I hadn’t read it that way. If you’re right, then I’m sorry I was a bit cranky in my last response.
I don’t know why you want to continue insisting that I don’t understand the things
that I am talking about. You seem to want to raise yourself above me and that
is your starting point of arguing with me. As if I was inferior to you to begin with
so I must be wrong, a some kind of abusive ad hominem attack against me.
And you try to prove your case so that by admitting that 1=0.999999…shows
that the person understands. And if someone says that 1≠0.99999…he
does not understand, and so he must be inferior to him who understands.
Somehow you ridicule me and my understanding, although what you
understand and what you do, is you try to force people to accept your way of thinking. That is called brainwashing.
Ok, I have included a reference to a document by James Tanton, and I mostly agree
what he has written, especially that …999999=-1, so it is a bit arrogant to say that Tanton says the opposite to what I am trying to prove.
I disagree with Tanton only when he writes that …999999.999999…=0
I repeat what he writes:
…999999.999999…=…999999 + 0.999999…= -1 + 1 = 0
so that he writes here 1=0.999999…and I agree with you that here Tanton seems to actually say the opposite to what I am trying to prove. So in this sense
you are right, but why should I agree with everything Mr. Tanton writes, why
I should agree with you that 1=0.999999… Why I should not be allowed
to think otherwise?
Do you think that 1=0.99999…is the absolute truth?
Prove that …999999.999999…=…000000.000000…and you prove me wrong, show that you understand the things you are arguing about. Now every digit is different, but can these numbers still be the same? What is the context
in which the are the same?
I already said it a few posts ago.
If 1/3=0.3333333333…then 1/3 is equal to 0.33333333…
If 1/3≠0.3333333333…then 1/3 is not equal to 0.3333333…
Having read the document by James Tanton, I am somewhat confused as to what you do and don’t believe about the nature of the document. The characters, Albert, Bilbert, and Cuthbert, (A,B and C) are not all espousing truths as believed by James. They are engaged in a discourse, and are arguing about the question. Nowhere does James anoint one of the three as exposing the “truth”. Indeed you never see anything other than conversation attributed to either A B or C. What James has written is an example argument about the problem, and some of the common ways of arguing it. There is nothing to say he believes any of these arguments to be correct.
Indeed, towards the bottom we read
So James has written that the characters don’t think …99999 is -1.
I am at a loss as to why you think James is somehow suggesting otherwise.
The argument that Tanton makes, that …99999999 = -1 is crucial to his proof
that 1=0.9999999…and this fact you are using against me.
Tanton writes that 0 = -1 + 1 = …99999999 + 0.999999…
which seems to be valid to him. He is assuming that he is right. But does
assuming to be right prove oneself right? Does assuming that 1=0.9999999…
prove that 1=0.99999999…? That is what you are doing.
Now, prove that
…999999.999999…=…000000.000000…
Well, we can reason for this in a perfectly analogous way to how Albert originally reasons for …999 = -1. In fact, Cuthbert presents the argument that …999.999… = 0 almost immediately afterwards.
Why do you so readily agree with Albert but not Cuthbert?
It seems to be a full-time job defending the notion .999… = 1. Since it’s now graveyard-shift in North America, let me take a turn from another hemisphere.
7777777, would you answer these questions?
(1) If 0.999… is a notation for 1-d (where d is an infinitesimal) then what is a comparable notation for (1/3)-d ? If you answer 0.333… then I’ll ask what is the notation for 1/3? Is it the same as (1/3)-d ?
This question demonstrates that the 1 = 0.999… peculiarity is less about the nature of numbers than about the peculiarity of repeating specifically 9’s in a decimal system. In base 2, (1/3) has but a single representation, albeit a repeating one. (1/2) has two representations: 1/2 = 0.1 = 0.0111111… This situation reverses in base 3.
(2) David Hilbert once wrote "“Mathematics is a game played according to certain simple rules with meaningless marks on paper.”
7777777, is it your claim that
(a) Hilbert et al play their game poorly,
(b) A different set of simple rules would be “better,” or
(c) Hilbert is confused.
(3) Explicitly summing the infinite series .9 + .09 + .009 + … to get 1 is impossible in a finite world. Zeno pointed this out almost 25 centuries ago. Was Zeno right? After 25 centuries, are you bringing something new to the debate?
(4) It is possible to design a number system with infinitesimals. Levi-Civita did it and you can play with a Levi-Civita field calculator here. Levi-Civita’s purpose, however, had nothing to do with the “mysterious” .999… = 1
I continue to wonder if distress at the modern definition of real numbers comes from its modernity; greats like Copernicus and Galileo may have barely understood the notion of “real number.” That’s why I like to refer to the Axiom of Archimedes, phrased in a translation of Euclid as “Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.”
A simpler way of stating this Axiom might be: Given 0 < a < b, there is an integer N with Na > b."
