Then I move from 0 to 0.333… in one step, 0.333… to 0.666… in the next, and 0.666… to 0.999… in the next.
Where is 0.333… you say? It’s right there at 1/3. That’s the definition.
You are making distinctions that don’t exist.
We’re always on the number line. How can we not be on the number if we never moved off it? Really, you’re starting to make less sense now.
Did you count your steps? It’s 3, not infinitely many. Try again.
“Nowhere” means “not at a specific point” in this context.
Yes, I got to 0.999… in finitely many steps. Defining right where it is on the number line.
Now at the start of your participation I thought maybe you were fine with that, but you’ve kept insisting that 0.999… doesn’t have a defined position. Are you going to backtrack on that now?
Does 0.999… have a defined position, just not one you can get to in infinite steps?
No, it doesn’t. In this context we have a very good idea of where it is, namely infinitely close to 1. Insisting it is “nowhere” is a deliberate and stupid obfuscation of the facts.
netzweltler seems to think the number line isn’t continuous … a terrible misunderstanding … a line has no gaps, that’s basic high school geometry … he wants to have a dashed line instead … but that’s plainly not the case … the longer he goes on the more disconnect he has with mathematical reality …
Let’s look at a simple example … (2 + 3 + 1 + 7 + 3) … we move our pencil two spaces, then three spaces, then 1, then 7, then 3 … we evaluate where our pencil is and we find the number 16 … and this is the correct answer …
What netzweltler is trying to do is move the pen 2, 3, 1,and 7 and THEN evaluating where our pencil is and finding 13 … then demanding we prove 13 isn’t the right answer … he didn’t add all the numbers together …
He’s doing the same thing with (0.9 + 0.09 + 0.009 + …) … he stops to evaluate where the pencil is and finds it’s not at 1 … and this is because he isn’t adding all the numbers yet … there’s still terms he left off … anytime he stops to evaluate, he’s left off numbers … of course he gets the wrong answer every time …
At some point, we will be moving our pencil an infinitely short distance … then and only then can we stop and evaluate … and we do indeed find ourselves at 1 … if I may put this crudely … (0.9 + 0.09 + 0.009 + … + dn/dt)
netzweltler has already stated he hasn’t the foggiest idea what an infinitesimal is so it’s very doubtful he understands what dn/dt means … it also seems he doesn’t want to learn … and that’s sad, these boards at the SDMB are so full of great information and learning opportunities it’s a shame he’s choosing to miss out …
No. Why should I do that? No one has shown a solution to that:
If the claim is that we can reach a point 0.999… on the number line if we move 0.9 + 0.09 + 0.009 + … digit by digit, then the task is to define a procedure (a list) that shows how this point can be reached.
“infinitely close” means “not at a specific point” in this context. Otherwise we would be able to name the position of the point on the number line.
No. There is no last infinitesimal term after infinitely many terms { 0.9, 0.09, 0.009, … }. 0.999… does not contain a last term.
At NO point we are moving the pencil an infinitely short distance. The distances we’re moving are well-defined and are { 0.9, 0.09, 0.009, … }.
It isn’t. It never was. That’s your invention. Equivalent to Zeno’s.
So what’s your statement? We don’t reach 1 if we move 0.9 + 0.09 + 0.009 + … digit by digit?
This discussion continues? What is it about 0.999… is exactly equal to 1 isn’t understood? It isn’t close to 1, it is 1. I submitted the proof back in the first few posts in this thread, and we’re noww over 500 posts here?
Bullitt, actually, your proof is quite incomplete.
The step where you subtract 0.999… from 9.999… supposes that we can concatenate, reorder and regroup two infinite series. Generally that isn’t true (turns out it’s true in this case, but that would need to be demonstrated).
netzweltler: I’m going to assume you are sincere and try one more time.
We use English to communicate. Paragraphs or essays start with a thesis, go on to elucidate that thesis, possibly using metaphors, and conclude with a demonstration.
Yet all we get ever from you is a strange incantation that seems nonsensical. It may in fact be a sound argument, for whatever you’re trying to prove but …
I’m afraid nobody knows what, if anything, you are claiming or trying to prove. You keep babbling about a pen that is somehow moving on a number line. Is it the pen or the number line we are supposed to be interested in? Is the pen a metaphor for something mathematical? If so, what?
What is your thesis? Are you claiming that 0.9999 and 0.3333 are not numbers? Or do you agree they are numbers, but somehow don’t exist on some “number line” you conceive of?
One more point before I ask you to answer. Some of us aren’t sure whether you even realize that 1 = 0.9999. If your “paradox”, or whatever it is, applies equally to 0.3333, it would be better to use it for your example. (BTW, do you know that 0.3333 = 1/3 ?)
So has that been much of the nature of the discussion?
Thanks for that. I remember that proof from a university math professor who showed it on the chalk board. When he subtracted the two infinite series, none of us questioned it. And because none of us questioned it, I’m guessing he didn’t feel the need to explain it. Or even mention anything at all at that step, and although that was 25 years ago I do vividly remember this.
I didn’t finish at a very good school (where that happened). In fact I’ve contended that my junior college offered a better education.
Great. :dubious: And sincerely, thanks again.
OK, here’s another one: What’s the limit of the sequence {1.1, 1.01, 1.001, 1.0001, 1.00001, …}? How would you write that? Is it a number?
NM - my initial guess was wrong
ETA3: Trying again, wouldn’t that limit be 1?
ETA4: 1, that’s my final answer. 
Oh really … did you just make that up … is this some middle school oblation … do you have a citation for your claim here … how in the Name of God can say the infinitesimal is ill-defined and yet still come up with this very explicit definition … one that just happens to fit this screwy set of mathemagical principles you’ve been foisting on the world … wait for it …
Oh … look look see see … now we have yet another made up definition of the infinitesimal … so for something ill-defined it sure does have quite a few definitions … all explicit … all that just happen to re-enforce whatever wrongness you’re handing out this particular sentence … “there is no infinity in infinity” … I could cry …
As I said, it was a crude description … we generally don’t mix discreet terms with non-discreet terms … however, that’s no excuse to completely ignore my contention (which I happen to think was quite rude of you) … every time you move your quill and stop, you have not finished … you can never stop … you must keep moving your quill or you will never find the right answer … “0.999… does not contain a last term” …
Please … explain yourself …
At the point 1 we are moving our quill an infinitely short distance … or are you still stopping after a very very large finite number of times?