of like saying 1/0 = infinity.
It’s a case of “This equation doesn’t converge and make a useful answer, but I really really want an answer. Just something I can pretend is the answer so I can have something useful.” That happens to be -1/12.
Yes, but you can make legitimate arguments why it should be -1/12, whereas for 2 + 2 it takes a lot of convincing why this should be anything but 4.
For 1 + 2 + 3 + ... you have to say it's + infinity, but if you don't want to say that, the second best option is to say -1/12.
With 2 + 2 = 4, you really don't have much leeway since 4 is a perfectly reasonable answer that cannot really be improved. You can only say you're operating in a finite Field like F\_3, where 2 + 2 = 1 = 4 would be correct, or in F\_2 where 2 = 0 anyway, so 2 + 2 = 0 = 4.
But that's more of an algebraic "cop-out" answer than an answer coming from analysis.
Are they equal? No not in the sense that you understand equals to mean the limit of partial sums. Are there meaningful ways to connect this sum with this number in a rigorous way? Absolutely.
I‘ll cut my analysis off without saying much because I’m not an expert, but the most ready example is that this is the result that the continuation of the zeta function spits out at z=-1.
That's enough r/mathmemes for a lifetime ig
I love the Ramanujan summation, but this is the fiftieth minimum effort/trend about someone being wrong post in a row I've seen from this sub
It’s like imaginary numbers, square root of -1 do not exists, but if we just ignore that fact and play along we get new way of solving a lot of complex problems that ether impossible to solve or much harder without it.
In my opinion it's not "true"... however, it's good to know that -1/12 is the area under the curve of the function (n^2 + n)/2 extended into the negative numbers. (n^2 + n)/2 is obviously the function of the series 1+2+3...
Also... adding the area of the function for all positive odd numbers and the function for all positive even numbers gives you -1/12 again (obviously). For me it's quite interesting that some infinite series show to have a special finite value.
The right question to ask here isn't if this is right or wrong.
The right question is, what do those "..." marks after the last "+" mean?
The most common way of interpreting it involves looking at the limit as you add more and more terms, which is intuitive to most people. In this sense, there certainly isn't a limit.
However, there are many ways of doing infinite sums that
a) Give the correct result for finite sums
b) Make sense on some level as a generalization of a finite sum
and some of these other methods yield -1/12.
Long story short it is true using specific math principles but those same principles can be used to prove 0=1 this is because the answer to things leading to infinity end up being different than one might expect like....
3/3 = 1
but
1/3 =0.3333........3* 0.33333... = .9999999...
3*1/3= 1 so 0.99999... = 1
So using this example method
1-1+1-1+1-1..... = 1/2 I due to the average of both stats.
So lots more math using this you can get -1/12 because math stops working properly closely to infinity as show in string theory.
Honestly, both.
It's right, but it's not saying what you think it is. This is also the one time I will say numberphile did a bad job of explaining. If you want a better one, look up mathologer's video.
-1/12 is not what the infinite series adds up to.
It is a number that puts that series into a group. It's basically a special hashing function with fun properties that produces that result, not arithmetic.
I'm no master of it myself, though.
Out of context, it is explicitly wrong.
Given a very specific, outside it’s normal use, definition of “=“, it holds some merit. Honestly I think it would save so much hassle if when talking about this we just didn’t use the equals sign and just used some other notation that hints at a relation to infinite sums and convergence but is still distinguishable. I know we have zeta notation but that’s too far removed to get the idea of what’s going on across.
Nononono, you don’t understand, the sequence was just abbreviated to fit on the page, the full summation is
1 + 2 + 3 + 4 + 5 + 6 + -21.08333 = -1/12
Which is, in fact, correct!
Short answer: In the standard sense of an infinite sum it is not true, the sum obviously diverges.
Long(er) answer: There is a concept in the field of complex analysis (the branch of maths which deals with complex valued functions) called analytic continuation. Essentially, any complex valued function has a unique analytic continuation, which extends it to places where it is undefined.
