.999 = 1?

Factual Questions
501

Yes. (Actually, there is a sense in which the hyperreals I’ve been discussing and the surreals are essentially the same thing; see Philip Ehrlich’s answer here)

Well, it’s up to us to define what the ellipsis is to be interpreted as, but this is a very salient issue.

As I mentioned in my last post, if we’ll be ignoring infinitesimals, then it actually will never matter. Any which way, the result will be the same (in this case, 0.000…1 = 0 no matter how many 0s, as does 0.000…2, 0.000…11, 0.000…100001, and 0.0000… anything). If we’ll be ignoring infinitesimals, then everything with an ellipsis in it followed by more will be exactly the same as if it wasn’t followed by anything more and was simply interpreted conventionally. (Or, put another way, any assignment of transfinitely indexed digits will be exactly the same as if one focused on only the finitely indexed digits and used the conventional interpretation)

If we are paying attention to infinitesimals, then there very much is a question of exactly how far out the ellipses are supposed to go. And I don’t think there is any cleaner answer than to say “Whenever anyone writes one of these, they must indicate exactly how far out they want to go”. In other words, they give a prescription for a decimal string as a function of some variable n, and then the interpretation is the value when n takes on some particular canonical infinite value (in the surreals/hyperreals).

502

Also noted here and demonstrated here, though that thread gets unduly acrimonious.

503

I am definitely out of my depth, but it seems to me that 0.000…1 is the equivalent of “Follow that line forever and when you come to the end, turn left.”

For instance, how about, using “pi” for the familiar ratio of circumference to diameter, the number “pi…1” What the hades would such a thing even mean? pi + 0.000…1? How could it possibly be distinguished from pi?

Could there be some sort of two-dimensional notation, like complex numbers, that would help out here? We’d write something like (0,1) or (pi,1)? The first number is just a regular number, and the second number is multiplied by ten to the negative infinity before it’s added? You’d manipulate such numbers in a plane?

I’m thrashing here! Somebody give me a straw to cling to!

504

The interpretation I am noting would take “…” as meaning not “forever, with no end whatsoever”, but simply “for a very long time, and then ending”. Then there is no problem.

“But then haven’t you lost the whole idea of the ellipsis representing infinitely many digits?”. Not necessarily. That very long time might be longer than any particular finite number of digits (and thus infinite) and yet still come to an end.

Here’s an analogy, only to illustrate the idea that “goes on for infinitely long” and “comes to an end” needn’t be in contradiction: Consider strings ordered alphabetically. First there’s “a”, then there’s “aa”, then there’s “aaa”, then there’s “aaaa”… This goes on infinitely. But eventually, after all of those, there’s “ab”. (And then “aba”, then “abaa”, then “abaaa”. This goes on infinitely again, then there’s “abb”. And after infinitely many more infinite stretches, there’s “abc”. And so on and so on…)

One can give a two-dimensional analogy: Suppose we order the points on a grid in row-major fashion, so that (0, 0) comes before (0, 1) which comes before (0, 2) which comes before (0, 3)… but that whole infinite sequence comes before (1, 0) which comes before (1, 1) which comes before (1, 2)… all of which comes before (2, 0), etc. There are these infinite spans which nonetheless are followed by something else.

[One can even give the same analogy one-dimensionally: it’s just like how the infinite sequence 1, 1/2, 1/4, 1/8, 1/16, … keeps decreasing, yet nonetheless 0 is even lower than all of them.]

The common idea in all of these: Going on infinitely doesn’t necessarily mean nothing else can come later. There’s no conflict between those two.

As for the interpretation of “pi…1”, this could have, on one interpretation, a value depending on how many digits of pi are intended to fall comprise the “…” preceding the 1. If only n digits beyond the decimal point are intended, then this would mean "pi rounded down to the nearest 10[sup]-n[/sup], + 10[sup]-n - 1[/sup]. Of course, we all agree that’s what it means when n is finite, but we can use the same definition and plug in an infinite value for n, in a context such as the hyperreals that has such infinite values. It would basically act just like it does for finite but very large values of n. It would, in fact, just be a fancy way of talking about the asymptotic properties of the function which sends n to “pi rounded down to the nearest 10[sup]-n[/sup], + 10[sup]-n - 1[/sup]”.

