What step numbers do you want them to have? I can make them anything.
Okay, I’m going to take a stab at resolving all of the confusion. (I know I’m not the first to try!)
It’s standard to interpret the number 0.999~ as the series 0.9 + 0.09 + 0.009 + 0.0009 … I think we’re all on the same page there. Is this a “process”? No, it’s a series, but there’s an obvious process related to it, namely to start adding up the terms one at a time. If you do this, you get partial sums of the series:
0.9
0.99
0.999
0.9999
…
Loosely speaking, this is good enough to describe the entire set of partial sums, which is infinitely long. As netzweltler has pointed out, the number 0.999~ (or, for that matter, the number 1) does not appear in this set, which makes sense because these things are only partial sums of the series. Even though the set has infinitely many elements, all of them have some finite number of 9’s. By the same token, the set of natural numbers is infinite, yet all of them are finite. Infinity itself is not one of the natural numbers. To emphasize the point, even an infinite list will not get you there, because infinity does not appear in the list anywhere. It’s not a number, and the list is only numbers. And, if we want to talk about 0.999~, infinity is where we need to go.
That’s why this process does not say anything about whether 0.999~ is equal to 1. But that’s okay, because that’s not how the value is calculated. Infinite series are not calculated by literally adding up an infinite amount of terms: that’s impossible. We need to sort of add up all the terms at once. Here’s one way to do it:
The original series:
S = 0.9 + 0.09 + 0.009 + 0.0009 …
Since this is a geometric series, the terms have a simple relationship that we can take advantage of. Here’s the same series, but divided by 10 on both sides:
S/10 = 0.09 + 0.009 + 0.0009 + 0.00009 …
Note that the new series is the same as the original except it’s missing the first term. We can subtract them to leave only the first term, thereby putting it in terms of the entire series:
S - S/10 = 0.9
Solve for S and get:
S = 1
And there you go. Another way to do it is by calculating the limit of the sequence of partial sums, which I’m sure has been mentioned before.
Elegant …
Yes, most people would accept this. And most of everything else you said, since that has been said numerous times already. One point, though. This particular process repeats what Bullitt said way, way back in post #3. The Great Unwashed, in post #535, correctly said that this is not rigorous.
Since it *is *true, the confusion was resolved with the very first substantive response in this month- old thread. The confusion is netzweltler’s and I don’t know what would reach him. I can’t imagine what the word “reach” means in his posts. I reached my limit in interacting with him some time ago. He’s repeating much older posts than Bullitt’s, namely Zeno’s. Zeno knew that the moving pen stuff netzwelter keeps referring to was wrong; he just didn’t know why. Now that we know why, our mystery is why anyone would repeat Zeno’s confusion. I find that confusing. My brain lists.
Thank for pointing this out. My main concern was to show that neztweltler’s point (or what I think it is) is correct but that it does not suffice to determine whether 0.999~ is equal to 1. Of course I’ll be thrilled if they turn up and tell us that my post helped!
My calculation of 0.999~ was very much secondary to the point, but I’m interested in a discussion of its rigors, and I’m aware that it is not rigorous as stated. Actually, I haven’t been able to find the missing pieces yet through research.
I can’t quite figure out what you meant by “the first substantive response”, but in any case, I’d appreciate being pointed in the right direction for a few things, if you know the proper resources.
For one thing, as intuitive as it is, I can’t even find a proof that the limit of the sequence of partial sums is equal to the series sum! It’s always just defined that way. But that’s why I didn’t use that method in the previous post.
Also, I don’t know how to determine a generating function for the partial sums without already knowing what the series converges to. (E.g. in this case it’s 1 - (1/(10^n)) but I don’t know how to derive that other than by eyeballing it.)
As for the type of calculation I used, I’m personally happy with the operations involved as long as the series converges. But, yeah, that’s not rigor.
You definitely refer to this list (written in simple English):
t = 0: I move my pen from point 0 to point 0.9 of the number line
t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line
t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line
…
There is nothing unmathematical about it. This is nothing else than the addition of distances which are elements of the set { 0.9, 0.09, 0.009, … }. And it is about the question what the state of an object (in this case the pen) is as soon as it has been moved all these distances (where does the pen point to?). In case of a finite number of distances it can be clearly defined:
t = 0: I move my pen from point 0 to point 0.5 of the number line
t = 0.5: I move my pen from point 0.5 to point 0.75 of the number line
t = 0.75: I move my pen from point 0.75 to point 1 of the number line
This is nothing else than the addition of distances which are elements of the set { 0.5, 0.25, 0.25 }. After the pen has been moved these distances it is pointing to point 1. This is simple mathematics - nothing else.