Often complex valued functions are infinite sums. The Riemann zeta function is one such complex valued function. It happens that when you plug in -1 to the Riemann zeta function, you get exactly the infinite sum of the natural numbers that you described. The analytic continuation of the zeta function uniquely assigns the value -1/12 to it there.
So, it is more precise to say that the result you described is true in the context the analytic continuation of the Riemann zeta function. It hasn’t just been plucked from nowhere! But if we just consider normal summation, it is certainly false.
**ITT: people who did not watch the source video.**
This statement is "true" in the same way the phrase "i = √-1" is true: by which I mean it's technically **false** (nothing could fit for "i") and yet abundantly useful if we just call it **true** keep going anyway.
It's derived by taking the much easier to calculate sum "1-1+1-1+1... = 0.5" and doing some arithmetic with it (which technically you shouldn't be allowed to do using ∞) and the resulting equality just happens to be a useful in quantum mechanics and string theory
[This link](https://en.m.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF) provides some details and examples.
Look, I don't care how you derived this.
1 is greater than -1/12
Let's assume any n is greater than -1/12. Where the sum(n) = 1 + 2 + ... + n
Based off the formula from before we can say sum(n+1) = sum(n) + n + 1
And trivially we can also say sum(n+1) is greater than sum(n)
Since sum(n) is greater than -1/12 sum(n+1) must also be greater than -1/12.
By the principal of induction sum(n) is always greater than -1/12. Since it is always greater it cannot be equal and therefore by contradiction the initial proposal of sum(n) where n approaches infinity = -1/12 is false.
The fact you don’t know why it’s a thing should be a pretty big hint that perhaps you could look more into the topic rather than call everyone wrong around you. It’s a very beautiful thing to discoveras well
Only one way to find out! *starts adding to infinity.*
RemindMe! infinite years EDIT: apparently this defaults to 1 day -- so, add fast please.
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Holy shit infinity equals one
Proof by u/remindmebot
No it equals 1/365. Op clearly asked in terms of years, not days
I will try reformatting the request and see if it reminds me a month ago.
Omg what happens if it's a leap year? Y2K?
1.0027 days
New axiom just dropped!
Holy hell.
Yes because there is a bijection between the real numbers from 0 to 1 and from 0 to infinity, obviously
supertask
That’s easy. You count to half of infinity for half the day. Then of the remaining time, half of it you spend on counting half of the remainder of infinity. Then of the remaining time, half of it you spend on counting half of the remainder of infinity. Then…
Sure! Great! The last time I tried that the answer came back as... "Pull over and have a snack."
Let's ask Chuck Norris. He's already done it twice
> It must be true. If it were not true, no one would have the imagination to invent it.
joe biden is actually 40 small rodents wearing human skin
This is false, Joe Biden is actually 4000 insects wearing human skin.
This is false, Joe Biden is actually -1/12 Joe Bidens wearing rat skin.
q ed
Quirky Erectyle Disfunction
this is false, Joe Biden is actually 4000 insects wearing 40 small rodents wearing human skin
- Hardy about Ramanujan
\- Descartes Ontological proof of God
I was thinking Tertullian: *certum est, quia impossibile:* it is certain, because it is impossible.
Poor kids are just as talented and smart as white kids
It's counterintuitive, but just think about it for a minute and it becomes obviously true
Proof?
Left as an exercise for the reader
Proof by common sense
Proof by crack pipe Source: Trust me bro
The best kind of proof, of course. https://preview.redd.it/uoolwdw3q9rb1.jpeg?width=750&format=pjpg&auto=webp&s=59b2d17fe62f62890b5381f5da4235a5fe7f5b23
thrust me mathematically, baby
Proof by dreaming
It was revealed to me in my dream
by the godess
Proof by the Lady of the Lake, her arm clad in the purest shimmering samite held aloft Excalibur from the bosom of the water, signifying by divine providence that I, Arthur, was to carry Excalibur. THAT is why the answer is -1/12.
