My pencil is not that skinny. And I don’t think it answers the issue anyway.
Both 1/3 and 0.333… are representations of the same thing – that is, unity divided into three equal parts and selecting one of these.
1/3 has the simplicity of representing the number as the result of the process, that is one divided by three.
0.333… examines this same number and same concept by placing it on a base 10 framework where none of the gridlines exactly align with the number we are representing. The “…” indicates an infinite process. Thus the representation is infinite. the actual quantity is well defined.
If we were to change the base to base six for example we would not have this problem. 1/3 would be 0.2[sub]base 6[/sub] and no further questions asked.
It is a feature of the real number system that we can change base or change representation to whatever form is most convenient to us without changing in any way the actual number that we are referencing.
Why should he do it that way just because you want to force him into something that can’t be done in that manner. It doesn’t prove your point.
There’s no problem directly locating some points on the number line that have an infinite number of decimal places.
Quoting from page 2 (seriously) of my calculus book where it is describing the reals and number lines it notes that: "Points associated with certain irrational numbers such as sqr(2) can be found by geometric construction.‘’
It had previously explained that rational numbers, like 1/3, can be found be found by subdividing equal line segments.
So locating 0.333… on a number line seems trivial since it’s rational and the fact that an infinite number of decimal places isn’t necessarily a deal breaker either in and of itself.
Why should he do it your way? That’s like asking him to walk across the room per Zeno’s paradox to prove that he can never get there when we all know that it’s trivially possible and we do it all the time.
Don’t assume 1/3 = 0.333…?
Why not ask him to prove that 1+1=2 without using the fact that 1=1.
Now, what I did was haul out my calculator, divide 1 by 3, and got .3333333333, limited only by the display. To a layman, like me, it would follow that .3333333333 multiplied by 3 would equal .9999999999, which it did when plugged in just those ten decimal places. However, my old Casio is smarter than I and knows that 1/3 does not stop at ten decimal places, but goes on forever, so when I keyed in (1/3)*3 it equaled 1 because 1 = .9999999999 taken out to infinity.
schooner26, I think part of the problem is that you didn’t have your mind blown by infinity when your were ten, like MsKaren’s kids. All for the best, I suppose; it’s used as capital punishment on the planet Frogstar B, but it’s made your universe less interesting.
Are we seriously talking about whether 1/3 = 0.333… on page 39 of a General Questions thread? At some point, you have to point people toward wikipedia, a math textbook, or an elementary school teacher and wash your hands of it.
0.333… and 0.999… are real numbers. They are not sequences or processes, except trivially (or via their construction from Q, although clearly anyone arguing that 0.333… != 1/3 or 0.999… != 1 has no idea what that means). The real number 0.999… is not the same thing as the sequence of real numbers (0.9, 0.99, 0.999, …). The real numbers contain no infinitesimals by definition (take your pick of the definition of them as the completion of Q, as Dedekind cuts, etc.) Real numbers are not required to have an algorithmic process for computing their digits. The fact that 1 - 10[SUP]-n[/SUP] < 1 for all positive n does not mean that 0.999… is less than 1. The string “…999” is not a well-formed decimal representation of a real number and is therefore meaningless.
And so on. If you don’t get it at this point, just accept that this is something you don’t understand and move on. Math does not work the way you apparently expect it to.
Whoops, trying to be clever or some such thing, I actually subtly screwed this up (ironically foundering in my mistaken rephrasing on the very difference that makes infinite games so distinct from finite games: one may play in such a way that at every moment, one can still launch a guaranteed-win strategy, without this in itself being a guaranteed-win strategy). It’s probably possible to salvage a correct description of the above flavor with some alteration, but let’s just give a more straightforwardly faithful rendering for now:
The Axiom of Determinacy says that for every function f from countably infinite bitsequences to {even-happy, odd-happy}, either (there is a function g from even-length bitstrings to their children such that any bitsequence s in which each even-length prefix is followed by the child prescribed by g has the property that f(s) = even-happy) or (the same thing with “even-” replaced throughout by “odd-”).
A famous proverb says “Those of you who think you know everything are annoying those of us who actually do know everything”.
