U then believe in a contradiction. If you accept a contradiction to be true then you prove anything to ne true and not true. It is a meaningless belief.
erik150x, I’ve learned things here on the SDMB. One thing I learned was that there are no other numbers between .999… and 1. That means they are the same number. Try looking for other threads on this subject.
You can also prove anything by starting your argument with an incorrect premise. Here I go:
A. All mammals are animals.
B. Carrots are mammals.
C. Therefore, carrots are animals.
Unassailable, sound logic, isn’t it? A syllogistic thing of beauty.
My point is that you don’t understand how infinities work, or what infinity means.
Believe me I have done much reading on it. You say there are no numbers between .999… And 1, therefore they are the same? Can you prove that no numbers exist between them? Even if true, how does it follow that because there are no numbers between two numbersthat they are the same? How many natural numbers exist between 1 and 2? Are they the same? I have never seen -5 apples, does that mean negative five can’t exist?
If you would like to point me to the exact proof you mention, perhaps I too could be enlightened.
Thanks.
You tell me you believe in a contradiction, i tell you thats meaningless, you parrot back just what siad and then claim I don’t undrrstand infinity. Okay. You got me.
1/infinity isn’t defined because infinity isn’t a number (at least not in the same way that the reals are numbers). However, it is undeniably true that the limit of 1/x as x approaches infinity is 0. It is also true that the limit of the sum of 9/10^x with x going from 1 to infinity is 1.0. 0.999… is just shorthand for that limit.
Let me add I am trying to say that I don’t believe .999… = 1. And i also believe the question can be rephrased as what is 1/infinity, (zero or not). Your state they are not essentially the same. Ok. I think your wrong. I have explained my view and why. I’d be interested to here your view and why they are not the same questions. You say I don’t understand infinity or its workings. Please explain what leads you to this conclusion?
[quote=“Dr.Strangelove, post:26, topic:27517”]
1/infinity isn’t defined because infinity isn’t a number (at least not in the same way that the reals are numbers). However, it is undeniably true that the limit of 1/x as x approaches infinity is 0. It is also true that the limit of the sum of 9/10^x with x going from 1 to infinity is 1.0. 0.999… is just shorthand for that limit.[/QUOTERight
Right. And a limit is basically saying if one can come arbitrarily close to some nimber, we will just say it is that number. For practocal purposes this is fine, but this is just an accepted assumption in calculus. It is not an absute truth. Show me the proof that the limit of a function is exactly the same as the result of the function?
[quote=“erik150x, post:28, topic:27517”]
[quote=“Dr.Strangelove, post:26, topic:27517”]
1/infinity isn’t defined because infinity isn’t a number (at least not in the same way that the reals are numbers). However, it is undeniably true that the limit of 1/x as x approaches infinity is 0. It is also true that the limit of the sum of 9/10^x with x going from 1 to infinity is 1.0. 0.999… is just shorthand for that limit.
[/QUOTERight
Right. And a limit is basically saying if one can come arbitrarily close to some nimber, we will just say it is that number. For practocal purposes this is fine, but this is just an accepted assumption in calculus. It is not an absute truth. Show me the proof that the limit of a function is exactly the same as the result of the function?[/QUOTE]
Limits were invented to get rid of that nasty 1/infinity. Essentially what is assumed (more or less) is that 1/infinity = 0 in calculus. Therefore to use calculus to prove .999… = 1. Is circular.
None of your counter arguments make any sense.
1-1=0. It’s an identity function. If there are no other numbers between .999… and 1, then they are the same. If there are other numbers between, then 1-.999… does not equal 0.1
If you add any number to .999…, you will get something greater than one. It’s just an anomaly of notation that there can be more than one representation of the same number.
Now if you have some new definition for a numeric system that allows them to be different numbers without breaking down elsewhere, good for you. Nobody else is using it, and it wasn’t presumed to be used in the question here. So go open a thread on your new topic, or find one that is more on the topic of your new system. The mathematicians on this board will sense the shift in the force that results and you will be able to participate in a thorough and detailed analysis of your theory.