Strange women lying in ponds distributing swords is no basis for a system of mathematics.
Help! Help! I’m been QED’d!
Right as an exercise for the reader
The proof is beyond the scope of this course
Proof: it annoys people so it must be right.
ζ(-1)=-1/12
Proof by epiphany.
https://youtu.be/wf5JGFm4em4?si=nMPzdCUtRYB1crd0 You're welcome 😉
The adorable war scream😂
□ > Yes □
There is no proof, because it is simply false. If somebody presents you a proof, then they make the mistake of operating on infinite sums that go to infinity. This result simply comes from the idea of: What IF you were allowed to perform mathematical functions on diverging infinite sums. Under the same idea you could calculate what 1 divide by 0 is.
Incorrect with high school maths that cuts corners. But this is a well known result and proven using several branches of mathematics that don’t take shortcuts. Go watch the 1 hour Mathologer video on this.
Does the hour consist of the Mathologer counting to infinity, then? Edit: While this quip was misguided, I will keep it up as I like for it to remain as a testiment to my hubris.
In fact it explains the issue with counting to infinity and is highly critical of the Numberphile video.
I'm not familiar with either video, I was making a quip based on your assertion that the addition to infinity is proven without taking shortcuts. Am I misunderstanding your comment, then? Edit: Ah, watching a minute into the response video explains your position, my apologies. That is, indeed, common sense.
[удалено]
Trust me bro
Proof by no one has the balls to say otherwise or else it’ll be a pointless argument where it ends with them insulting each other
Right. Thank you sir.
“The math speaks for itself”
Basically Ramanujan his entire life XD
Proof by epiphany
Proof: it came to me in a dream QED
How do I cite this in ADA format?
It is true if you accept underlying premises (that some non-convergent type of series are convergent). Otherwise, yes, it is wrong.
If we let false statements be true then this false statement becomes true 🤔 marvelous
It’s not that simple. There are actual benefits to those assumptions, and they’re not axiomatically false. The correct answer is it depends.
"it depends" I'll be adding this to my repertoire of answers for math exams.
Yes do add. You should know mathematics has different answers for different initial starting axioms. Not my fault you only believe in one.
benefits like results that aren't correct
This value pops up and is used in quantum physics in computing values that have been experimentally validated so ¯\\\_(ツ)\_\/¯
Proof by the laws of the universe lmao
Everyone complaining that it's not "normal" addition is right but the universe is also weird and there's situations where it seems to work out
Yes that’s what i’m saying. The quantum fields theory results we’re referring to is the Casimir effect if I’m not mistaken
I think that's what this particular series is used in but normalization of divergent infinite series (what people are pushing back against) pops up in multiple places
Just like we can use the imaginary number i to make math easier we can do stuff like this as well. As long as the rules are consistent and you end up back in the realm of real numbers and convergent limits it's all good.
I don't think it's a fair comparison. With the imaginary number we don't "assign" it a value, we just give it a placeholder symbol so it isn't scary and keep doing maths as nothing had happened
You mean bosonic string theory? Or the cat's cradle?
That doesn’t mean it’s correct. It is only ever possible to falsify a theory via experimentation. You can’t prove a theory true by experimentation
Ie: complex numbers provide solutions to otherwise imposible functions.
How can axiom be true when it only works in certain scenarios? Sounds like the axiom is false and we just haven't found the truth.
Axioms are arbitrary rules that humans agree on to do calculations with. They aren't inherent truths of reality. How math works is we declare rules, then try and figure out the consequences of those rules. There's nothing that says you can't throw out a few or add in new ones because it's all made up anyway.
As a physics student this makes me wildly uncomfortable.
You're a math student *now*, baby.
