Timeline for Why does an argument similiar to 0.999...=1 show 999...=-1?
Current License: CC BY-SA 3.0
31 events
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| May 14, 2018 at 14:02 | comment | added | ntg | I agreed 100% with your answer(+1)... Then checked this video (youtu.be/0Oazb7IWzbA?t=6m38s) It seems there are mathematics where the sum might be -1, but not -1 as we know it, and not with the mathematics I know and understand... The argument is similar to " what is the root of -1"... If asked long time ago, one would explain it does not exist... at least for real numbers... | |
| May 13, 2016 at 8:08 | comment | added | Paramanand Singh♦ | I liked the overall discussion about writing stuff which is "blatantly false" but can be interpreted as true in some very specialized context. I have to agree here with @AsafKaragila If the context of a statement is not immediately obvious then it is necessary to mention explicitly. Standard contexts which are obvious to a large population of users may be left out as they can be inferred without any mention. | |
| Apr 3, 2016 at 11:21 | history | edited | Aerinmund Fagelson | CC BY-SA 3.0 |
added 3 characters in body
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| Jan 31, 2016 at 20:52 | audit | First posts | |||
| Feb 1, 2016 at 2:29 | |||||
| S Jan 29, 2016 at 11:32 | history | suggested | Kyle | CC BY-SA 3.0 |
I just fixed a little typo, I hope you don't mind
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| Jan 29, 2016 at 11:07 | review | Suggested edits | |||
| S Jan 29, 2016 at 11:32 | |||||
| Jan 29, 2016 at 0:24 | comment | added | Asaf Karagila♦ | @bartgol: If someone walked to you, up from the street, and said that 1+1=3, you would tell him that this is blatantly false. No? If someone would have written that 0=1, that would be blatantly false. No? But wait, those are just symbols in a formal language and we are free to interpret them however we like, no? So there is a context in which 0=1 or 1+1=3, or both. But for some reason you don't stop to think about those. Why? Because some symbols and contexts have been standardized to be treated as particular cases unless explicitly stated otherwise. Convergence of a series is one of them. | |
| Jan 26, 2016 at 22:45 | comment | added | bartgol | I think this conversation had gone too far. Yes, there is a context where Geinmachi formula is correct, so we can't say that what he said "is false", per se. On the other hand, given the question, its tags, and the context, I doubt bringing up Ramanujan results can help the user who asked the question. To be more clear, a link right next to the formula would have explained the context. Or at least, I would have put a winky face, to give more of a "joke" connotation to the comment, which, as it was, I reckon could have been confusing for the person who asked the question. | |
| Jan 26, 2016 at 19:21 | comment | added | Asaf Karagila♦ | @Kyle: I'm terribly sorry. But there's really no point bringing in Euler into the mix. "Meaning of symbols" has completely changed over the past 150-200 years. Let's bring the ancient Pythagoreans and just argue that the term "number" is senseless, since everything is just length of a multiplicity of a unit line. Sure. And if you want to believe I'm acting in bad faith, go ahead. But I think this discussion has ran its course. | |
| Jan 26, 2016 at 19:01 | comment | added | Kyle Strand | @Asaf Even restricting the view to math using modern notation, different formal theories can use "$=$" (e.g. see formal theories for lambda-calculus or combinators, which us "$=$" to mean "$=_{\beta}$" or "$=_{w}$", respectively). As for "bad faith," yes, it's certainly bad faith to "assume that someone...has no idea bout" what they're talking about unless you have evidence that this is the case (not posting anything on Math.SE isn't "evidence", it's a lack thereof). | |
| Jan 26, 2016 at 18:59 | comment | added | Kyle Strand | @AsafKaragila What...? I don't expect the set of statements acceptable as comments here on Math.SE to be exactly identical to the set of statements acceptable as "things I expect first-year analysis students to expect right off the bat," and I don't see how that's relevant. And it's simply not true that "$=$" only ever has one meaning, especially if you look at old math writing (e.g. Euler's work on convergent series). | |
| Jan 26, 2016 at 18:40 | comment | added | Asaf Karagila♦ | @Kyle: And a word on "bad faith", I think it's not bad faith to assume that someone whose only activity on this site is that comment has no idea about the fact the context in which this equality is interpreted is not the standard context. Not to mention, that if I decide tomorrow that "Hasudorff space" means that every open cover has a finite subcover, and a "compact space" is a space where every two points are separated by disjoint open sets, I can still do mathematics just like everyone else, but saying something like "$(0,1)$ is compact but not Hausdorff" will make no sense to the rest. | |
| Jan 26, 2016 at 18:35 | comment | added | Asaf Karagila♦ | @Kyle: And it's also blatantly false that = has different meanings in mathematics. It has exactly one meaning. Two terms are equal if they are the same object. The context in which we interpret the terms changes, but the meaning of = remains the same. | |
