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Oh, God. Please stop. Do not mix mathematics and linguistics, because you will end up in etymology, and then you will go to ontology and epistemology, and by then you're in philosophy, and philosophy is basically the antithesis of math.

Yet I'm not sure if that's quite correct. We round because we're willing to deal with positive error. But .9999... equals 1 exactly, with no error at all; the trick is proving that to stubborn people.
For instance, if you're willing to have an error of .001, then .999~1. This is an instance of rounding. If you're willing to have an error of .000001, then .999999~1, another instance of rounding.

But .99999.... is exactly 1, with 0 error; this is not rounding.

So the relation = really means is functionally indistinguishable from?

How small is nothing?

Well it's just if .999999=1 then the difference between them must be nothing. So is the difference really nothing? or is it just so small that it can be considered nothing?

Quick abusive version of why ∑_{n=1}^{∞} 9/(10^n) =1:

Let ∑_{n=1}^{∞} 9/(10^n) = S.
.9+.09+.009... = .9 + .1(.9+.09+.009...) <= This works but only very carefully
Thus S = 9/10+S/10.
Multiply both sides by 10 to get: 10S = 9+S
Subtract S from both sides to get: 9S = 9
Divide both sides by 9 to get: S = 1.
biggrin
Actually building the reals topologically is unhappy.
It starts with the reals being Hausdorff and ends in tears.

[Warning: lots o' text]

The reals are a field, which means that you can take any two real numbers and multiply them together to get another real number. It also means that if you take two numbers a and b, and b is not 1 or 0, then ab is not equal to a.

So, suppose you have two numbers a and b. We can denote the distance between these numbers as d(a, b) = |a - b|, i.e. the absolute difference between them. Note that |a-b| is a real number, so we can multiply this value by 1/2, thus getting a different real number |a-b|/2.

Suppose that a does not equal b. In that case, we can construct a pair of sets such that set A contains a, set B contains b, the two sets are disjoint (i.e. share no numbers) and both are of positive size (in this case length).

What we do is we make set A all numbers c such that |a-c| < |a-b|/2, and similarly set B is all numbers d such that |b-d|<|a-b|/2.
|a-a| = 0<|a-b|/2, and |b-b|=0<|a-b|/2, so a is in A and b is in B.
For c in A, |a-c| < |a-b|/2, and by the triangle inequality |c-b|+|a-c| > |a-b|, |c-b| > |a-b|/2, so c is not in B. Similarly, if d is in B, d is not in A. Thus A and B are disjoint.
Finally, set A has length |a-b|, as does set B.

So now we have two sets that fit the criteria we want.
Since we can construct these two sets, we call the set of real numbers "Hausdorff".

Now, suppose that we had a sequence that converged to a. What does this mean? This means that if we denote the sequence as a_1, a_2, a_3... then for any positive real number e, there exists an N such that for all n > N, |a_n-a| < e. So the sequence gets as close to a as we want it to, no matter how small the distance (as long as the distance is greater than 0).

But suppose that we tried to make a sequence that converged to both a and b.
Suppose that we've chosen e < |a-b|/2. Then for some N, for all n > N,
|a_n-a| N. But then what about b? For all n > N, |a_n - b| > |a-b|/2 by definition, and thus we can't get it any closer to b. Therefore it can't converge to b; it can only converge to one number.

So what does all of this mean?

Well, look at the sequence a_1=.9, a_2=.99, a_3=.999, etc. It obviously converges to .9999.... since the difference between a_n and .9999... is one tenth of that for a_{n-1}, so going far enough along will give us a difference of any smallness we want.

But it also converges to 1, since |a_n-1| = .1^n, and thus we also have the difference being divided by 10 each time.

So therefore, since it converges to both 1 and .999..., and we just showed that any sequence can only converge to one thing, we get that 1 and .999... must be equal.

And before anyone says anything, yes, I should have showed that ∑_{n=1}^{∞} 9/(10^n) is a meaningful number, and a real number, and included something about Cauchy sequences, but feh on you. I don't have to worry about Cauchy because ∑_{n=1}^{∞} 9/(10^n) is rational, and I'm not going to bother showing that either.

In this post, i will attempt to explain the "problem" with proofs stating .999... = 1. If this debate has shown anything, it has shown people do not understand the nature of mathematics.

The most common proof i see is .333... x 3 = .999..., and we know 1/3 x 3 = 1. To be fair, i think the proof should be changed to a number less optically pleasing, such as 0.14285714... × 7 = 0.999.... In either event, the result is meaningless, as provided by A. N. Walker:



In a similar manner, the proof in which we multiply .999... by ten is visually attractive, but is justified only by making special cases. At which point there is no "proof." The other common example was explained in my post immediately prior to this one.

Every proof offered to prove .999... = 1 is nothing more than an optical delusion. They do well to convince high school students, but nothing more. Each is inherently self-referential to the system for which is defined.

This is not to say the two numbers are different. In any system following Archimedean mathematics the two numbers will be the same. The reason is in standard analysis infinitesimals, infinitely small numbers, do not exist. This is axiomatic to the system, meaning it is accepted as true without proof. A different system may use different axioms, which makes it quite possible for .999... to be different than 1. Such systems do exist, with the most known being surreal numbers.

This debate is not one of proofs, but one of mathematical systems. In the system taught in elementary school, the two numbers are equal. Because this system is so ingrained into the mind of society, it is treated as the the only truth. This problem is so predominant those who teach mathematics and science often do not know better. When students encounter this problem, they are given these "proofs" which do not answer anything. This is a grave injustice, which can easily be avoided by encouraging an understanding of non-standard mathematics.

(Other proofs exist, using Cauchy Sequences and Dedekind Cuts. Both methods derive their validity directly from the "standard" system, making them just as tautological as the three proof i already mentioned.)

Edit: If you want information on Xeno's paradoxes, i would be glad to do so. However, i think it would be best done in another topic.