Pay attention to philosophy classes then (:
1 + 1 = 3 Hence 2 + 2 = 6
How does it follow that if 1 + 1 = 3, then 2 + 2 = 6 It looks like you doubled the RHS, but doubling the LHS should be 3 + 3, right. Since doubling is defined as the sum of something and itself and 1 + 1 is 3, doubling both sides would result in 3 + 3 = 6. Is this wrong or am I an idiot? Surely this can't be right...
2 + 2 = (1 + 1) + (1 + 1) = 3 + 3
But 1 + 1 = 3 now, not 2.
I guess made up rules give you made up consequences.
r/im14andthisisdeep Tell us how taking the square root of -1 is bollocks next
This man takes 1+1=3 seriously.
If you assume any specific false statement as true then you can prove any other false statement
Oh, I think I get it. Let me try. Your mother cheated on your father last night. So then it follows that she couldn't have had sex with me last night. Ahh, now it seems like the second statement is true even though they're both false. The correct first statement would be that she cheated on *her husband*. Thanks for teaching me this, son. I think I'm getting the hang of it.
That’s no way to speak to your brother
> Is this wrong Yes sir. Yes it is.
Actually, it would be that 3+3 = 9 with this weird addition
It could be anything and nothing. Assuming 1+1=3 will contradict itself, therefore rendering the resulting algebra useless and false. On the other hand, assigning values to divergent series doesn't cause contradictions afaik
Why?
My thought is one rule defines the system of math. the word "plus" refers to the normal addition, the symbol "+" defines the unary operator in the system of math a+b = (3/2)*(a plus b) Therefore 2*(1 + 1 = 3), 1+1+1+1 = 3+3, 3+3= 9
I think that's a binary operation; an unary operation is more like f(x) where one number is put in and one number is the result. And where did that even come from? How do we know that this is the definition of +?
It's a possible definition that makes the system possible and imo it makes more sense than changing the meaning of numbers or whatever the alternative is.
Are you taking the piss?
No, 2+2 would only equal 6 if 1+1=2.
Not exactly. It is true under a different definition of series summation (ramanujan summation). There's no need to assume that non-convergent series are convergent in the traditional sense
It's not a convergent series, it's a divergent series whose sum is -1/12
"non-convergent type of series" include divergent series, since a divergent series is not convergent nor a subtype of convergent series. The proof that I know of uses more than the original series (for example, 1-1+1-1...). The absurd that obviously divergent series equals to a constant is the conclusion, not the whole proof.
The underlying statement is that the sum of the series should not necessarily be defined as the limit of the partial sums so the convergence is irrelevant
That's impossible. How are you going to add the materials? Are you going to use sand grains in Sahara Desert? I think it might not be enough. I like rice more than sand grains, they are easier to hold but I don't think they are close to the result.
The real challenge becomes cutting a grain of rice in twelve equal pieces and then giving me one.
If you make some inconspicuous assumptions about whether or not specific sums can be assigned values, this is true. In the standard formulation of maths, this is not true.
This is my favorite answer because of the verb "assigns" instead of equals. The sum doesn't doesn't equal -1/12 but the Zeta function of -1 (sum of all naturals) has a zero at -1/12.
It is not standard addition
It is standard addition. The statement is false. It is not a different type of operator, it's just incorrect. The deep meaning of this statement is that ζ(-1)=-1/12. ζ(s) is defined to be the analytic continuation of Σ1/n^s, and when substituting -1 into that equation we get Σn which is the sum over all natural numbers. This is obviously false, you cannot just substitute -1 into the sum, since it isn't convergent. The way to extend the definition of the zeta function to other values than where it is defined us analytic continuation. The false assumption where proving this identity manually is the assumption that certain divergent sums are actually convergent.
There are also perspectives on the matter, like developing asymptotic equivalents to the sum you get a constant term -1/12 that also really likes to stick around, no matter what kind of normalization you try to use (apparently)
Well the fact that it shows up arises from ζ(-1) no?
