Talk:0.999...=1

sorry, but 1, 2 and 4 are rubbish. 1 and 2 are completely useless - why is 0.333.. = 1/3 a truth if 0.999.. = 1/1 is suspect? unless you expand them with full division algorithm, and even then it's kinda dodgy, since you have to really formulate it in terms of induction to justify going to infinity. And 4 is invoking density of real numbers to prove this basic fact.. seriously.. Why is this true for reals, but not true for integers, for example? I mean, I know why, but it's pretty far from an intuitively obvious step..

Good Point

Think they should be taken down, or edited?

I vote for making a note of the issue but leaving the proofs. Maybe move 1 and 2 to the bottom, so the more convincing proofs come first. And density of real numbers is perfectly valid, even if it seems like overkill.--Cynic 23:04, 27 April 2008 (UTC)

I'd say keep the 2 formal proofs - as a sum of geometric progression, and the 'multiply by 10' version at the top, and indicate increasing informalness as we go further down. It really depends on your definition of the reals whether density is a basic and/or valid concept to invoke in this case.. If you defined the reals to be those numbers that we can write in (possibly infinite) decimal expansions, then you are more or less done with the proof, but your definition is flawed, since you never actually explained what you mean by an infinite expansion (that's what we are trying to prove, after all)... If, instead, you only have axiomatic completeness of the reals - ie, every Cauchy sequence has a real limit - then it's already a non-trivial fact to prove that between every 2 distinct reals there's a third one (ok, it's trivial, their mean, but you need to invoke several axioms to prove that 1. it's different from either and 2. it's real) - and then you still have to explain why that third one must have a decimal representation, which must be higher than 0.9999... This is a pretty hard concept for people to swallow - a lot of them seem to think in some non-Archimedean space, with infinitesimal numbers and the like. Ivancho 14:20, 28 April 2008 (UTC)

More explanation

It would be nice if proof 6 had a little more of an explanation. --Joe 14:41, 11 August 2008 (UTC)

Proofs 5 and 6.

Proof 5 wasn't a proof, so I removed it. While it is a compelling argument to help convince someone that .999... = 1, the person first has to prove that there does, in fact, exist no real number between .999... and 1.

Proof 6 seemed to be a circular argument, saying "if .999... lies on [0,1) then 1 lies on [0,1)" which assumes .999... = 1.

I think we established that 5 was a valid (if not fully explained) proof above. Assuming we can prove the infinite density of real numbers (which we can, even if we don't here) this we can use it here. However, you are right on 6, that seemed more like a proof that $.999...\neq 1$ since it put 1 on an interval that is defined as not containing 1. Of course, it assumed .999 was both less than and equal to one, so it failed at even that. --cynic 02:36, 15 August 2008 (UTC)

Proofs 2,3 and 5

Proof 5 was hardly valid. By definition, between any two unequal reals there exist infinitely many reals. Saying "obviously there exists no number between .999... and 1" is begging the question, as it supposes the equality in the first place.

Proofs 2,3 and 4 are all equally terrible. We first need to prove .333... = 1/3 and .111... = 1/9 in order to conclude that .999... = 1. With proof 4, we need to show that 10(.999...) = 9.999...

In reality, proof 1 is the closest to rigorous, and simply needs to reference the "geometric series" test to explain why the series converges. We should refer to the definitions of decimal expansions. We should explain that the decimal (where a_n represents a single digit) .a_1a_2a_3... is equal to a_1/10 + a_2/100 + a_3/1000 + ... and that .999... is equal to the infinite series in proof 1 by this definition.

I agree that some of them have not been rigorously proved, but they are nevertheless still correct. I think that one of 2 and 3 should be deleted, since they are basically the same thing using different numbers. The fact that 0.33333... = 1/3 can quite easily be shown. This brings rise to a good question, what should we allow our base assumptions to be for any one proof, and how rigorous do we have to be? The fact that 10*(0.9999...) = 9.99999... probably does not have to shown, but would of course be nice if it were to be proven elsewhere on its own page. In terms of the last proof, I think that it does need a little bit more of an explanation, showing how and what ideas it uses. --Joe 12:59, 15 August 2008 (UTC)

"... rigorous do we have to be?" Fully, in my opinion. Every line of every proof ought to have a justification somewhere else in the database. If such a justification does not yet exist, leave the tag for it and leave it for later or for someone else to do it.

Ultimately everything ought to be able to be referenced back to the axioms (ZF or Propositional Logic or whatever). I'm doing what I can to build up from the bottom (it'll take me a while) but once I've done what I've got planned we'll have an axiomatic basis for the real and complex number fields and thence continuity and a basis for rigorously proving the validity of calculus. Ambitious? Yep. --Matt Westwood 17:56, 15 August 2008 (UTC)

I agree. A fully axiomatic approach from the bottom up is really the best way to ensure the validity of each proof. By the way, you're doing an excellent job Matt. --Spoon737 18:13, 15 August 2008 (UTC)

Thx. Feel free to comment on specifics - I know there are plenty places where it can be improved. --Matt Westwood 18:53, 15 August 2008 (UTC)

While I agree that a fully axiomatic approach is an excellent long term goal, we do need to realize that building up all of calculus is going to take a looooooong time. On an unrelated note, I prefered proof 2 to proof 3 for absolutely no reason, so I took down three since they were identical (and renamed the proofs so we don't have to keep renumbering them every time we make a change).--cynic 02:57, 16 August 2008 (UTC)

Proof 5

The proof still fails to show that there exists no real between .999... and 1. You can't simply assume that there exists no real number between .999... and 1. It has to be shown.

Just to be clear, you're not actually arguing that $0.9999\neq 1$ or that there are two distinct reals that are not separated by some number, are you? If you don't like a proof, fix it. That's what the whole concept of a wiki is about, anyone can edit it. The rest of us will stick to the sections of the site that we enjoy working on more, or doing other things with our time. That said, I will take a solid look at the proof after I get back from vacation.
P.S. add your signature by clicking on the 2nd button from the right above the edit box so we know who's leaving what comment.
--cynic 03:52, 17 August 2008 (UTC)

No, it's true that .999... = 1. But the "proof" is taking for granted the fact that there exist no real numbers between .999... and 1. It's entirely reasonable to use this as an explanation for someone in doubt; asking a layman "can you think of a number between .999... and 1?" in an effort to persuade him. But it's not a proof, as the premise, that there exists no real between .999... and 1, is not proven. In order to say ".999... = 1 because there exists no real number between .999... and 1" we must first be able to say "there exists no real number between .999... and 1." And realistically, the only reasonable way to show that is to prove the equality in the first place.

--Kal 18:21, 17 August 2008 (UTC)