Nonsense. 0.999… doesn’t make sense without a limit definition. There’s nothing circular about using the definition of a notation to prove properties about it.
When you can name a number between 0.999… and 1.0, you’ll have a case.
Okay. 1st why is it a necessary conclusion that if no number exist between 1 and .999… That they are the same. If it breaks your or our number system, add to the list of paradoxes known to man.
Just for fun though… (1-999…)/2.
Sorry that should be (1-.999…)/2 + 1 - .999…
.999… Doesn’t make sense without out a limit? Maybe in your mind. I’m pretty sure i could explain it to many people who have never heard of a “limit”.
9/10. + 9/100 +9/1000 … Just a sum of an infinite sequence, no limit involved.
This assumes a property of “number.” Note that if by “number” we mean “integer” then there is no “number” between 16 and 17 yet they are unequal.
I find it best, if only “for old time’s sake” 
, to appeal to the Axiom of Archimedes, which can be paraphrased as
For any positive ɛ there is a finite integer N such that Nɛ > 1
Notice that for any N, you can make N(1 - 0.999…) < 1 by providing sufficiently many 9’s.
It’s not about 1/infinity or limits or sums or what have you. It’s all about notational convention.
Here’s my standard reply to this debate:
Keep in mind that one must distinguish between notation and what that notation represents. Different notation can represent the same entity, as in, for example, the equality of “1/3” and “2/6”: they are not equal as notation, but the fractions they denote are equal.
Now, does “0.9999…” denote the same thing as “1”?
Well… first off, a disclaimer: of course, one could invent an interpretation of this notation on which they denoted different things, just as one could invent an interpretation of notation on which “1/3” and “2/6” denoted different things (for example, they denote different dates…). But I’m not going to talk about that sort of thing right now. Instead, I’m going to talk about the standard, conventional interpretation of infinite decimal notation, the one that mathematicians mean when they use this notation, and the one which justifies the claim that “0.9999…” denotes 1.
When a mathematician gives an infinite decimal as notation for a number, what they mean by it is this: the* number which is >= the rounding downs of the infinite decimal at each decimal place, and <= the rounding ups of the infinite decimal at each decimal place. This is the definition of what infinite decimal notation means; it’s true because we say it is, just as the three letter word “dog” refers to a particular variety of four-legged animal because we say it does.
So, for example, when a mathematician says “0.166666…”, what they mean, by definition, is “The number which is >= 0, and also >= 0.1, and also >= 0.16, and also >= 0.166, and so on, AND also <= 1, and also <= 0.2, and also <= 0.17, and also <= 0.167, and so on.” What number satisfies all these properties? 1/6 satisfies all these properties. Thus, when a mathematician says “0.16666…”, what they mean, by this definition, is 1/6.
Similarly, when a mathematician says “0.9999…”, what they mean, by that same definition, is “The number which is >= 0, and also >= 0.9, and also >= 0.99, and also >= 0.999, and so on, AND also <= 1, and also <= 1.0, and also <= 1.00, and also <= 1.000, and so on.” What number satisfies all these properties? 1 satisfies all these properties. Thus, when a mathematician says “0.9999…”, what they mean, by definition, is 1.
[*: Of course, when one says a thing like “THE number which is…”, this may be taken to involve an implicit claim that there is a unique such number. So when mathematicians use infinite decimal notation, they also generally have a very particular number-system in mind in which these uniqueness claims are all justified. But, there are many other number-systems (just as useful ones, or even more useful ones, for many purposes; the world is diverse and our mathematical analyses needn’t be shoehorned into “one size fits all” form) in which there may be no number or many different numbers satisfying such systems of constraints; in such contexts, infinite decimal notation is generally less useful as a way to denote numbers, though it can still be used in essentially the same way to denote certain intervals instead.]
Arrrrgghh! Not this old stuff again! I am reminded of the Petunias in Hitch-Hiker’s Guide to the Galaxy saying “Oh no, not again!” Wasn’t this settled in a much earlier thread, like in the days of Archimedes?
Okay, I’ll make an attempt to add something to the discussion.