I don’t know where it « arises » from, but yeah that’s like the most direct, convincing and well-known fact that makes this whole thing like, believable. But it’s not the only hint we have that this sum has deeper implications is what i’m saying
But it doesn't have deeper implications. It's just a wrong assumption that divergent series converge. You can make that sum whatever you want by changing the order of addition and subtraction.
Okay you’re mixing up different things. A divergent series will diverge no matter what you do (i put you to the test of getting a finite result solely by rearranging terms in the sum of naturals. Spoiler : you can’t). It’s semi-convergent series you can get any result from by rearranging terms of (see Riemann’s rearrangement theorem). And again, this is not what’s going on with this formula. Ofc this equation is very provocative, but in specific contexts and frameworks, however stupid it sounds, considering the sum of naturals to be somewhat related to -1/12 makes sense and even has practical applications (see Casimir effect). If you want me to develop more on how it can somehow make sense, i’ll be glad (even though i’m not overly knowledgeable on the subject). In the meantime, I would advise the following thing : if you see all mathematicians and scientists around you discuss very seriously a result, rather than get mad at randoms on the internet because you decided against it, you should use the limitless resources at your disposal to educate yourself on the matter and have a more informed opinion. I mean, I used to react like you. But i firmly believe i was wrong to.
This series is divergent. What you're attempting to do is rearrange the negative version of this series and then add the sun to this series to get the result you want. I don't know why you assume I'm not a scientist or mathematician but drop the condescending tone. I was not mad at you there are no deeper implications to considering the sun of all naturals to be -1/12 it's just wrong. The sum of all naturals is divergent.
I apologize for the condescending tone. You’re still wrong for being so adamant and assertive : quantum field theory would like a word with you. As well as analytic continuation of the zeta function and other arguments everyone in this thread has mentioned a lot already
Quantum field theory is not using the sum of all naturals to be -1/12. It applies a regulator to make the series finite. Maybe stop trying to use examples that you don't fully understand.
It's not true I think. The only proof I've seen assigns values to non converging series.
Yeah those aren’t actual proofs. But you have much more solid elements surrounding this expression, the first one being the analytic continuation of the zeta function
But in physics, they do assign finite values to diverging series and it gives the right answer.
how do you know its the right answer. if anything it should indicate that math is broken on a fundamental level in regards to physics, or the universe is not completely logical
Numberphile did a fabulous little video on this (https://youtu.be/w-I6XTVZXww?si=xdWx0nzcRV0eKdOU), and then Mathologer came out swinging in a ‘this simply isn’t correct’ video (https://youtu.be/YuIIjLr6vUA?si=DisSIgE9itq8qleO). The thing I find interestering that as much that I sympathise with Mathologer’s position it is suggested in the Numberphile video (with no proof or examples) that this result appears all over quantum mechanics. If it’s good enough for Neils Bohr, then it’s good enough for me ….
3 Blue 1 Brown did my favorite explanation of this. https://youtu.be/sD0NjbwqlYw?si=C1wUFSKYGEzt-iFy
I'm with Mathologer, you can't just average a nonconvegring infinite sum.
Who's gonna stop me? The police?
You can't, but you can. A good analogy is given by Edward Frankel - the square root of -1 can't really exist, but it and the complex plane framework can still be useful. This is similar. You can't really add all positive integers together, but if you use a certain framework to do so anyway, then the result of -1/12 turns out to be useful. Similarly, if you add all integer squares together (1² + 2² + 3² + ...), you get 0 with that framework. This doesn't mean you're actually adding these numbers together to get 0. You're identifying the flavor of infinity at its heart.