My challenge to 7777777, unless he wants to claim Archimedes was wrong, is to write
0 < 1 - .999… < 1 and find the N with N*(1 - .999…) > 1.
In fairness, Archimedes would not have recognized a decimal expansion like “.999…” either. It would not be claiming Archimedes was wrong to decide that whatever “1/(1 - 0.999…)” was to denote, it was not the sort of thing Archimedes was thinking about. [It would be an erroneous understanding of how others use this notation, but it wouldn’t be pretending oneself smarter than Archimedes, per se]
Don’t underestimate Archimedes! He developed his own novel numeric notations and, whatever the notation, certainly was very familiar with notions like .9 + .09 + .009 … (BTW, he credits the Axiom of Archimedes to Eudoxus.)
Anyway my purpose was not to shut down debate but to give a fresh perspective. Many Dopers fall back on the 19th-century work of Dedekind and Weierstrass to defend .999… = 1. I point out that it has a much more ancient origin. If infinitesimals were “real,” many constructions of the ancient Greeks would be invalid.
It seems that it is Cuthbert who is inconsistent.
He says that …999999.999999…= 0
And then he says that …999999 + 1 = 0
These can’t both be true (unless, of course, if one assumes that 1=0.99999…
is true, but since we are proving whether it is true, we cannot think that
assuming it true proves it true).
What is true is:
…999999 + 0.999999…= …999999.999999…
and
…999999 + 1 = 0
They can both be true, and as a result one might conclude that 1 = 0.999…, not by circular assumption, but as a consequence of the reasoning that had originally led one to …999.999… = 0 and …999 + 1 = 0. Observing that X and Y lead to Z does not mean one ought say “X and Y can’t both be true (for their truth would have us assume Z, and, of course, we can’t assume Z, since we are interested in seeing whether we can prove Z).” That would be silly.
This isn’t an answer. It’s just two tautologies.
What is the decimal expansion of 1/3? Is it 0.333… ? If it isn’t, what is it? Does 1/3 not have a decimal expansion?
Only if you’ve already pre-judged the “correct” solution. Your own biases are getting in the way here.
BTW, you never did provide the context for your notion of equality. Mathematical discussions are pointless if you start with different definitions from everybody else but never bother elucidating. And if you do have different definitions, it’s no surprise you come to different conclusions.
Worse, you still never really explained how or why your proposed cyclic number line (the one with …999) is supposed to be applicable to the number line everybody else is using (the one without it). As I’ve noted a few times now, show your work. I’m not asking for ultimate rigor, but more than zero would be good.
During the span of this thread the participants have posted dozens of proofs that in fact, 1=0.99999… It’s well accepted mathematical knowledge because there is lots of good, accepted, logical reasoning that backs it up. There are number systems where it isn’t true, but those number systems aren’t in common use and they really don’t apply here.
You don’t accept that 1=0.999999… Several people in this thread have come in with the same idea. Some have been convinced that they were in error, some have not. But the arguments you are presenting don’t hold up under scrutiny. You’re starting from a document that doesn’t say what you think it says, and building a whole story on a shaky foundation.
There are many knowledgeable mathematicians in the thread that are trying to show you the flaws in your argument. Please read their comments with an open mind and you might be shown the light.
Or not, it doesn’t seem to often happen for the new contributors to this thread. But we’ll see. I have a good feeling about you.
I’m curious. Tanton gave several arguments to show that …999 = -1. Which ones convinced you?
How hard is this? Three one-third fractions = 1.
This thread got to 25 pages?
It’s probably already been done in this thread, but what the hey:
Lemma: 9/10 * 0.99999… = 0.9.
Proof: 1/10*(0.999…) = 0.0999… (since for decimal numbers in base 10, division by 10 and shifting everything one decimal place to the right are equivalent). And 0.9999… - 0.0999… = 0.9, by virtue of the fact that 0.09 - 0.09 = 0, 0.009 - 0.009 = 0, etc. all the way down the infinite string of nines. So the only place without a zero in it is the 9 in the tenths place.
So .9999… - (1/10)(.9999…) = .9. So .9 = 9/10(.9999…).
Proposition: .9999… = 1.
Proof: Start with the Lemma, and multiply both sides by 10/9.
That probably won’t convince anyone, but that’s life. 
Let’s put it this way: if you look at the banner of the main Straight Dope webpage, the words “Fighting Ignorance Since 1973” appear.
Now notice the words in the parenthetical directly underneath. 
It’s a little harder than you think. You’ll note that not everyone automatically accepts 1/3 = 0.333… either.
The problem with this approach is that if you say:
1 = 3 * 1/3 = 3 * 0.333333333… = 0.999999999…
You are implicitly assuming that the algorithm you use for multiplying finite decimals, also works for infinite decimals. It does work, but that fact is not “obvious” and it gives people who don’t want to believe the conclusion an out, hence 25 pages.