like I said above – 1/3 is a specific value, 1/4 is a specific value. 0.25 is a specific value, it terminates. 0.333… is a non terminating repeating decimal, therefor how can it have a specific value if it never stops?
That’s exactly the point, Kinoons–it doesn’t have a specific value when represented as a repeating decimal. But it is equal–exactly equal, in your sense–to 1/3. When dealing with the repeating decimal, we refer to the definite value of it as a limit. The limit of 0.999… is 1. It really is just a difference of representation.
Let me try one more proof: between any two distinct real numbers (meaning numbers with decimal expansions) there are an infinite number of real numbers (actually, a non-denumerable number of real numbers–an uncountable infinite). There are no real numbers between 0.999… and 1.000…, so they’re the same number.
Since the anti-equaltarians have admitted they don’t care about pesky proofs, I suggest this be moved to Great Debates.
I second that.
All right, one more time with emphasis.
Do you remember how to do long division by hand? Good. Now get out a piece of paper, and divide one by four. First you get a two, then a five, then a zero. Your teachers told you to stop when you got to zero. You know why? Cause you’d just keep getting more zeroes forever. Same way that you keep getting threes when you divide one by three. Not just similar, mind you. Not even just close. Exactly the same thing’s going on here. Just cause we don’t normally write the zeroes out doesn’t mean they aren’t there.
.25 is equal to 1/4 in the limit the same way that .333… is equal to 1/3 in the limit, except that one series converges after a finite number of terms and the other doesn’t. This difference is absolutely inconsequential.
You still haven’t explained why 1/2 = 2/4 doesn’t bother you. How about why 1/3 = 3/9?
I’d like to offer an analogy (I wish I could call it a proof but it really isn’t one) to have one more go at convincing the anti-equalitarians.
Influenced by Xeno, take a look:
Take 1 + 1/2 + 1/4 + 1/8 + 1/16 and keeping going forever. When you are done, what is your sum? I mean exactly. If you can’t say exactly what the sum is, I haven’t gotten through to you.
I will grant that this infinite series is a different concept from 2 (oops, gave it away 
) but it is not a different number.
Okay here is something, maybe closer to a proof. Humor me a little - I’m going to break some rules but I hope it will be worth it.
First, let’s do some subtraction (and in all cases, my ellipses mean “continued to infinity”).
1.0… - 0.9… = 0.0…1 = Z (this is where you have to indulge me, I know you can’t really follow infinity with single number but if we temporarily forget that you’ll see where I’m going)
Now let’s recall that
10A = A
only if A = 0
If A is non-zero it cannot equal a product of itself and non-one.
So let’s return to Z.
0.0…1 x 10 = 0.0…10
Look at that. 0.0…10 Drop the trailing decimal …
Z = 0
QED
But I bet I’ve broken so many rules with that that the real mathematicians are reaching for the Rolaids. 
There were some issues with the dots being surrounded on either side earlier in this thread (basically, it’s a no no.). Other than that, it’s not bad.
All of the pro-equalitarians proofs say that the difference between 1 and .999… eventually gets so small that it doesn’t count?
Because if you take away infinity you lose the possibility of equality. And if you keep infinity you’re stuck with infinite fractions.
If you go out a gazillion fractions (9’s) you’re no closer to infinity than when you started. And the fractions still don’t equal zero.
Or is infinity approachable? Do I need to revisit infinity?
Peace,
mangeorge
You got a problem with omega + 1? (This is in fact exactly what happens when you work with surreals.)
I’m thinking of starting a MPSIMS thread to discuss the Axiom of Infinity. Anyone interested?
Here’s another way to think about it (utilizing the 1/3 comparison)…
When rounded up, the decimal shorthand for 1/3 is often written as .334 (or somesuch). When rounded up, the decimal shorthand for 2/3 is written as .667. And, finally, when rounded up, the shorthand for 3/3 (the .99… that is giving us so much trouble) is written as 1.
Why? Not because it’s rounded (necessarily), but because it’s easier for people to quanitify it that way. Because, ultimately, the .333… or .666… will ultimately become something MORE than .333 or .666. Same with .999… IF it continues on for infinity.
Hmmm, well maybe you do and maybe you don’t. Let’s take a look at 1/3 = 0.3rep as an example. As you say, if you add up the values of a gazillion decimal placeholders (3s in this case), you’re no closer to infinity than when you started. And the sum of the things you added up don’t equal 1/3. Yep, correct.
However, where you’re going wrong is in thinking that you have to add up the values of an infinite number of decimal placeholders one at a time. You might want to revisit the concept of an infinite series, where an infinite (and not just “large number”) of terms can be added together to produce a discrete value.
This question is kind of like Zeno’s paradox in reverse. What you are saying is this: “If you move your finger on the number line from 0 to 1/3, first you have to cross 0.3. Then you have to cross 0.33, then 0.333 and 0.3333 and 0.33333 and so forth, infinitely. Since you have to do an infinite number of things in a finite length of time, your finger never reaches 1/3.” Which is of course ridiculous; it’s easy to touch 1/3 on a number line.
Of course you can substitute 9s for 3s and 3/3 for 1/3 in the argument above, should you so desire.
Therefore, kinoons = Arthur Dent? 
And Dr. Matrix, you are the king of cool sig lines.
You can finish the problem quite easily. You have to realize that there is a pattern to the numbers in the answer. That pattern is: “All numbers are three.” No matter how long you keep up the long division, you will always get another 3. Just because the list is infinite doesn’t mean I don’t know what the next number will be, or what all the numbers are. 1/3 has a result. The result is that it is equal to a infinitely long decimal where every number after the decimal point is a “3”.
You’ve just described a converging series, which was nicely proved to be equal to 1 quite a few posts up. Yes, this will only reach 1 in the infinite limit, but we’re dealing with infinity all over the place here anyway. Your disbelief doesn’t invalidate the multiple proofs.
(also, the particle thing is a terrible analogy, as there is no correlation between the two, and particles can occupy the same space…after a fashion)
Nah, Omega + 1 is fine by me. Other people had a problem with it, and I’m not sure it’s the clearest way to explain this.
Go ahead and start that MPSIMS thread.
Fun stuff!
It seems both the equalitarians and the anti-equalitarians have reached one point of agreement: that “.9rep” and “1” are understood as distinct (and thus different) concepts.
It may be that even the anti-equalitarians are willing to concede that our system of mathematics mandates, beyond a shadow of a doubt, that the two symbols represent the exact same quantity.
The disagreement at this point appears to be:
Is “a certain number” a reference to (a) precisely one concept; or (b) precisely one quantity.
If (a)–> Kinoons and his followers have a good case.
If (b)–> They don’t.
I’ve explained that. Perhaps you’d care to read these 150+ posts again…
sounds like I’m a cult leader.
I did. I went back through and read all the posts. You have not even addressed anything remotely like this. If you have an explanation, share it. If you explained it before, quote it. But don’t count on my laziness to carry you.
I will not agree that .9… and 1 are different concepts. Both are infinite series of base 10 coefficients. They converge on the same quantity.
“A certain number” is multiple concepts, depending on whether we’re talking about naturals, rationals, reals, complexes, or other things. There is also no single notion of “quantity” that applies to all these things, because the equality relationship denoted by “=” is different on each set.
Assuming we restrict ourselves to the reals, it would be more accurate to describe “a certain number” as one quantity.