I seem to remember that the proof of some basic property of rational numbers depends on whether 0.9999… (infinitely repeating decimal) is considered equal to 1 or is considered less than 1. I’m thinking that the property had to do with a one-to-one mapping of the rational numbers to positive integers, but I’m not sure.
If that’s it, does the proof require the less than or the equal to property, and why?
Yes, because of the betweenness property of points. For two points to be distinct, there must be at least 1 point between them, that is, greater than point A and less than point B. That’s impossible in the example you cite. It behooves me to point out that repeating decimals generally happen when you divide a number by another number, e.g., 1/3=.3333 repeating. In the case of .9999 repeating, there’s really no way to achieve this. Really, .9999 is an artificial construct built to make a point.
We’ve discussed why 0.999… = 1 before (several times). However, there are a couple of confusions in the question as posted.
Firstly, rational numbers are fractions, i.e., the ration of one integer to another. When you are talking about 0.999…, you are talking about infinitely long decimal fractions, i.e., the limit of a series of finite decimal fractions, i.e., the domain of real numbers. So when you say that 0.999… = 1, you are saying that the real number expressed by 0.999… is the same as the real number 1. That this maps to the integer 1, or to the rational number 1/1, is irrelevant, really.
Secondly, I don’t understand what is meant by the “less than or the equal to property”, so I’m not sure how it’s relevant here. Proofs about limits generally involve less-than (<) or greater-than (>) relationships, but could work with less-than-or-equal-to or greater-than-or-equal-to relationships as well. But the question needsto be more specific.
Incidentally, I think the last time we discussed this was about 2 years ago.
It looks like a simple argument, but the devil’s in the details on this one. It’s not immediately obvious that there’s no point between the two, and you have to get into some deep facts about real numbers to show it.
Every rational number has two representations in base 10, one that appears to terminate (but that really ends 0000…) and one that ends in 9999… Neither is more valid than the other.
.9999… = 1 follows pretty much immediately from the definition of a decimal representation and some basic facts about geometric series. I’m not sure why you’d want to get into any counting arguments to do it.
There’s nothing mysterious about 0.9999… = 1
“0.9999…” is definied (by mathematicians) as the limit of the sequence 0.9, 0.99, 0.999, 0.9999, …
The word “limit” itself has a precise mathematical definition, and using this definition one can prove that the limit of the above sequence is indeed 1.
So if you’re using the standard mathematical definition of 0.9999…, then the statement that 0.9999… = 1 is uncontroversial. So far as I can tell the only reason people tend to think there’s something mysterious here is that they are (perhaps unintentionally) using 0.9999… to mean something slightly different (and perhaps less clearly defined) than what mathematicians use it to mean.
I have read this question many times before. I am no mathematician however and most of the detailed stuff just flies right over my head.
The response that was good enough for me was.
1 divided by 3 = .3333…
Multiply .3333… by 3. It isn’t 1 or is it? It has to be because you just divided it by three to get that result.
They are the same. It is an artifact of the decimal system.
Look, it’s simple algebra.
x = 0.999999999…
That’s the original number
10x=9.9999999999…
we multiply by 10
10x-x = 9.9999999… - 0.999999999
we subtract the original number from each side
9x=9
we divide by 9
x=1
It’s been since then. Can we just automatically close these with standard references to old threads?
You should think of it as being analogous to 1/2 = 2/4. You can think of that as two notations for the same thing or else you can delve deeply into the actual mathematical meaning of the notation, in which case you can prove it.
Incidentally, infinite decimals are not the best way for doing arithmetic of real numbers. Cauchy sequences are much better, even Dedekind cuts are. I am not going to try to define them here; see Wiki for definitions. But try to imagine the complications involved in multiplying .99999… by itself, all the infinitely many carries needed.
Obligatory “this applies to the real numbers”, and “it’s possible to construct a number system where this isn’t true” comment.
Obligatory “unless you didn’t need to be told that because you were already aware of at least one field where this isn’t true, don’t try to use this as support for your disbelief in any proof given here” follow-up.
Well, 0.9999 does equal 1, because “infinitessimal” equals zero. Or at least it kind of does, the way trans-infinities also equal infinities in many ways. However, they don’t exactly. And infinitessimals don’t equal zero exactly either. (I mean they “equal exactly”, but they have other properties that make them distinct). Integrals, for example, are about adding infinite quantities of infinitessimals to get finite answers.