505

You might instead say “No, that’s not what I want the ellipsis to mean. When I use an ellipsis, I mean for it to indicate a pattern of digits going on and on in such a way that there is no room left for anything else ever to follow. It goes all the way to The End, so to speak”. Which is fine, and perfectly standard. In that case, yeah, it would be, tautologically, nonsense to follow the ellipsis with anything else. In that case, “pi…1” would be a load of bosh.

It’s all a question of what you intend. I’m pointing out various things you can sensibly intend, if you’d like; concepts which fall closely enough to how people use ellipses and so on in other contexts that it’s not unreasonable to describe them with the same language. But if you don’t like using that language for them, you don’t have to. You can demand that those concepts be described only with markedly different language, or ignore them altogether.

506

That’s a sharp idea. That works. But you’d need more than two coordinates once you started doing multiplication. (0, 1) * (0, 1) = 10[sup]-∞[/sup] * 10[sup]-∞[/sup] = 10[sup]-2∞[/sup]. We might write this as (0, 0, 1). (So the first coordinate is multiplied by 1, the second coordinate is multiplied by 10[sup]∞[/sup], the third coordinate is multiplied by 10[sup]-2∞[/sup], and so on, and so on. The whole thing is essentially written in base 10[sup]-∞[/sup], or, read the other way, as a “decimal” in base 10[sup]∞[/sup]).

You might understand these strings of coordinates more fluently by thinking of them as instead strings of coefficients of a polynomial (that, after all, is what writing in base B amounts to: setting up the coefficients of a polynomial, and then plugging B into the polynomial).

And, in keeping with the idea that we’re working with an infinitesimal base, we’d consider a polynomial to represent a positive value just in case its lowest-order coefficient was positive. So, for example, (0, 0, 1) < (0, 1), because (0, 1) - (0, 0, 1) = X - X^2, which is positive because the coefficient of X is. A polynomial would represent a finite value if it was constant, and infinitesimal value if it was non-constant but had a constant term of 0.

In particular, (pi, 1) would be the polynomial pi + X. And we could do arithmetic with this quite readily. It would square to (pi^2, 2pi, 1), the polynomial pi^2 + 2Xpi + X^2. This would be infinitesimally greater than pi^2 (their difference would be 2Xpi + X^2, which is a positive infinitesimal). And so on.

Perhaps I haven’t explained that very clearly right now, but sleep beckons for now. The gist is, yes, that’s a good idea of yours, which can be made to work very well (it handles +, -, and * immediately; and if you extend to infinitely many coefficients (a la Taylor series), you can handle a hell of a lot of other things just as easily).

507

I’d like to thank FV here for at least attempting to find a fault in my informal proof, that there is indeed an infinite’th decimal place. And maybe he has. FV, you do not really explain how one can move from A to B without completing an infinite number of halves.

One could accomplish this if space was not infinitely divisible. In other words you get to down to 1x10^(-trillion) of an inch just for example, and for some reason physical space simply cannot be divided any further. If this were true then the assumptions of Zeno’s Paradox are incorrect and my proof is wrong. This would be counter intuitive of I think most of notions of continuous space. What would make it indivisible??? I don’t totally discount that possibility, but it also not my belief and it is also to my knowledge unproven. I also can’t prove space is indivisible except perhaps by inductive logic. You would have to give me some reason that the nature of space changes at some point. Or at least some evidence showing this.

Others here I suppose argue by saying it leads to contradictions to ordinary arithmetic. Perhaps, but one, I could argue ordinary arithmetic must the be flawed. Or maybe you just can’t apply ordinary arithmetic to infinitesimals.

People who are trying to make sense of infinitesimals with Limits… you still don’t understand the nature of Limits. The very purpose of Limits is to work WITHOUT infinitesimals, not to work WITH them. It seeks to ignore them. It’s a perfectly fine thing to do if you do not care to talk about them or they are not relevant to your problem. But as I think we and not just my self have made perfectly clear, Limits do not prove .999… = 1, they simply assume they are equal and thus ignore what value (if any) lies between .999… and 1.