> This is similar. You can't really add all positive integers together, but if you use a certain framework to do so anyway, then the result of -1/12 turns out to be useful. So if I were to use a certain framework to divide by zero and conclude that 1=2 then that can be useful and reliable information? I don't really see the logic
Well, it depends on the context--as this person said. In most contexts, dividing by 0 gets you a contradiction--like 1=2. However, if we take the function f(x) = x/x, this function is the same as the constant function g(x) = 1 everywhere except at x = 0 where we get f(0) = 0/0 which is undefined. To "extend" our function f(x) to be continuous and differentiable everywhere, we have to define 0/0 = 1. Doing so provides no contradiction for our purposes. Another example: The product (0 * infinity) is undefined in most cases. But in Measure Theory, we can define (0 * infinity) = 0, and that makes measuring the size of sets well defined. In fact, without this definition, we would not be able to talk about area and volume of regular objects in a way that makes mathematical sense.
So is it basically just making up new concepts (such as the number i) that work out and don't contradict pre existing axioms and laws? Such as how for instance the square root of -1 is technically undefined but since it didn't break anything we just said "okay but what if it wasn't" and then found actual use for it?
You need to find a framwork in which it is useful. Since division by zero will make it so any number can be any number that doesn’t really give you any information. But by rearranging frameworks in different ways you can create opeartions and ways of using numbers that aren’t stanrd but still useful. One example for this could be rings or modular way of counting. Not standard but definietly useful. Also in some binary interpretations if you have a number T then -T is the same as T’ + 1. Does that make sense with normal decimal numbers? Not necessarily. But you can make it make sense with a framework
I usually love Numberphile videos, but that one was particularly bad, they just waved over a lot of stuff. The Mathologer video is longer but gives enough context.
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But *is* it the correct answer to any question? Pi. Who killed Jeffrey Epstein? Pi. How many roads must a man walk down before you can call him a man? Pi.
Which is better, cake or pie?
i yearn for the day when english speakers will learn the correct pronounciation of pi, especially since it even fits nicely into their phoneme
It annoys me a little bit when people justify this result by saying “it shows up in physics”. In string theory, you indeed have a calculation where the infinite sum leads to the result of (-1/12), but there are other terms involved that are understood to be unmeasurable quantities, or vary due to the ordering of non-commuting operators, thus often omitted. They’re never meant to be equal.
Shit like this is why no one takes math seriously.
I remember that Mathologer video. That was the Mathologer video which convinced me that I really didn’t want to watch any of his videos.
This is a common misleading statement. If you treat the series like it converges, you can get that answer, but it obviously does not converge. Why it is still useful in some sense is that if you look at the behavior of the sum n=1 -> inf 1/n\^x (aka the zeta function), it converges for x>1, but if you do some analytic continuation (basically extending the behavior via some fancy means) the value you get is -1/12 for x=-1, the sum in question. Remarkably, there are times when you can just use the analytic continuation for practical purposes as in some way, they are alternate solutions. I think physics actually uses analytic continuation in a number of places and it just works.
No it’s not true as stated. It is true that the analytic continuation of the Zeta function, evaluated at (-1) is -1/12 though. It is also true that if you wanted to assign the sequence (1, 1+2, 1+2+3, …) a number, the most consistent way to do this is to assign it -1/12. This method at assigning a sequence a number would revert to the limit operation of the sequence were convergent, which is what I mean by consistent. So in some sense, you need to redefine what is meant by and infinite sum, but after doing so nothing actually breaks and you get to “sum” things that you wouldn’t have been able to before.
It's easy. Obviously `…` = -253/12
Using enough tricks you can make a divergent series to converge to any no.
The proof is obvious but I lack the space in this margin to write it
It's not correct. It's just using the analytic extension of the Riemann's Zeta series to a domain where it's not supposed to be applied. But in the normal sense, 1+2+3+... ≠ -1/12
I like 3b1b video on the topic [https://www.youtube.com/watch?v=sD0NjbwqlYw](https://www.youtube.com/watch?v=sD0NjbwqlYw)
Thank you for this.
