Thursday, January 16, 2014

Sum of All Positive Integers = - 1/12

This is so amazing it's hard to believe:

 1 + 2 + 3 + 4 + 5 + 6 +.... = -1/12


(via: This simple version of the proof does have a bit a of a fudge, because the sum

1 + x + x2 + x3 + ...

only equals 1/(1-x) for the absolute value of |x| < 1, not (as the video stated, for x < 1). There's a longer version of this proof on the Numberphile Web site, using the Riemann Zeta function and what's called analytic continuation:

The thing is, this analytic continuation makes both mathematical and physical sense -- physics experiments have confirmed it. For example, the well-known Casimir effect, which is the electrical attraction of two parallel, uncharged metal plates due to quantum effects, is calculated to involve the Zeta function ζ(-3), which by similar reasoning equals 1/120, and the predicted Casimir force has been verified by experiments. That is

1 + 23 + 33 + 43 + ... = ζ(-3) = 1/120

is experimentally verified.

Wikipedia has an entry on this sum, including a brief one on its relationship to string theory, and this wonderful excerpt from a letter from the Indian genius Srinivasa Ramanujan's to the mathematician who eventually brought him to Englaand, G. H. Hardy:
"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. …"


William Connolley said...

I'm a bit disappointed with your reporting here. 1+2+3+... doesn't have a sum. Its a divergent series. All this and similar proofs show is that *if* you allow yourself to attempt to sum divergent series, you can get inconsistent results. But you can get inconsistent results much more easily by insisting that 1=2, if you want to be inconsistent.

David Appell said...

OK, how about this wording: the series diverges, but it can be assigned the value -1/12 in a way that is not mathematically ridiculous, and that agrees with the results of physical experiments. To a physicist it makes perfect sense.

William Connolley said...

The point is similar to that which surrounds people wondering if 0.999... = 1 or not. Of course it is. But its not a textual identify; you need to know what 0.999... *means* and similarly you have to know what 1+2+3+... *means*. It has a formal definition; or rather, something like 0.9+0.09+0.009+... fits within a formal definition; 1+2+3+... doesn't.

So it depends what you mean by "ridiculous". If you mean "not immeadiately obvious to the layman", then yes. But fundamentally it makes no more sense than 1=2.

David Appell said...

But it's difficult to attach "meaning" to an infinite sum because the number of things you're adding up is...well, infinite.

Meaning in this case means, I guess, analytic continuation. It surely bothers physicists a lot less than mathematicians because in quantum field theory one is all the time taking the sum and differences of quantities that are infinite and getting finite results, that agree with experiments to amazing numbers of decimal places (i.e. for QED's prediction for the g-factor of the electron).

Victor Venema said...

The proof in the video sounds strange to me. Series S1 does not converge, it is either 0 or 1. To show that you can simply average that to 0.5 would need additional work.

There is an interesting proof on Wikipedia. That should be convincing to WMC. :)

I cannot see an error in that proof. Except maybe that that the result is just correct if you assume that the series is generated by that generating function. I could imagine that there are more generating functions that in some limit would lead to a series 1+2+3+4+5+6+.... and that those function would have different solutions as -1/12.

David Appell said...

Victor, yes, I too thought that claiming S1 = 1/2 was very hand-waving.

Even in the 2nd video with the longer proof, there's a point where he takes the Taylor series for 1 + x + x^2 + x^3 + ..., differentiates it, and substitutes x=-1. But that series is valid only for |x| < 1, not, as he writes, x < 1.

It's only justification is that it works!

William Connolley said...

VV: I'm not sure what you mean by "proof" on the wiki page.

1+2x^2+3x^3+...=x/(1-x)^2 is fair enough for some x, but its not valid at x=1, obviously.

Brandon Shollenberger said...

The discussion here (primarily William Connolley's second comment) led to me having a discussion with a couple people about whether or not 0.999... equals one. That led me to writing a post on the issue. I figure it is only appropriate to share a link here since it was the trigger for the post.

William Connolley said...

Oh dear. Your post is hopelessly wrong. I'm very doubtful you have enough maths to even begin to understand why.

might help. You need to begin by understanding what the textual string "0.999..." represents; this is by no means trivial.

Brandon Shollenberger said...