I don’t have any good answers to how we work with infinitesimals, how precisely there properties and operations should be defined. I also don’t know what it means for one infinity to be bigger than another. That doesn’t mean they don’t have meaning or exists.

I believe there is an ultimate set of rules which make sense of all of these things. Limits to me are not some universal truth we have discovered, they are are tool for avoiding the the very things we do not yet understand for practical purposes. Limits were created because ordinary arithmetic breaks down and is inconsistent in the very matters we are discussing. Limits are not a solution, they are as we say in the computer biz… a “work-around”.

All some very interesting thoughts on the consequences and possible interpretations of .000…1 and nice to see people actually discussing. These are things that have baffled the greats ever since they were posed. I certainly am not proposing I have any type of new knowledge to add here. Only to remind people in my humble opinion Limits do not provide us any real truth in the matters here.

FV, I very much look forward to hearing your view on how one can proceed from point A to point B without progressing through such an infinite division of distances.

As far as “You have still not provided a useful definition of the notation 0.9999…0” - I assume you mean here by the last 0 here the same decimal place I refer to when I am talking about the 1 in “0.000…1” - Well I don’t know how you define useful? It is enough for me to say when you have fully traversed an infinite number of decimal places the first one you come to is this one.

As far as “The proof is internally inconsistent and thus fails”… how is it inconsistent? It is not a formal proof, so perhaps there are weak spots.

508

[0,1) is an infinite series with a first element but no last element.
[0,1] is an infinite series with a first and last element.

The notation “0.9~” is more like the former than the latter.

Infinity is weird; read all about it.

509

Indistinguishable provided the proof earlier. Or rather a counterpoint to the fallacy. Of course I can move though an infinite number of points trivially at any time. I do it all the time. The fallacy is that there is some notion that you can’t have an infinite number of points in a finite interval. In Zeno’s case, the idea that you can’t cover an infinite number of points in a line in a finite time. But the paradox itself shows how you can have an infinite number of points in a finite interval. It simply blandly asserts, with no justification that whist you can have them in space, you can’t have them in time. There is no justification for such an assertion. If I can have an infinite number of points in a finite interval on the X axis, I can have an infinite number of points in a finite interval on the Y axis. That is how I am able to draw diagonal lines. This implicit, unjustified assumption is why worrying about Zeno and anything derived from it is simply not relevant. Time flows, and covers an infinitude of instants in time in any interval of time. So to does movement cover an infinitude of points. Neither prevents the other.

I will also point out that the halving of distance algorithm for generating the infinite number of points is but one of an further infinitude of ways of generating other sets of an infinity of points in that same finite interval. Why just halving? I can come up with an infinitude of infinite series that converge to one. I cover all of the points in all of them as I cover that same finite interval from zero to one. They all converge to one. So, how does your algorithm for finding the last infinite digit of 0.9999… find that particular point amongst the infinitude of “last” points that belong to all those other infinite series? They are all the last infinite point of a series that converges to one. Given that I can choose infinite series with real terms, not just integer terms, there are more than just Aleph Null series here too.

The issue with the inconsistency of the arithmetic has been belaboured here. You can choose a system where your particular notation and idea works. But it isn’t the real numbers. The real numbers are what we assume the decimal notation of the infinite series 0.999… is addressing. All conversations, in this and all other threads assume you mean the reals. You mean the reals too. That is clear. But the reals have a very clear set of properties. The provided notation for 0.999…9 and the operations you define upon that - in particular the property of the “infinitieth” digit can be shown to lead to a contradiction of the properties of the reals. Thus we have proven that whatever various systems of numbers your proof covers, the real numbers isn’t one of them.

510

This difference between the treatment of space and of time is a strikingly clear way of pointing out the fallacy! Thanks; I’ll be using that from now on.

511

Missing negation sign re-inserted in bold.