It’s all about the strange way that there may exist different scales of numbers, separated from one-another by nearly-impassible infinities, which all end up kind of falling into the terms “infinity” and “zero” when looking at them from any particular scale, but which also have their own properties and their own relevance when looking at the big picture. (And when using methods that take away the intractability of examining them head on.)
So yes, 0.9999… does equal 1. Maybe it “completely” equals 1 if the concept of one minus infitessimal doesn’t hold any interesting properties. However, it also might not quite equal 1, in that there might be such properties. (For example, 1+infinitessimal to the infinite power equals e).
But, if you ask most mathematical philosophers, they’d rather not talk about infinities at all. The subject is quite closed to intuition (and intractable head-on), and they’re right it is often much better to talk about it from a different angle. For example, usually they’ll say that we can never perform any infinite operations (such as infinite additions), and that we only take the limits of them approaching infinite procedures. A useful, practical way of looking at things. But I do not believe it is often the best or philosophically accurate way to treat the topic. For example, would anyone like to start talking about Godel? (Actually, don’t answer that, I’m about to go start a thread in GD).
NOO!!! Besides entertainmnet value, there is IMMENSE REASON that a society cannot just devolve to answer every question by referencing some previous answer. We have to keep reexamining questions anew and actively continuing discussion (even if we don’t add anything on the third time we reanswer the question, but on the 20th). This is a huge threat that is dawning on the information age.
Alex_Dubinsky: There are a few things wrong with your post that I, a mathematical layman, can see offhand:
[ul]
[li]“Infinitessimals” don’t exist in the Reals. They do exist in other fields, as panamajack has implied, but for the purposes of this thread the Reals are of the most interest. I doubt jebert is aware of any other field, or even that fields, as such, exist.[/li]
(If I’ve offended you, jebert, smack me down.)
[li]Because of the above point, the concept of 1+infintesimal has no meaning.[/li][li]I also doubt that raising something to an infinite power has any meaning, either.[/li][li]“Mathematical philosophers” might not like infinities, but mathematicians have plenty of ways of dealing with them. Cardinalities spring to mind, as does the calculus. They may not operate ‘intuitively’ (the set of the even integers is the same size as the set of all integers evenly divisible by one trillion is the same size as the set of all integers) but they are amenable to rational analysis.[/li][li]This issue is closed. That is, no discussion can possibly bring about a new answer, or even any really new insights. The prior threads were exhaustive studies into both the mathematics of the situation and the tenacity of some people’s ignorance, and repeating that here would be a huge waste of time for no good end.[/li]
Sometimes, referring someone to prior work is the best possible answer. That’s why we keep it around.
[/ul]
Derleth pretty well covered what I was about to say. Clearly Alex_Dubinsky was referring to the fact that lim(1+1/n)^n = e (taking the limit as n goes to infinity). However, summarizing this as “1+infinitessimal to the infinite power equals e” is inaccurate. For one thing, so far as I know 1+infinitessimal can’t be rigorously defined if we’re restricting ourselves to real numbers (as opposed to hyperreals). Furthermore, the phrase “1+infinitessimal to the infinite power” is ambiguous. It could describe lim(1+1/n)^2n equally well (or equally poorly), but this limit is e^2, not e.
In fact, 0.9999… does equal 1, it can’t be not quite equal to 1, and it doesn’t have any properties (“interesting” or otherwise) that distinguish it from 1. It is simply an alternate representation of 1. To claim otherwise is to misunderstand the meaning of “0.9999…”.
Lastly, Alex_Dubinsky’s “NOO!!!” seems to be misinterpreting Mathochist’s request as applying to all previously-answered questions, when it is in fact pretty clearly meant to apply to this one in particular (for which I think the suggestion is well justified).
WTF?
Since this discussion was originally An infinite question: Why doesn’t .999~ = 1?, a
column by Cecil, let’s put it over there.
Also, here are a few additional threads on the boards on this subject.
http://boards.straightdope.com/sdmb/showthread.php?t=187902&highlight=0.9999 (this one only had 381 replies!)
http://boards.straightdope.com/sdmb/showthread.php?t=15863&page=1&pp=50
http://boards.straightdope.com/sdmb/showthread.php?t=179452&highlight=0.9999
Please take the time to go back and read some of these and it may help. Or, it just might convince you to sell your computer and go live in the woods.
I’m going to lock this one until everyone has time to go read some of these threads. There were probably ten others I could have linked to, but I’m old already and don’t have much time.
If anyone really feels like this one should be opened to present new information, appeal to bibliophage, who is the moderator of the forum where this thread now resides(heh heh! Hey bib. I owe you. 
)
samclem General Questions moderator