Accordingly, I think that we Dopers who’ve spent 1950.999… posts explaining this topic, ought to get together and agree upon some standard response, terse and to the point, that we will all post in response to any further disputes about the value of 0.999
Something like:
This topic has been discussed fully, and settled to the satisfaction of a consensus of mathematically fluent Straight Dope Message Board members. That consensus has agreed that 0.999 is equal to 1. This consensus was established in Thread Number 32759.999…, to which the interested reader is now referred for further enlightenment on this subject.
schooner26, it seems like the only thing you actually disagree with here is the notation.
You agree that limit n->∞ Σ 9/10^i for i=1 to n is equal to 1. You just disagree that 0.999… represents limit n->∞ Σ 9/10^i for i=1 to n.
But the thing about notation is that it doesn’t have any intrinsic meaning. It means what people agree it means. The other people in this thread, and most mathematicians, use “0.999…” as shorthand for “limit n->∞ Σ 9/10^i for i=1 to n”. It’s intended to represent exactly the same thing, just more concisely.
So, my question for you is why do you disagree with this shorthand? Is there any reason not to use this shorthand?
Yeah, we could just close the thread down and refer all dissidents to the answer.
However, there are many who have gained a lot from its longevity, the to-ing and fro-ing and side discussions, the assortment of cranks who have come and gone, the not-so-cranks who have a genuine intellectual objection that they can’t seem to clear.
Here is what I have got from it…
An insight into the nature and range of mathematical stumbling blocks that people have (and persist with). As a high school Maths teacher, this is not a bad thing to see.
A chance to put forth a reasoned and cogent response to address specific issues that are presented.
Exposure to a range of proofs and mathematical ideas which I may not have gravitated towards on my own.
Exposure to concepts and discussion that are beyond my normal mathematical fluency – Dedicand cuts, Borel summation, finer points of rigour, hyper-reals, surreals, and infinities of black and white hats on prisoners’ heads.
An opportunity to raise questions for further discussion. I have a short list of spin-off questions that deserve their own thread. (I’ll get to it soon.)
The fact that it has attracted so many views suggests that it has been a popular thread and has some merit as it stands at present.
Sure there have been grounds for getting feisty with a few wilfully ignorant individuals. And there have been some mathematical giants among us who have elected to step out of the conversation and resist wasting their time and patience. But I doubt a generalised terseness is warranted.
Let it sink in the morass of the archives until some crank wishes to tangle with it and then those who want to can go around the carousel again.
Assume next that 3=2.9999999…
so that 1/3 = 1/2.999999… ≈ 0.333333333…
because of the assumption that 3=2.9999999…
The result, 1 divided by 3:
1/3 ≈ 0.333333333…
Conclusions: this is a proof by assumption. It is assumed that 3=2.999999… is true. Similarly one can assume that 1=0.9999999…
But assuming it true does not prove it true.
And those are cool things, and I’d like to read the thread of spin-off questions, if you set it up. My frustration here is seeing an otherwise interesting discussion getting repeatedly derailed by a few people asking the same insistent questions and just refusing to accept the obvious answer for no real, defensible reason. There’s nothing that will satisfy them; I certainly haven’t seen anything in this thread that would indicate that those people understand mathematical proof in general, the construction and definition of real numbers, or even what mathematical notation means.
7777777, you lack cogency. This has not gotten any clearer since the last time that you cut and pasted it. If you are being ignored it is because you do not make sense.
You contradict yourself here.
[ul]
[li]It is proof by assumption (whatever that means)[/li][li]Assumption is not proof[/li][/ul]
How exactly is a reasoning person supposed to respond to this?
Let S denote the space of sequences a = (a[SUB]1[/SUB], a[SUB]2[/SUB], …) of hats. (We can identify S with [0, 1] in the obvious way, but there’s no real point in doing so.) For a given configuration a in S, prisoner n can see the (n+1)st, (n+2)nd, … hats a[SUB]n+1[/SUB], a[SUB]n+2[/SUB], …n. Define an equivalence relation on S by setting a~b iff a[SUB]n[/SUB] = b[SUB]n[/SUB] for all but finitely many n. Using the axiom of choice, choose an element f([a])\in S of every equivalence class [a] in S/~. The prisoners memorize the set of f([a]) before being led into the courtyard. Each prisoner can identify the class [a] for the configuration a they’re all in, since each one can see all but finitely many of the hats. By definition, having each prisoner respond with the element of f([a]) he corresponds to will result in all but finitely many prisoners’ going free.