Er, I switched back and forth between thinking of base 10[sup]∞[/sup] and base 10[sup]-∞[/sup] here. In base 10[sup]∞[/sup], a Laurent polynomial would represent a finite value if it had no terms of positive degree, and an infinitesimal value if had only terms of negative degree. In base 10[sup]-∞[/sup], a Laurent polynomial would represent a finite value if it had no terms of negative degree, and a finite value if it had no terms of positive degree.

512

Argh… that very last bit should say “In base 10[sup]-∞[/sup], a Laurent polynomial would represent a finite value if it had no terms of negative degree, and an infinitesimal value if it had only terms of positive degree.”.

513

Zeno has a bunch of paradoxes, more than have been touched on in this thread. The paradox referenced above is (a version of) Dichotomy. Invoking a corresponding time series is a maneuver anticipated by Aristotle. See the SEP.

514

I admit that I didn’t wade through all of the pages before this so I may be repeating others arguments:

Eric, I understand that the last 1 was written at exactly 1 hour, what time was the second to last number written?

I am confused that you t that you accept the solution to Zeno’s paradox that 1/2+1/4+1/8+…=1, while having problems with 9/10+9/100+9/1000=1. It is exactly the same thing, except instead of each step getting you half way to the end, each step gets you 90% of the way to the end.
It seems to me that you are trying to create a new number system that is distinct from that used my mathematicians. This number system includes all of real numbers, and also includes a number, alpha, such that alpha is greater than 0, but alpha is smaller than any non-zero number. You might be able to make a consistent system based on this number, for example defining nalpha=alpha for any non-zero n, but in the process you would likely lose some of the properties of the reals. For example, 3alpha-2*alpha = alpha-alpha = 0 which is not equal to (3-2)*alpha= alpha. So you lose the commutative property of the reals. If you want to do so for you won amusement you could play with this idea and let us know what sort of number system you end up with and what properties do and don’t hold, but whatever you come up with, it isn’t the reals.

515

Just want to point out for the record that Knuth didn’t come up with surreal numbers (nor did he claim to), although his book popularized them. John H. Conway (who also invented the Game of Life) did and wrote about them in “On Numbers and Games,” a (somewhat) more conventional academic book.

516

As Indistinguishable has been tirelessly pointing out, there are several such number systems that mathematicians have developed. They are not the conventional real number system but they’re perfectly respectable and used in, for example, nonstandard analysis.

517

Right. For what it’s worth, my point isn’t just to invoke the corresponding time series. The point is to note that few people feel any discomfort with a finite distance containing infinitely many points, that they are correct in that lack of discomfort, and that there is no reason to feel any differently when speaking of finite intervals of time having the same property.

The paradox asks us to rush from “There are infinitely many points” to “You can’t cover them all in a finite stretch”, and this is completely fallacious. Pointing out the difference in people’s naive attitudes towards space and towards time can help clarify that.

I’m not one of those guys who’s boorishly dismissive of philosophy. But I am sort of dismissive of the amount of attention Zeno’s paradoxes of motion have gotten, particularly when people come to regard them as needing sophisticated mathematical treatment. It’s the analogue of having spent centuries continuing to debate “…But that’s only $29 between the bellhop and the guests. Where’s the missing dollar?”, and coming to a sense that the problem was only soundly resolved with the development of measure theory.

518

I thought it worth noting for interests sake, since you took an interest in that approach.

I don’t know enough about the history of what has been written on this topic to say much about it’s worth.

I had imagined it would be more like; how can we fully model what is going on; and given some sophisticated mathematics, we can show exactly what is going on with some (perhaps modified versions) of Zeno’s paradoxes. That is to say, one need not have a worry that the paradoxes showed anything, but one could still think that there was something to say about them (viz. how to model them).

519

Oh, yes, I definitely found it interesting, and appreciate your mentioning it.

520

This is how I’d always been taught it in math classes.

Fascinating to see that there are non-standard (“non-Euclidean,” so to speak) alternatives and variants.

The first time in my life I blindly and drunkenly stumble on to something that works, and it has to be something I don’t believe in!

Oops! Very right; if I’d re-read Surreal Numbers more recently, I would remember that. The book very clearly gives Conway the credit. Knuth just published it in a cheerful and relatively informal manner. Who says math isn’t fun?