Well, at least Wikipedia thinks it’s rigorous. There you can read: 0.999… = 9(⅒) + 9(⅒)² + 9(⅒)³ + …
0.999... - Wikipedia…
Wikipedia doesn’t include the whole explanation and refers to a convergence theorem for convergent series. You can read about that here, but to summarize it equates convergent series with their limits. Since that’s the step you’ve consistently refused to accept it would be wrong to simply state that 0.999… = 9(⅒) + 9(⅒)² + 9(⅒)³ + …
0.999… is either equal to 9(⅒) + 9(⅒)² + 9(⅒)³ + … or it is not. Everything in this equation is clearly defined. “…” means “(countably) infinitely many”. We are still doing math here, no? If it is simply not equal, what is the difference between 0.999… and 9(⅒) + 9(⅒)² + 9(⅒)³ + … then?
It is, but only if convergent series equal their limits. Which you have consistently rejected.
Interestingly enough, from that wikipedia article…
Sound familiar?
To me 9(⅒) + 9(⅒)² + 9(⅒)³ + … is simply another notation for 0.999…
All I rejected is that we can find a point 0.999… on the number line.
Guys,
I have only been casually following this thread. I looked up netzweltler, who joined SDMB on 09 October. Since joining s/he has posted 154 times to date, and all 154 posts to date have been in this thread. S/he has no other posts in any other SDMB thread.
First post, post #98, on 09 October, is here: http://boards.straightdope.com/sdmb/showthread.php?p=19686578#post19686578
Don’t know if this tells us anything, but it’s a little interesting. Thought I’d share.
Bullitt
That is equivalent to rejecting that convergent series equal their limits.
We have to be very very careful here quoting Wikipedia … we have a circular reference thing going on … if Wikipedia is citing SDMB for their information … then we can’t in turn use that information as a citation on SDMB …
I can’t say “the sky was yellow and the sun was blue” … and use this citation from my post #609:
Anyway, I’d like netzwletler’s opinion of the information in the subsection “Infinite series and sequences”, fourth equation …
Just as a friendly digression, one of my favorite demonstrations of how infinity is counter-intuitive is to take the curve 1/x and rotate it about the y axis. You get a long funnel shape. (Infinitely long…)
This shape has an infinite surface…and a finite volume.
You can fill it with paint…but you cannot paint it.
Historical note: This is Gabriel’s Horn (sometimes called Torricelli’s trumpet since Evangelista Torricelli discovered it circa 1644.)
That’s true. The limit must be a point on the number line. This point is 1.
There’s nothing new. “The last step, that (⅒)ⁿ → 0 as n → ∞, is often justified by the Archimedean property of the real numbers.” What last step? We are dealing with infinity, no?
To make some sense we should be able to write * (⅒)ⁿ = 0 as n = ∞*. This just doesn’t happen, since n ∈ ℕ. There is no last number n = ∞.
Yes, and according to math that point is also the value of the infinite series. Every single one of your posts equate to “But it isn’t.”
I’m going to try this approach once more, cause I’m a glutton for punishment. Try to look beyond your step-by-step-approach.
You are describing a process that never ends. It’s obvious that on it’s own this approach doesn’t tell us all there is to know about infinities. For instance if we move from 0 to 1 on the number line by any continuous process that actually gets us to 1, we have to pass by any and all numbers between 0 and 1, including those with infinite digits, such as 0.333…
Now please take the time to either accept this or refute it with actual mathematics and not the return to your monomania. Either we pass by 0.333… and pi to infinity digits, or we pass by some number with a finite number of 3s, or a finite expansion of pi, and the latter is quite obviously absurd.
Now looking at your process we can’t just say “well at t=1 we’re at 1”, because by the rules of your process there’s no reaching t=1. But due to what I’ve described above, we also can’t just say “you can’t ‘reach’” infinity or “0.999… is not defined”. We can reach infinity if we look at the continuous process (just as Achilles can in fact overtake the tortoise and the arrow can reach the target) and 0.999… has to behave the same way whether it’s an instance in time in a continuous process or the, somewhat miraculous, “endpoint” of an infinite process.
So try to grasp this. By the rule set you are operating on, there’s no reaching infinity. But the rule set is obviously limited, as it can’t reach 1, and we know 1 exists. So we have to look beyond your limited set of rules.
How do you suggest we expand the rules?
Or alternatively, refute the convergence theorem using something other than your tiresome repetition of zeno’s paradoxes.
I asked for YOU to comment on the equation, not repeat “heuristics” you don’t understand. This equation lays out the derivation of (0.999…) = 1 clearly, accurately and unambiguously … please state the exact step you think is in error.
You don’t give up your notion that 9(⅒) + 9(⅒)² + 9(⅒)³ + … is a continuous process, and I don’t give up the notion that 9(⅒) + 9(⅒)² + 9(⅒)³ + … is a step-by-step-approach. I think we stuck in a loop.
I think I did. Otherwise be more specific, please.
It’s addition … not a recipe …
You copy/pasted what’s posted at Wikipedia … I’m asking about the equation and nothing more … see there, nowhere in the equation is it specified that n equal infinity, only that n approaches infinity … there’s a big difference.
I understand you answered a question that wasn’t asked, so to be more specific would be to ask you to answer the questions that are being asked … which step in the equation’s derivation do you think is in error? That question is open to anybody, I make no claim that this is a perfect proof … …