Yes.
I just want to say that I do not like the proof that Libertarian gave. It’s the classic proof, but it’s informal, and you can use a similar-looking “proof” to “prove” things that are wrong. You shouldn’t be subtracting two series when you haven’t even proved they converge. But I won’t argue with the result. That’s all. 
Since the OP has been pretty well answered, can I expand the question a little…
0.9… = 1.0…
where … indicates the infinite repetition of the digit to its left.
Has been shown above.
But in considering the real number line, you can consider that 0.9… aproaches 1.0… as the repitition of the disgits tends to infinity, but can not become 1.0…
Is there any value in equating 0.9… and 1.0… to two adjacent real numbers? That is two real numbers that have no real numbers between them.
I find this a way to consider the real number continuum, and I wonder if this is terribly flawed.
1.0… = 0.9…9 = 0.9…8 = 0.9…7
where 0.9…8 would be the imagined number that begins 0.9… but whose ‘final’ ot infinitieth digit is an 8.
it can be seen that this set could have a countable infinite number of entities (0.9…7… would be a valid entity) each precisely equal to 1.0… Which at least to me gives me a sense for the difference between a continuum (the reals) and a countable infinity.
Please comment or point me to a web site that exposes the problems with this way of thinking.
Cheers, Bippy
There is no final digit in the standard theory of decimal representations.
Another one of the things for the web kooks to stay away from : surreal numbers. These are discussed in the first thread Wendall Wagner gave the link for; on the second page MrDeath gives a good explanation of what they are. Though 1.0 and 0.999… would be unique surreal numbers, again these are NOT real numbers (hence the name). As has been shown, these forms do represent the same number in the reals.
A brief quote may help explain a bit without having to pull up that thread :
Originally posted by MrDeath in another thread
Well, strictly speaking, a single number can’t “approach” anything; it just sits there having whatever value it has. A series or sequence of numbers (like .9, .99, .999, …) can approach a limit, and a repeating decimal like .9999… can be defined/understood to mean the limit of this sequence, which is 1.
Under the standard conception of the real numbers, there is no such thing as “two real numbers that have no real numbers between them.” If two numbers are not the same, then there are always infinitely many other numbers between them (like (a+b)/2, for instance).
You may be wanting to use “infinitessimals”: “infinitely small” numbers that are greater than 0 yet smaller than any other number. In the very early days of Calculus, mathematicians used this idea to explain and develop what they were doing, but it became apparent that this didn’t really make a whole lot of sense and didn’t hold water logically, so eventually the ideas of Calculus were reformulated in terms of the modern delta-epsilon definition of a limit, which put things on firmer ground.
Fairly recently, as I understand it, something called “nonstandard analysis” has been developed, which is an attempt to actually define and use infinitessimals in a logically consistent way.
Sorry about being unclear I meant by
you can consider that 0.9… aproaches 1.0… as the repitition of the disgits tends to infinity
The series 0.90…, 0.990…, 0.9990… continuing as the number of 9s tends to infinity.
Is there a form of logic interpretation where we can say that
1.0… = 0.9… AND 1.0… > 0.9… AND NOT 1.0… < 0.9… in a consistant fashion?
Clearly this doesn’t map all real numbers (as pi for minstance cannot be expressed in this way) but is there a sence in considering a last digit, or last set of digits after an infinite repitition of digits, even though they must have a value equal to zero or infinity.
( 1 integer as a real in decimal notation could be considered 0… 1. 0…, any number not starting 0… is numerically infinite, so we drop the 0… whn writing a finite number )
Cheers, Bippy
p.s. does anyone have a method to overline a number on this board to allow for the more usual symbol for repeating digits.
GAH!!! Must flush memories of Real Analysis from brain.
GAH!
I hate you ultrfilter! 
10 / 3 = 3.333…
3.333… + 3.333… + 3.333… = 9.999…
Yep. This exact topic is one of the things I often cite when asked whether I’ve changed my mind about anything due to my activities here at Straight Dope. The proof is both valid and sound. After seeing it, there was nothing else to consider. I had to change my mind — much like some atheists who have seen the modal ontological proof of God’s existence.
I must again disagree. 0.999… is not arbitrarily close to 1, or “as close to 1 as it is possible to get” or “close enough to 1 that you can use 1 in its place”. These imply that it is some sort of approximation. The math clearly shows that it is not, and that 0.9999… is identical to 1. They are the same number. It certainly is a quirk of the number system that this is so. In base 9, for instance, you can use the same reasoning to show that 0.88888888 … = 1
That’s why I said “loosely … in application” In application, you do NOT have an infinite series.
Look, .999… means the sum, as i goes from 1 to infinity, of 9/(10^i). If you want to call that a number, it’s fine with me. But it’s an infinite series. Loosely, in practice, if you want to think of it as a number, and not an infinite series, you can think of it as a number as close to 1 as you need it to be.
BTW, this is not a quirk of the decimal system, as the more formal representation of the infinite series above shows. You can replace 9/(10^i) with k/[(k+1)^i for any k and still get 1 as the value of the infinite sum.
No, not exactly. Exactly, it’s the limit (as n goes to infinity) of the partial sums for i=1 to n. And a limit is a number. It’s defined to be a number, with a definite value.
When we say “the sum from 1 to infinity” it’s shorthand. We really mean a limit of partial sums.
You’re right. My bad.
this is not true
2/9 = 0.22222(repeating) and 2/9th at the end as a fraction of the last digit in the repeating series as the number of 2’s goes to infinity
I’m not at all sure what you’re trying to say here.
So I know you warned the Web Kooks, but I felt that I had to comment here. By briefly reading about p-adic numbers, it seems that they are, in some sense, akin to writing numbers in a given base. Thus, if we consider 2-adic numbers,
1 + 2 + 4 + 8 + …
becomes
11111…
Which set my little computer-science brain ticking: If this were a register, and we were dealing with twos-complement numbers, once we filled the register with 1s, we’d have the representation of a decimal -1. So extending this to the magic “infinite register”, it seems obvious that the only way to represent -1 is by having an infinite series of 1s, or, in standard numbers, an infinite summation of powers of 2.
But then I figured that p-adic numbers probably came first, and so twos-complement notation uses the properties of 2-adic numbers to do its magic, instead of vice-versa. I always wondered how twos-complement worked (it always seemed sort of like magic), and this gives me at least some basis for understanding.
Is this important? Is this even correct? I dunno, but everything’s spinning around in my head, and it’s beautiful! Woo!
(oh, and Lib? Thanks for mentioning that argument. Now I’m gonna have trouble falling asleep tonight)
Say this to yourself over and over.
Infinity means unending. There is no last digit.
Infinity means unending. There is no last digit.
Infinity means unending. There is no last digit.
Infinity means unending. There is no last digit.
Infinity means unending. There is no last digit.
Infinity means unending. There is no last digit.
…
I think twos-complement notation owes more to plain old modular arithmetic than to p-adic arithmetic. A 8-bit integer register (for example) is just performing arithmetic modulo 2[sup]8[/sup], which is why -1 is represented as 2[sup]8[/sup]-1, or 11111111 in binary. There is probably some way in which 2-adic arithmetic can be thought of as the “limit” of arithmetic modulo 2[sup]n[/sup] as n goes to infinity, and a number theorist could probably explain it detail, but I can’t.
It’s certainly an interesting correspondence, though.
Is there any fancy mathematical theorum needed to justify:
.9999… - .9999… = 0?
Or is .9999… just another number? It’s been quite awhile since that kind of stuff was second nature to me.