.999 = 1?

Factual Questions
361

erik150x, I’ve done you the favor of explaining on what interpretation of things your account can be made to make perfect formal sense.

And you, in turn, understand and accept that there are definitions relative to which everyone else’s accounts of things make sense, yes?

So what is it that you keep looking for? You want some extra proof that 0.999… is or is not actually equal to 1? That infinitesimals do or do not actually make a difference?

This is a nonsensical thing to look for. It’s like saying “Look, I get that you’ve made up a rule that says kings can move one square in any direction. But can you prove that kings actually move one square in any direction? Don’t just point to the rule you made up; can you prove it without that rule?”.

In mathematics, everything is ultimately an appeal to definitions. Everything.

There’s nothing to argue about. In one system, the rules are chosen so that 0.999… is inarguably equal to 1 and infinitesimal differences count as zero. There’s no question of “How do you know you’re allowed to do that? What gives you license to do that?”. It’s explicitly allowed by the rules. Yes, the rules we made up. That’s what math is; you make up the rules you’re interested in and see where they go.

In another system, the rules are such that 0.999 is inarguably less than 1 and infinitesimals can’t be ignored. That’s fine, and maybe you find this system much more interesting. But it doesn’t mean there is anything illegitimate in choosing to look at the other one without lengthy justification.

There’s no sense in asking which is the REAL system of rules. There’s no sense in seeking proof as to which are the true rules. That’s a meaningless question. It’s like saying “I understand that checkers and chess are different games, but I want some proof as to which is the REAL, TRUE game. Proof, not just definitions!”. It’s a nonsensical thing to ask about.

So what is the continued argument about? What are you still looking for?

362

Indistinguishable has it. Math is a priori by nature. Pretty much all progress is made by proofs from existing knowledge. You need premises to start from, the non-existence of infentesimals is one of them (caveats out the wazoo that we’re only talking about Reals etc etc aside).

363

I’m sorry to jump in guys (I’ll be ignoring you, Erik), but AT infinity, there really is no infinitesimal. It’s not just a rule.

364

I’m just trying to understand things is all. I was thinking abou the fact that people are saying we can not ever “find” in a metophical sense the end of the string of a repeating decimal. A very normal assuption to make. But some other guy got me thinking about Zeno’s parodox. The it just occurred to me, we seem to be able to get to the end of an infinite string of points there the halves if you will and thought why not for repeating decimals.

So I am exploring this idea, it seems to lead to a contradiction somehwere, but not sure where. As I have said I am sure I have an error in my thinking. But would appreciate some help in finding it. If you don’t want to help that’s fine. Seems a more difficult puzzle than I first imagined it would be.

365

I have no idea what this first sentence means, but you’re wrong. Everything in math is just a rule. If I want to talk about the system of rules in which you can write 0.999… with ∞ many 9s, such that the difference from 1 is the infinitesimal quantity 10[sup]-∞[/sup], there’s nothing stopping me. If you want to talk about some different system where there aren’t nonzero infinitesimals, that’s also fine. Those are also nice rules. But there’s no external arbiter. There’s just our rules and their consequences.

366

Maybe you’ll appreciate this more than Erik did. I certainly hope so.

367

Here’s the only resolution of Zeno’s paradox you need:

The paradox: You’re walking down the road and you come to milepost 3, and you’re trying to get to milepost 5. Someone says: Hey! How can you ever get to milepost 5? First you have to get to 4 miles, then 4 1/2 miles, then 4 3/4 miles, then 4 7/8 miles, and so on, and so on. There are infinitely many points you have to get to inbetween here and there.

The resolution: Yeah, so what? There are infinitely many points between here and there; why would that be a problem? Just look at the very situation you are describing! There are infinitely many points between 3 and 5, yet all the same, the difference between them is a very finite 5 - 3 = 2. I’ll just walk at a leisurely 2 miles per hour and in the very finite timespan of one hour, I’ll have covered the distance, and all the infinite points inbetween. You, fair paradoxer, are claiming something to be a problem that doesn’t seem to be any kind of problem at all. You just have some reflexive phobia about “infinity” that makes you break down in discomfort whenever you see it… but if you just think for a moment about the very situation you are looking at, you’ll see that there’s no reason to be uncomfortable at all.

In short: The response to the paradox is “Huh? What are you talking about? What’s the problem? You think you can’t have infinitely many points in a finite interval? You’re wrong. Just look at the very case you’re pointing out.”. You don’t need to develop a theory of infinite summation or limits or any such thing. Even in contexts where there is no theory of infinite summation or limits, Zeno’s paradox is no problem, for the same reasons above.

(That having been said, if it spurs you to develop such theories, great!)

368

Indistinguishable’s point, I think, is that it’s perfectly possible to make a working mathematical system where that proof doesn’t work. There are useful systems not along the Natural/Integer/Real/Complex/Quaternion continuum. So depending on your starting assumptions, definitions, and rules, it’s possible to construct a system where, indeed .999… =/= 1 or a number of other things (though some might be difficult to pull off, like an equality operator that’s not reflexive or transitive).

369

That’s not the point of the exercise. Yes that was Xeno’s asinine point but it wasn’t mine. My point was infinite divisibility of the distance. M’kay?

I was trying to point out whether it was something infinitely long, infinitely small, whatever, it doesn’t matter. It’s the concept of infinity that is critical to understanding the equivalence.

Jesus. You’re worse than he is. At least he seems to be trying - some of the time anyway.

370

Yes, that image shows that, according to certain rules(/definitions), 0.999 … = 1. Like I said.