Ok. i will get on to it.
However, I would consider this at least a partial victory – I think Strinka has succinctly nailed Schooner’s problem to the wall. I await his reply and am not expecting any real change. This one boiled down to a difference of notational interpretation and not some crackpot conspiracy theory. But it did take some perseverance to isolate the essential point of contention.
You must also yourself do your thinking and not always suppose someone else is doing it for you and providing all the answers.
My proof above shows how you are proving that 1=0.99999…is true by assuming
it is true. I don’t prove things by assumption. I just show how you do it.
You must admit your errors, and learn form your errors.
You try to make mathematics a kind of politics where consensus decides the truth.
But you must admit, math truth is not and cannot be decided by human agreements.
That is what maths apart from politics. Maths deal with objective truth. Absolute truth. If you don’t learn from your errors you are going to end up at a dead-end.
There is no progress anymore after that. Science does not progress anymore.
What you are doing has serious consequences.
It does not matter if you are math teachers or not, you keep repeating what you
were taught. You repeat even if you were taught wrong things.
7777777, you lack cogency. This has not gotten any clearer since the last time that you cut and pasted it. If you are being ignored it is because you do not make sense.
I don’t know— it seems equivalent to the common complaint that 0.999… is not a real number itself, but simply some sort of ongoing process that never terminates and thus always stays less than 1. Or something. I haven’t been following the last few pages of thread in detail; maybe there’s something there I missed. At least the NSA’s secret program to suppress recursion hasn’t popped up again.
Not it’s directly relevant to the initial question, but saying that a real number x = 0.x[SUB]1[/SUB]x[SUB]2[/SUB]x[SUB]3[/SUB]… is the limit of the 0.x[SUB]1[/SUB], 0.x[SUB]1[/SUB]x[SUB]2[/SUB], 0.x[SUB]1[/SUB]x[SUB]2[/SUB]x[SUB]3[/SUB], … is true but mildly missing the point in discussions like this one. The rational numbers Q aren’t closed under limits; it’s in general not possible to take the limit in Q of even a bounded, monotonic series of rationals. R is defined to be the smallest field containing Q in which those limits exist (i.e., R is complete as a metric space). But it’s not a priori clear what that field should be or even that it exists. It’s begging the question to try to define real numbers as the limits of, say, decimal expansions; once you prove that such a field R exists, though (again, the usual constructions take Cauchy sequences modulo sequences that converge to 0, or equivalently Dedekind cuts, or etc.), then it’s totally valid to say that 0.999… = lim_{n o \infty} 1 - 10^{-n} = 1.
The underlying issue here is what exactly a real number is. But that’s not up for debate; and once you accept that definition, then it’s automatic and unarguable that 0.999… = 1. As has been pointed out many times here, it’s totally possible to construct a field with infinitesimals, even complete and ordered ones like R((X)). But R is not one of those fields.
Completion means that the limit of Cauchy sequences are well defined: sequences a = (a_n) such that |a_n - a_m| o \infty as n, m o \infty. Any such sequence is bounded, but boundedness isn’t enough: take a_n = (-1)^n. Bounded, monotonic sequences are Cauchy, though (and are equivalent for these purposes for the case of Q or R), and I was referring to those sequences specifically.
Great! I don’t know if it was related directly to your previous comments in this thread; I just wanted to talk about something besides the usual 0.999… = 1 thing.
I’m not sure what “closed under limits” means per se, but Itself did go on to elaborate on what he meant by the term. The real numbers are complete in a way that the rational numbers are not.
(The link is to the Wikipedia entry on the completeness of the reals. I suspect a full understanding of the concepts mentioned there would go a long way toward settling any remaining controversy in this thread. To me personally, it all looks at least vaguely familiar but it’s been a long time since I’ve studied it.)