I quite agree and apologise for any confusion I may have caused. It doesn’t help that I mistook my audienced and pitched at too low a level, for which I can only apologise.
pan
But seriously, couldn’t you just take this isomorphism and declare it to be an equivalence relation (between Z and the obvious subset of R)? Then, in matters of the relevant algebra, you could rightfully say that 1[sub]Z[/sub] = 1[sub]R[/sub] and also recognize that the two are not identical?
Does that tidy this up?
Nothing changes the fact that real numbers and integers are objects of different type, so they can’t be the same. Furthermore, the equivalence you want is already contained in the isomorphism, so why introduce something new? Lastly, equivalence relations only exist on one set, not between two, so you’re out of luck there.
Wow, this thread is awesome. Not only did someone already make any argument I could think of as to why .999… = 1, but some of you, mostly kabbes and ultrafilter, started discussing things I never even heard of. I figured I at least knew a little of most areas of math, since at my school a Computer Science major was only a few classes different from a Math major. I feel like Neo watching Morpheus jumping across skyscrapers.
If you divide 1 by 3, no matter how many decimal places you go out, you’ll never get an exact answer, however 1 divided by 3 does exactly equal 1/3, so:
1/3 != .333…
2/3 != .666…
3/3 != .999…
each decimal is infinitely less than the fraction, just the same .999… is infinitely less than 1.
emcee, read the rest of the thread it has been proved in several different ways that 0.999… = 1
I guess it depends on how you look at it, if you assume that if you continue the decimal place out forever it will eventually equal the limit, than there is no question .999…=1, however if you assume there is an infinitely small amount between a repeating decimal and a value, than most of the proof here is invalid.
I mistyped, this is what I meant to say:
I guess it depends on how you look at it, if you assume that if you continue the decimal place out forever it will eventually equal the limit, than there is no question .999…=1, however if you assume there is an infinitely small amount between a repeating decimal and the limit, than most of the proof here is invalid.
Th
There are people here who are far more qualified than me to address your post, but here goes:
There is no need to assume that “you continue the decimal place out forever.” The notation “0.999…” means an infinite number of decimal places. All of them are to be considered. There is therefore no “infinitely small” difference between 0.999… and 1. They are equivalent.
P.S. I’m not sure what happened with that last post. 
Yes, emcee, but it has even been shown that assuming that there is an infintessimal difference between 0.999… is false.
Actually I should rephrase that to “any difference” not “infintessimal difference”.
I’ve reread this many times, and the rest of the discussion, and as near as I can tell, it is not true. I’m not even sure what distinction you’re trying to make here. If you think about it, there is no difference, much less a crucial difference. Race Bannon, in his original post about it, was very careful to avoid discussing the even integers as a subset of the integers. He specifically called them out as separate sets.
There is a subset of the reals that are called the integers, just as there is a subset of the integers called the even integers. It doesn’t matter that there are also mathematical constructs with those same names.
Otherwise, 2[sub]Z[/sub] does not equal 2[sub]even[/sub], just as 1[sub]Z[/sub] does not equal 1[sub]R[/sub]. But they are equal if they are taken as subsets. There is no difference in the two examples, much less a crucial difference.
I’m with Race Bannon on this one.
You are almost there, you seem to fail to grasp however that within the “…” concept the idea that you “continue the decimal out forever” is already explicit. Therefore not “eventually equal the limit” but rather “defined to be the limit.” It only depends on how you look at it if you don’t understand what the “…” means in a mathematical context.
If there is no infinitely small difference between .999… repeating and 1, than by that logic .000…1 = 0, since this is the length of each side of a circle in euclidean geometry, than the circumference of every circle should be 0.
You have a misstep in your logic. When you say “than by that logic .000…1 = 0” you are not really following that logic. Notice, your decimal representation has a terminating 1.
Why, I don’t know…
0.000…1 is not mathematically meaningful. How many places after the decimal point is that 1?
Ah. At last I have something to add to the thread: The ‘1’ is, of course, an inifinite number of places after the decimal point. Where’s my banana? 
Exactly right. If you assume something false, you can conclude anything you want.
Under the standard construction, the even integers are a subset of the integers, but the integers are only isomorphic to a subset of the reals.
You’re right–it’s not a crucial difference, and if you ignore it, you’ll never suffer for it. But it’s there, unless you use an alternate construction. Most people don’t worry about things like this, but I do.