And according to different rules(/definitions), it doesn’t. It’s just a rule.

Specifically, that image first uses the rule that “a.bcdef…” = the limit, as n goes to infinity, of “a.bcdef…” truncated to n decimal places. This is just a rule. One could pick a different one.

That image also concludes by using the rule that zero is the limit, as n goes to infinity, of 1/10^n. This is just a rule (or consequence of certain rule choices). One could pick a different one.

And even the middle bits are just certain rule choices, although minor ones which no one would quibble with in this context. We could take decimals as having the least significant digit on the left, rather than the right. We could do anything.

That image does nothing to contradict the fact that everything in mathematics is relative to a choice of rules. There’s no external truth; there’s just the rules.

(On edit:

Right.)

371

Obviously you have never even seen a calculus book.

That is not even CLOSE to what it says.

Goodbye. You ARE the weakest link.

372

1 and 2 are “true” because they are either the usual axioms that we use for real numbers, or can be proven from them. If you don’t accept them, then you aren’t talking about the same real numbers that mathematicians are talking about. (You’re free not to accept them, but then don’t say that you are talking about “real numbers”.)

Point 1 means that you are talking about the usual axioms or theorems that can be derived for real numbers. (They are usually theorems, because you derive them from how you define the real numbers based on the rational numbers, and from the properties of the rational numbers.) I didn’t list them because they are generally things that you would think as obvious. But here goes:

The real numbers R are the unique complete totally ordered field. That means that R is a set of numbers with two arithmetic operations + and x, and an order relation <. (You can define other operations such as subtraction and division, and other relations such as >, from these.) (“Unique” means that there’s only one such object, in the sense that any complete totally ordered field can be put into isomorphism with R.)

Note that you need the order relation < to define “between”, i.e., “c is between a and b” means that:
((a < c) and (c < b)) or ((b < c) and (c < a))

I’m not going to list all the properties, but here are some examples:
For any a and b in R, (a + b) is also in R, and (a x b) is also in R
There is an arithmetic identity 0 such that for any a in R, a = (a + 0)
For any a and b, (a + b) = (b + a)
There is a multiplicative identity 1 such that for any a in R, a = (a x 1)
For any a, b and c, (a x (b + c)) = ((a x b) + (a x c))
For any a and b in R, (a < b) or (a = b) or (b < a)
For any a, b and c, if (a < b) then ((a + c) < (b + c))
For any a, b and c, if (a < b) and (0 < c) then ((a x c) < (b x c))
Any non-empty set of real numbers with an upper bound has a least upper bound in R. (This is not true for the rational numbers, and it’s what makes R “complete”, though you can define “complete” in other ways that will be found to be equivalent.) More formally, given any non-empty set S of real numbers, if there exists a real number b such that, for any number a in S (a < b), then there exists a unique real number c such that ((c < b) or (c = b)) and that for any number a in S (a < c).

Those are the sort of properties that you need to prove my statement that:
“if x<y then x<(x+y)/2<y, i.e., (x+y)/2 is a real number between x and y not equal to either”

373

Oh, how tangled this discussion has become… Wasn’t erik150x earlier just peachy with the idea of midpoints? See:

He (I will go ahead and assume they are male from the name) could continue to have both midpoints and infinitesimals, if he wanted. Of course, he wouldn’t be talking about real numbers, but then, he never was. (There would also be, in response to the context from which this whole branch grew, values which had no representation as infinite decimals, but that’s ok. It would be just like the fact that there are values which have no representation as fractions of integers, and all the other such things.)

374

:confused: :confused:
We’ve agreed all along that the question can be considered a matter of definition. No one thought the intuition of the ancient Greeks somehow overrode modern mathematics. For that matter, Archimedes didn’t even use the repeating decimal notation in which OP’s quandary arises in the first place. The “Axiom” does state, however, a common-sense truth whose acceptance would dispose of OP’s question, along with other benefits.

The ancient Greeks formalized an intuition about the lengths of geometric segments, and thus non-negative real numbers. That they had considered certain paradoxes and disposed of them so easily seems worth remarking.

And in particular, unless OP thinks he’s found a truth that Archimedes overlooked, it could have disposed of this thread hundreds of posts ago. :wink:

375

What I am saying is, in what sense is the Axiom of Archimedes (literally, the statement “There are no positive infinitesimals”) a “common-sense truth”?

It’s neither common-sense nor true for someone who is pondering nonzero infinitesimals. It will do nothing to sway them.

(And we’ve all found truths that Archimedes overlooked. Archimedes was just one man, and he lived a long time ago. There’s no reason we have to investigate only the things he did, in only the ways he did.)

376

I’d just like to correct my math for the record made what seems like days ago. I owuld have thought someone would have point out my error, but since it was reposted:

(1 - .999…)/2 + 1 - .999…

which is halfway between .999… and 1.

Should have been:
(1 - .999…)/2 + .999…

The extra one was a mistake. I’m sure no one took it serioulsy anyway :wink: But I might as well make my rediculous statement accurate.

377

Haha, yeah. Good (self-)catch.

378

Moderator Warning

allotrope, insults are not permitted in General Questions. This is an official warning. Do not do this again.

Colibri
General Questions Moderator

379

Oh my. (For it’s worth, I didn’t call down that wrath. I can take the occasional wild assumption and moldy reference.)

380

If I may all I would like to solve my own puzzle regrding mile attribution of Zeno’s Paradox and the repeating decimals.

The two different answers:

.000…1
and
.000…0

are not at all different according to the theory of limits. If someone plainy said this and I did not understand, my appoligies.

So I guess there is no contradiction after all here.