| Jan 26, 2016 at 18:29 | comment | added | Asaf Karagila♦ | @Kyle: And there is just one context in which $1+1=0$ makes sense. I dare you to tell that to your students next time you teach analysis. See if it makes sense to any of them right off the bat. Not to mention those atrocious Numberphile videos that originally made this into a global phenomenon didn't even set the context. They just showed unjustified trickery. So I honestly doubt if most people who give this equality are even aware to its different context. | |
| Jan 26, 2016 at 18:27 | comment | added | Kyle Strand | @Asaf "$=$" doesn't have the same meaning in every context, though. You're right that Geinmachi didn't specify the context for the comment, but there's exactly one context where the statement makes sense, and it's familiar enough (espcially given MichaelSeifert's link for those who aren't familiar with that form of summation) that it's obvious what Geinmachi meant. Arguing otherwise seems to be assuming bad faith. | |
| Jan 25, 2016 at 23:37 | comment | added | Simply Beautiful Art | @Joshua But then you would have it all equal down to $\infty$ in the end, which contradicts with the straight down decimal subtraction method. :D | |
| Jan 25, 2016 at 19:54 | comment | added | Joshua | ∞−∞/10 = 9∞/10 if both sources of ∞ were the same source; but this fact is not useful in most cases. | |
| Jan 25, 2016 at 19:18 | comment | added | Asaf Karagila♦ | @Voo: No. I'm saying that $1+2+3+\ldots=-\frac1{12}$ is blatantly false. The statement "The series $1+2+3+\ldots$ converges in the sense of Ramanujan to $-\frac1{12}$" might as well be true. | |
| Jan 25, 2016 at 19:12 | comment | added | Voo | @Asaf So are you saying that the concept of Ramanujan summation is "blatantly false" or just the particular notation? Because if it's the former, that's a rather dim view of mathematics. Formalizing new concepts to analyse something is at the core of math. If on the other hand it's just about the tongue in cheek joke, well de gustibus non est disputandum - I found it entertaining. | |
| Jan 25, 2016 at 19:04 | comment | added | Asaf Karagila♦ | @Voo: Just like you can't write something like $\pi=\frac{22}7$, because it's blatantly false, you can't write $1+2+3+\ldots=-\frac1{12}$. Because the equality there has a very specific, very technical meaning, and it fails here. The fact that physicists do something is ever more the reason not to do it "just because". Just like every time you say "Quantum mechanics teaches us that everything can happen at any given moment!" is blatantly false (even if formally you can interpret it that way). There is a technical meaning behind statements in mathematics (and physics) that you can't ignore. | |
| Jan 25, 2016 at 18:59 | comment | added | Voo | @Asaf I don't see anyone arguing that the series does not diverge - just that there's one way to assign a value to a divergent series which happens to have "practical" applications in at least theoretical physics. | |
| Jan 25, 2016 at 17:21 | comment | added | Asaf Karagila♦ | @Michael: Only that a "number" is not a well-defined mathematical notion. It is incorrect to say there is a rational number $x$ such that $x^2=2$. On the other hand, the convergence of a series is a well-defined mathematical notion, and while there are other notions of convergence, there is exactly one notion of convergence where one can omit the type of the convergence. And let me tell you, it ain't Cesaro, Ramanujan or otherwise. And $1+2+3+\ldots$ does not converge in that type of convergence. You want to argue that it does in some other type? Sure. But you need to be explicit. | |
| Jan 25, 2016 at 15:54 | comment | added | Michael Seifert | @AsafKaragila: I'll concede that I did appeal to authority there in order to make a pithy joke; mea culpa. That said, mathematics makes progress by finding useful methods by taking things that are "formally incorrect" and formalizing them "through particular and non-trivial methods." If all you know about is the rational numbers, it's formally incorrect to say that there's a number for which $x^2 = 2$, but it's still a useful notion that (once formalized) allows for a lot more interesting math to be discovered. Should we dismiss the existence of $\sqrt{2}$ as being "formally incorrect"? | |
| Jan 25, 2016 at 14:53 | comment | added | Asaf Karagila♦ | @Michael: I can't, he's dead. And just because he said something that is formally incorrect and can be formalized through some particular and nontrivial methods, doesn't mean that everything he says is true. That's a common fallacy known as appeal to authority. | |
| Jan 25, 2016 at 14:51 | comment | added | Michael Seifert | @AsafKaragila: Tell that to Ramanujan. | |
| Jan 25, 2016 at 1:01 | comment | added | Asaf Karagila♦ | @Geinmachi: That is no more true than $0=1$. | |
| Jan 24, 2016 at 23:49 | comment | added | image357 | There are ways to manipulte divergent series and assign meaningfull numbers which in the special case of convergent series just give the limit. Those manipulations make total sense in the right context. However, I agree that this context is not the usual topic of finite limits. | |
| Jan 24, 2016 at 20:49 | vote | accept | CommonToad | ||
| Jan 24, 2016 at 15:25 | comment | added | Geinmachi | 1 + 2 + 3 + 4 + ... = -1/12 | |
| Jan 24, 2016 at 10:52 | comment | added | CommonToad | Thank you what a great answer | |
| Jan 23, 2016 at 19:51 | history | answered | Aerinmund Fagelson | CC BY-SA 3.0 |