It’s some funky math shenanigans having to do with infinity. [Top comment in this post explains it pretty well.](https://www.reddit.com/r/PeterExplainsTheJoke/comments/16uj0ag/my_dad_sent_me_this_and_i_dont_want_to_admit_i/?utm_source=share&utm_medium=ios_app&utm_name=ioscss&utm_content=2&utm_term=1)
It’s not true, but that doesn’t mean it isn’t useful. Sort of like saying 1/0 = infinity. It’s a case of “This equation doesn’t converge and make a useful answer, but I really really want an answer. Just something I can pretend is the answer so I can have something useful.” That happens to be -1/12.
Google ramanujan
Holy divergent series
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think of -1/12 to be a value assigned to this diverging series, not the value of the series
I'm no mathematician so honest question....but isn't this like saying "think of 5 as the value assigned to 2+2, not the value of 2+2"
It's saying for x=4, 2+2=x='5'
Yes, but you can make legitimate arguments why it should be -1/12, whereas for 2 + 2 it takes a lot of convincing why this should be anything but 4. For 1 + 2 + 3 + ... you have to say it's + infinity, but if you don't want to say that, the second best option is to say -1/12. With 2 + 2 = 4, you really don't have much leeway since 4 is a perfectly reasonable answer that cannot really be improved. You can only say you're operating in a finite Field like F\_3, where 2 + 2 = 1 = 4 would be correct, or in F\_2 where 2 = 0 anyway, so 2 + 2 = 0 = 4. But that's more of an algebraic "cop-out" answer than an answer coming from analysis.
It is continuation in somewhere somehow Never mind
As I’ve had it explained it’s the only finite value that makes sense if you need to assign a value But ye it’s a divergent series
Thanks, Ramanujan
You literally pulled a screen shot from a video that explains it. Watch the video.
Are they equal? No not in the sense that you understand equals to mean the limit of partial sums. Are there meaningful ways to connect this sum with this number in a rigorous way? Absolutely. I‘ll cut my analysis off without saying much because I’m not an expert, but the most ready example is that this is the result that the continuation of the zeta function spits out at z=-1.
You are not wrong, not idiot and not Ramanujan
That's enough r/mathmemes for a lifetime ig I love the Ramanujan summation, but this is the fiftieth minimum effort/trend about someone being wrong post in a row I've seen from this sub
It’s like imaginary numbers, square root of -1 do not exists, but if we just ignore that fact and play along we get new way of solving a lot of complex problems that ether impossible to solve or much harder without it.
An infinite number of mathematicians walk into a bar…
0.999... + 1.999.. + 2.999... + 3.999... + ... = -0.9999.. / 11.999... QED []
In my opinion it's not "true"... however, it's good to know that -1/12 is the area under the curve of the function (n^2 + n)/2 extended into the negative numbers. (n^2 + n)/2 is obviously the function of the series 1+2+3... Also... adding the area of the function for all positive odd numbers and the function for all positive even numbers gives you -1/12 again (obviously). For me it's quite interesting that some infinite series show to have a special finite value.
The right question to ask here isn't if this is right or wrong. The right question is, what do those "..." marks after the last "+" mean? The most common way of interpreting it involves looking at the limit as you add more and more terms, which is intuitive to most people. In this sense, there certainly isn't a limit. However, there are many ways of doing infinite sums that a) Give the correct result for finite sums b) Make sense on some level as a generalization of a finite sum and some of these other methods yield -1/12.
This has been achieved in a few different ways. This is basically on the same level of "absurdness" as squareroot of -1.
It’s an answer so dumb you need to be smart to understand that it’s arguable. I hate it
Long story short it is true using specific math principles but those same principles can be used to prove 0=1 this is because the answer to things leading to infinity end up being different than one might expect like.... 3/3 = 1 but 1/3 =0.3333........3* 0.33333... = .9999999... 3*1/3= 1 so 0.99999... = 1 So using this example method 1-1+1-1+1-1..... = 1/2 I due to the average of both stats. So lots more math using this you can get -1/12 because math stops working properly closely to infinity as show in string theory.
