This is a fun one, because of course the kneejerk rejection of the statement "1+2+3+4+5+... = -1/12" is correct Under everyday, ordinary grade-school arithmetic, the answer can't be "-1/12" -- but the answer can't be any other number either, the operation itself is not possiblehttps://twitter.com/tomgabion/status/1289857027381002241 …
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So people's intuitions about this are correct but they don't take it far enough Caught between the dueling intuitions between "Well the answer to this question can't be any ordinary number" and "It must HAVE an answer though"
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All new forms of math are built on that second part "Okay, I get that this is a stupid-ass question and Pythagoras or whoever would've just said 'shut the fuck up' if I asked him but WHAT IF you COULD add up numbers an infinite number of times"
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All summations of infinite series are based on making up new rules and saying "Okay let's pretend you can do this, what happens if you do, what new stuff do you discover if we just fuck around and act like this makes sense"
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So like, let's be clear This classic series: 1+1/2+1/4+1/8+1/16... Is, from a pure old-school POV, just as bad as the other one Even though this one looks like it has an answer (it adds up to 2 in the end)
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*You can't do things infinite times* If we imagine the addition as a real-world activity that takes time to do -- a second to write the new sum on a piece of paper, a fraction of a microsecond for a processor to encode it in memory -- then the Sun will go out before you get 2
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Ancient Greek philosophers knew about this shit This is, famously, Zeno's paradox Getting around this and saying "Well let's just pretend you CAN do it infinite times" is not *answering* Zeno, it's just telling him to shut the fuck up
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Specifically, saying Zeno was wrong, and I *can* just wave a magic wand and say "skip to the end of something I just said was by definition endless", is formally inventing the idea of a "limit" Which has made many people very angry and been widely regarded as a bad move
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The fact that being able to say "1+1/2+1/4+1/8...=2" is very useful, and all of calculus is based on it, doesn't actually mean we were *right* Whether "infinite converging series" are, like, a real thing that exists in the world of atoms is this big heavy question (probably not)
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There's a famously sexist quote apocryphally attributed to Shaw that I'll paraphrase as "If you wouldn't sell out for $10, but you will sell out for $10 million, then you are a sellout and you're just haggling over the price"
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When you agreed you could just shrug off the paradoxical-ness of doing anything an infinite number of times because pretending like you can is useful, you left the original rules of arithmetic behind Now we're just arguing over how weird we want to get
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This is, in fact, a *whole field of mathematics*, and the summation of infinite series can be done according to any number of different methods, which mathematicians invent at their pleasure, designed according to different criteria
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(I'm verging into stuff I only half-remember here, full disclosure) 1+1/2+1/4+1/8+... is one of the easiest series to just up and say has a solution, that it's 2 Because you're only adding, not subtracting, so you can move around all those numbers at will
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This is called "absolute convergence" If the numbers change sign, you have "conditional convergence" Paradoxically, for people used to doing things a finite number of times, it suddenly matters how the numbers are arranged
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1-2+3-4, if I stop at 4, is always going to give the same result (-2) even if it's -2+3+1-4 or -4-2+1+3 (commutative property of addition) This doesn't work with infinite sums (which doing things an infinite number of times is a filthy lie)
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The classic example is the alternating harmonic series, 1-1/2+1/3-1/4+1/5..., which converges on the number ln(2) If you rearrange the numbers differently, so it's positive+negative+negative and not positive+negative+positive, 1-1/2-1/4+1/3-1/6-1/8+1/5-1/10-1/12... it's ln(2)/2
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Without writing out the proof, you see what kind of dirty trick I'm pulling here, right? For each positive fraction with an odd denominator, I'm "borrowing" an "extra" negative fraction with an even denominator from "the future"
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(Dug up a more in-depth discussion here http://larryriddle.agnesscott.org/series/rearrang.pdf …) If this were a finite set of numbers, then I would eventually run out of "future numbers" to borrow from, and this trick to make the sum constantly be lower wouldn't work, and the sums would come out the same
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But it's an *infinite* series, so I *never* run out And so the same set of numbers rearranged converges to 1/2 the original sum This is why doing things an infinite number of times is, again, a filthy lie
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(Riemann proved you can rearrange this series to converge on ANY sum, at all, or to not converge and instead spit out ∞ or -∞ Which is one of those things mathematicians do that really pisses people off)
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Anyway There are series that are absolutely convergent, and series that are conditionally convergent, and series that are divergent The series 1-1+1-1+1... doesn't converge, it doesn't "get closer" to anything the longer you do it, it just flips from 1 to 0
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So does 1-1+1-1+1... equal anything at all? According to the method people used when they started talking about this ("classical summation"), no, absolutely not There is no way to rearrange this series so it converges on anything, it's a divergent series
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A Norwegian guy named Abel (a deeply weird, extremely smart dude who invented all kinds of new math before he died of TB at the age of 26) was like "Sure you can, it's 1/2" You probably looked at that and said "Yeah it's 1/2", if you thought about it at all
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The thing is in math you are *completely allowed* to say "Okay the rules say you can't do it but it looks like it should be 1/2 so I'm gonna say it is" You just have to go on to explain to everybody what that *implies* if you decide to do it, which is the hard part
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I.e. by the delta-epsilon definition of a limit, which is "It has to get closer to the number every time you do the next thing", 1-1+1-1 by definition is not approaching any limit You do it once, it's 1, you do it again, it's 0, then 1, then 0
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The "distance" between each new sum and the answer 1/2 is exactly the same every time, you *never* get closer A so-called Abelian summation method involves using the idea of *averaging* instead of the idea of *limits*
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Does that make sense? Should that be allowed? The authorities haven't come to a moral conclusion on the matter, but hey it's fun and cool prizes come out when you do
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(Philosophically, taking an average of an infinite number of sums is worse than adding numbers together an infinite number of times, because it means you have to "divide by infinity", and that's a big can of worms Which Abel happily opened)
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It goes on from there Every new bleeding-edge summation method has more scary problems with it -- or, rather, it lacks certain properties that were assumed under classical summation Abelian summation does not "work" in many ways, but we accept that
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It "works" in other ways and that's okay Cesaro summation and Abelian summation are two methods that give you 1/2 for a "mildly divergent" series like Grandi's series (1-1+1-1)
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Abelian summation is slightly more "powerful" and can also handle weirder cases like 1-2+3-4+5, which according to him = 1/4
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