William Connolley, you're welcome to believe that. However, I find haughty responses that contribute nothing of substance unconvincing. You didn't even say in what way I'm wrong. Obviously, I'm not going to be convinced by that.

If you want to have an actual discussion, you're free to comment on the post. If not, we can just leave it at, "You think I'm a fool and mock me."

William Connolley said...

Sigh. I just know this is going to be a waste of time. DA: tell us when you get bored of taking over your comment thread.

You say: "Consider this common “proof” the two equal one another...". But you're wrong: its not a proof. Its a heuristic. Wiki handles this poorly; I somewhat regret pointing you at it.

First of all, you need to understand what "0.999..." is. It isn't self defining. Calling it "0.9 followed by an infinite number of 9's" is meaningless.

Probably the best thing is to say that it is the limit of 0.9, 0.99, 0.999, ...; which is to say: lim_{n->infinity}[sum-from-k=1-to-k=n-of(9*10^-k)], if that limit exists. "lim" (i.e. limit, but it has a technical meaning so I'm avoiding the English word otherwise we might get confused) also needs defining, but we can skip that for now.

Also, lim(a*f(n))=a*lim(f(n)) is (provably) true for a constant "a", and reasonably obvious, as is lim(f+g)=lim(f)+lim(g) and other simple linear combinations.

From that, you can get a version of the "10x-x" proof that works.

You write: "0.000…1. It’s infinitely small, but it is real". But "0.000...1" is a textual string devoid of meaning. "It’s infinitely small, but it is real" is also meaningless (even if you substitute "non-zero" or "positive" for "real". "real", when used in maths, means "one of the real numbers", which zero certainly is. But that's not what ). The only meaning you can assign to "x is infinitely small" is "for any given number y, x is smaller than y". The only number, x, satisfying that restriction (on the non-negative reals) is 0.

Victor Venema said...

VV: I'm not sure what you mean by "proof" on the wiki page.

Me neither.

1+2x^2+3x^3+...=x/(1-x)^2 is fair enough for some x, but its not valid at x=1, obviously.

That is why they make a substitution and compute the limit to x=1. That seems allowed.

However, I can imagine that with another generating function or substitution you can get some other value as solution as -1/12. In that case you should really talk about the generating function having -1/12 as solution and not the shorthand that 1+2+3+... has -1/12 as solution.

Let's see. The video has an enormous amount of viewers. So I expect that a mathematician will soon explain matters better.

William Connolley said...

> That is why they make a substitution and compute the limit to x=1. That seems allowed.

Ah, I see. The key word there is "seems". What they've done by that is just disguise the invalidity, not remove it.

lim{x-tends-to-0}(f(x)) != f(0) for all f. Put that way, its probably obvious. See-also

Brandon Shollenberger said...

William Connolley, it's convenient you point me to a link then only after I discuss it do you say it has issues. I suppose you'll also tell me it handles things poorly when it directly contradicts you. For example, in discussing the use of limits, it says:

A different definition considers the equivalence class [(0.9, 0.99, 0.999, ...)] of this sequence in the ultrapower construction, which corresponds to a number that is infinitesimally smaller than 1.

What you call meaningless is directly stated by your own source to be a true thing. Moreover, your own source quotes multiple mathematicians supporting my argument, such as:

Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about 0.999... < 1 are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.

It will now be evident that .9999... does not equal 1 but falls infinitesimally short of it. I think that .9999... should indeed be admitted as a number ... though not as a real number.

You act as though your position is so obviously true while linking to a source which directly contradicts it. In fact, your source basically offers the same argument I offered, quoting respected mathematicians in the process.

Quite frankly, it appears you are the one who doesn't "have enough maths to even begin to understand" this argument. That's the only reason I can imagine you would offer a source that directly contradicts you.

William Connolley said...

Sigh, again. All that stuff you're quoting comes from infinitesimals/non-standard analysis. You don't understand it, all you're doing is quoting it.

The key is "non-standard": that's not the usual definition of numbers; it isn't part of the reals.

But it still doesn't help you because you *still* need to assign a clear meaning to the textual symbol "0.999...". If all you're saying is "there is a schema in which the textual symbol "0.999..." does not have the value 1 then I'd say: so what? That's trivial. You don't need high-level maths. Just use a schema in which it is given the value 7.