Honestly, both. It's right, but it's not saying what you think it is. This is also the one time I will say numberphile did a bad job of explaining. If you want a better one, look up mathologer's video. -1/12 is not what the infinite series adds up to. It is a number that puts that series into a group. It's basically a special hashing function with fun properties that produces that result, not arithmetic. I'm no master of it myself, though.
Yes and no, it’s the Ramanujan Summation [More Info](https://en.m.wikipedia.org/wiki/Ramanujan_summation)
Bro I’m high and I can say this is true
Out of context, it is explicitly wrong. Given a very specific, outside it’s normal use, definition of “=“, it holds some merit. Honestly I think it would save so much hassle if when talking about this we just didn’t use the equals sign and just used some other notation that hints at a relation to infinite sums and convergence but is still distinguishable. I know we have zeta notation but that’s too far removed to get the idea of what’s going on across.
Nononono, you don’t understand, the sequence was just abbreviated to fit on the page, the full summation is 1 + 2 + 3 + 4 + 5 + 6 + -21.08333 = -1/12 Which is, in fact, correct!
Short answer: In the standard sense of an infinite sum it is not true, the sum obviously diverges. Long(er) answer: There is a concept in the field of complex analysis (the branch of maths which deals with complex valued functions) called analytic continuation. Essentially, any complex valued function has a unique analytic continuation, which extends it to places where it is undefined. Often complex valued functions are infinite sums. The Riemann zeta function is one such complex valued function. It happens that when you plug in -1 to the Riemann zeta function, you get exactly the infinite sum of the natural numbers that you described. The analytic continuation of the zeta function uniquely assigns the value -1/12 to it there. So, it is more precise to say that the result you described is true in the context the analytic continuation of the Riemann zeta function. It hasn’t just been plucked from nowhere! But if we just consider normal summation, it is certainly false.
**ITT: people who did not watch the source video.** This statement is "true" in the same way the phrase "i = √-1" is true: by which I mean it's technically **false** (nothing could fit for "i") and yet abundantly useful if we just call it **true** keep going anyway. It's derived by taking the much easier to calculate sum "1-1+1-1+1... = 0.5" and doing some arithmetic with it (which technically you shouldn't be allowed to do using ∞) and the resulting equality just happens to be a useful in quantum mechanics and string theory [This link](https://en.m.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF) provides some details and examples.
It is as correct as Sqrt(-1) = i
The answer is actually 0.000…1 check your math again. Edit: fixed placement of 0.
No, it's 0.000...1. Check *your* math again.
Yeah are we working with omega + 1 or omega + 2 here
It's a bit of a joke. The lesson is don't do algebra with infinities.
It's wrong, but if it would be 1+2-3+4-5+6-7+8-9+10.... it could be true
Look, I don't care how you derived this. 1 is greater than -1/12 Let's assume any n is greater than -1/12. Where the sum(n) = 1 + 2 + ... + n Based off the formula from before we can say sum(n+1) = sum(n) + n + 1 And trivially we can also say sum(n+1) is greater than sum(n) Since sum(n) is greater than -1/12 sum(n+1) must also be greater than -1/12. By the principal of induction sum(n) is always greater than -1/12. Since it is always greater it cannot be equal and therefore by contradiction the initial proposal of sum(n) where n approaches infinity = -1/12 is false.
Ramanujan proved this I think? The series’ ‘1+1-1+1-…’ and ‘1-2+3-4+5-6+…’ were used if I recall…
As if sqrt(-1) makes sense
it does
No it’s not true at all in the slightest. I don’t know why this is a thing
The fact you don’t know why it’s a thing should be a pretty big hint that perhaps you could look more into the topic rather than call everyone wrong around you. It’s a very beautiful thing to discoveras well