If you think using 3s instead of 9s does make it more obvious:
t = 0: I move my pen from point 0 to point 0.3 of the number line
t = 0.9: I move my pen from point 0.3 to point 0.33 of the number line
t = 0.99: I move my pen from point 0.33 to point 0.333 of the number line
…
The thesis is that none of the actions on this list does move the pen to point 1/3. So, what is the pen pointing to at t = 1? Since we haven’t reached a well-defined point on the number line the position of the pen is no longer defined at t =1. Even if we have completed 0.333… at t = 1 we haven’t completed reaching a specific point on the number line which could be named 0.333…
Definitely 1. 1 is a number.
How do infinitesimals come into play? We are moving the pen an infinite number of distances which are elements of the set { 0.9, 0.09, 0.009, … }. None of the elements is an infinitesimal!
Can you also make a complete list of the steps you are suggesting?
Let me try this: I am adding the terms of your first equation (S) and subtracting the terms of your second equation (S/10) term by term:
t = 0: I move my pen from point 0 to point 0.9 of the number line
t = 0.9: I move my pen from point 0.9 to point 0.81 of the number line
t = 0.99: I move my pen from point 0.81 to point 0.9 of the number line
t = 0.999: I move my pen from point 0.9 to point 0.891 of the number line
t = 0.9999: I move my pen from point 0.891 to point 0.9 of the number line
t = 0.99999: I move my pen from point 0.9 to point 0.8991 of the number line
t = 0.999999: I move my pen from point 0.8991 to point 0.9 of the number line
t = 0.9999999: I move my pen from point 0.9 to point 0.89991 of the number line
…
To which point of the number line is the pen pointing to at t = 1?
:smack:
For the bajillionth time, this is not a useful or productive way of looking at this problem. It’s like trying to calculate the area of a triangle by measuring and adding pixels when you have two sides and an angle given - no wonder you’re going to get stuck!
You’d have to know what an infinitesimal is first … which you don’t … I’m not spending the time to instruct you …
I’m politely asking … how wide is your pen?
But the sequence
{1.1, 1.01, 1.001, 1.0001, 1.00001, …}
is identical to this sequence
{2 - 0.9, 2 - 0.99, 2 - 0.999, 2 - 0.9999, 2 - 0.99999,…}
If the first converges to 1, so does the second.
I’m calling that a win for Chronos.
0.9
Ladies and/or gentlemen, I laud thee for thy seemingly infinite patience.
AFAICT, netzweltler’s one and only point is that ∞ is not finite. In a list of pencil locations like (.3, .33, .333, .3333, .33333, …) the point .3333 cannot be reached in a finite number of steps. Whippee!
This is exactly the same insight as Zeno’s Achilles and the Tortoise paradox; Mr. Netz should have just been cut-pasting Zeno’s essay the whole time. At least we might have learned some ancient Greek; as it is we learn nothing.
In a way it is “amazing” how big ∞ is! A googol, 10[sup]100[/sup] ? No, ww’re not getting close yet. One of Knuth’s numbers like 5↑↑↑↑↑5 ? Much bigger than a googol, but ∞ is bigger yet. And Graham’s number can’t even be written simply in Knuth’s notation, yet it is still finite. Think about these huge but finite numbers for a while and you’ll relish a nice simple ∞ !
On the matter of jepflast’s proof: Why the insistence that it is “non-rigorous”? For rigor, isn’t it sufficient to prove that 0.999999… is absolutely convergent? And is not that easily proven, especially by those of us who prove such things easily?
Oh no. The pencil plods along, from patient, to patienter, to patient+0.999 and even patient+0.99999. But we’ll never get to infinite patience. netzweltler has deomonstrated that.
I wasn’t talking to you, septimus. 
Complete off topic, but regardless of what netzweltler writes in response I will read it in the voice of Captain Queeg.
The second post in the thread was merely “Which column is this about?” Bullitt’s post was next and it addressed the question so it was the first substantive post.
The most informative thread here on the Dope is the ancient and venerable .999 = 1?, which started in 2000, got bumped repeatedly and wound up at 2177 posts. Lots of heavy math in that one. And lots of the same tired non-arguments we see in this thread.
#134
#165
#216
#256
#281
#556
Ho hum.