Brandon Shollenberger said...

Willaim Connolley, are you kidding me? I explicitly said those things were taken from other mathematical systems than the real number system. That was the entire point of the post. I'll quote it:

Only we don’t say so. A person who thinks 0.999… doesn’t equal 1 obviously believes infinitesimals exist. They don’t accept that axiom. They’d use a different one, like many mathematicians who work with infinitesimals on a regular basis.

That’s right. There’s an entire field of math which uses infinitesimals. It’s just as valid as the real number system. Which one you use is merely a matter of preference. Whether 0.999… and 1 are equal is based on the completely arbitrary choice of whether one uses the real number system or a different one.

The entire point of the post is the choice of which mathematical system to use is arbitrary. There is no inherent reason one must work within the real number system. As such, there is no inherent reason .999... must equal one. You're criticizing me for not understanding something despite the fact it is the entire point of the post you're criticizing.

As for your question, "So what," this is not some trivial observation. Many mathematicians work in frameworks which allow for infinitesimals to exist. People who express doubt the two values are equal should be told about those mathematicians. They should be told about the fields of math which say their instincts are right.

People shouldn't be told they're wrong simply because they instinctively think in a framework other than the real number system. People shouldn't be told math says only one system is right when in reality it says any number are. People should be told what the nature of math is, and how axioms affect our conclusions.

Put simply, people should be taught to think, not just mindlessly parrot what they're told.

William Connolley said...

You can use alternative axioms. You can't just make up magic as you go along. Your symbol "0.000…1" remains meaningless even in the infinitesimals and you don't even realise this.

> There’s an entire field of math which uses infinitesimals.

Its a minor sideshow. No-one uses it. People use the reals, for the obvious reason.

> because they instinctively think in a framework other than the real number system

Don't be silly. People who don't like the std proof that "0.999... = 1" aren't thinking in the infinitesimals - they're just not thinking.

Brandon Shollenberger said...

William Connolley, you blatantly misrepresented my post to such an extent it is clear you weren't aware of what the post said. When your error was pointed out, you did nothing to correct it or apologize for the insulting tone you used while making it. This continues the incredibly rude pattern of behavior you've exhibited in every response you've made. To excuse your false statements, you now make the silly claim:

Its a minor sideshow. No-one uses it. People use the reals, for the obvious reason.

There a large number of papers published about the use of infinitesimals (and at textbooks for it). There are numerous courses teaching it. There are problems which were first solved by people using them. There are entire sets of problems nonstandard analysis is used for instead of "the reals." You've just hand waved away entire branches of mathematics many people work in and get degrees for on a regular basis.

On top of that, it wouldn't even matter if what you say were true. The popularity of a mathematical framework has nothing to do with its logical coherence. Everything I said would be correct even if everybody happened to use the real number system.

Don't be silly. People who don't like the std proof that "0.999... = 1" aren't thinking in the infinitesimals - they're just not thinking.

This is complete and utter nonsense. Infinitesimals are a common intuitive understanding amongst people. People who say the two values aren't equal often say things like, "There's a difference; it's just really small." There are even papers which specifically examine this intuitive understanding.

You are flat-out making things up. You have nothing but your arrogance and hostility to justify anything you've said to me. You've been wrong in every response you've made. For all your comments about my supposed lack of knowledge and understanding, you've displayed far more ignorance. Worse yet, while I've displayed an open mind, you've proven yourself extremely close-minded.

Feel free remain in your close-minded ignorance. Feel free to continue mocking me. All you're doing in this exchange is demonstrating what I said in my post:

0.999… does not equal 1. People who offer “proofs” otherwise don’t understand math.

Unknown said...

Brandon, I wrote a comment on your post regarding 0.9999... = 1, and here is the fatal flaw of your reasoning:

You claim that there is an infinitely small number between 0.9999... and 1.

But if there exists an infinitely small positive number, that MUST also mean that there does not exist a SMALLER positive number. How can something be smaller than infinitely small? But we all know that there is no smallest positive number (yes, there are different sizes of infinity, such as the real numbers versus the natural numbers, but that has to do with cardinality, not comparing sizes of two objects in the same set)

Which means your number does not exist.