I think I see a problem with many people’s understanding of .99999… like erik150x that I don’t think I’ve seen mentioned yet. I’ll mention it, but I don’t feel much like expounding in detail.
What does an infinitely long decimal fraction mean in the very first place, anyway?
Consider a finitely long fraction, like .357 – In our decimal number system, that means:
3 x 10[sup]-1[/sup] + 5 x 10[sup]-2[/sup] + 7 x 10[sup]-3[/sup]
Everybody agree so far? It’s the sum of a finite number of terms, one term for each decimal digit.
So how do we define the meaning of an infinitely long decimal fraction? Well, it’s the sum of an infinite number of terms! Kaboom! We have a problem right there, Houston. Adding up an infinite number of terms is NOT like normal addition, and the very concept of a sum of infinitely many terms needs to be carefully defined right from the start.
It’s defined in terms of convergence to a limit. You create a sequence of sums of finitely many terms. That is, you create this sequence of sums:
– Sum of just the first term.
– Sum of the first two terms.
– Sum of the first three terms.
– Sum of the first four terms. --and so on.
Now you stand back and look at your sequence of sums, and ask yourself: Does this sequence of partial sums seem to be approaching a limit? (And if so, can you prove it? That’s even better!) If the sequence of partial sums approaches a limit, then the sum of your infinite series is defined to be that limit. And that is exactly how we define any meaning at all for infinitely long decimal fractions.
Once you get your mind wrapped around that, you can begin to deal with things like .99999…
Your sequence of partial sums is none other than:
– .9
– .99
– .999
– .9999
– .99999 --and so on.
Does this sequence approach a limit? Yes, it approaches a limit. It approaches 1 as a limit. That’s why, by definition, .999999… equals 1.
It is reassuring to notice that there is consistency in the mathematical world. You’ve seen all those proofs that .99999… = 1, for example by shifting the digits and subtracting, and tricks like that. Those are all tricks, but it’s comforting to know that they all give answers consistent with our definition of infinitely long decimal fractions in the first place.
There. How about that. I got started and ended up giving the full lecture after all. So there.
Is .999… + (1/infinity) > 1? Or perahaps undefined? If it’s undefined does that mean its a pointless question? Unkowable? Or just not understood?
My point here has been and remains that I have yet to see a convincing proof that .999… = 1
Every proof I have seen has hidden somewhere in the assumptions that 1/infinity = 0. When you use limits for purposes of proving .999… = 1 whether you are aware or not you are making an assumption that for the purpose of this question, 1/infinty = 0. Which in itself leads to contradivtions. Is 0 * infinity = 1?
Another way of looking at it limit use here without saying 1/infinity = 0, is that the limit of the sum of the series 9/10^x as x aproaches infinity is 1. However a limit is an inuitive assumption here, not a proof. It is not a mathematical identity like 1 - 1 = 0.
I’m honestly not trying to be stubborn here (though I can be and may seem to be). I really just haven’t seen a proof that convinces me. I have shown the holes in the algebraic proofs and the assumptions made in the limit process. Are the no definitive proofs?
Again you can say that the limit of 1/x as x aproaches infinity is 0, .but that is very different than saying 1/infinity is zero.
Please helo a lost soul sonewhere between exit 10 and exit 9.999… On the number highway.
Okay, fine, here’s your proof. “Just a sum of an infinite sequence, no limit involved”.
Now, 1/10 = .1, agreed? And 1/100 = .01 = .1[sup]2[/sup], agreed? So
.99999… = 9/10 + 9/100 + 9/1000… = 9 * .1 + 9 * .01 + 9 * .001… = 9 * .1 + 9 * .1[sup]2[/sup] + 9 * .1[sup]3[/sup]… =
∞
Σ { .9 * .1[sup]n[/sup] }
n=0
This is a geometric series, meaning of the form:
∞
Σ { a * b[sup]n[/sup] }
n=0
Which can be reduced to:
a/(1-b)
So
.9/(1-.1) = .9/.9 = 1.
19th century mathematicians, especially Dedekind and Weierstrass, defined real numbers and then proved statements about limits. Those suggesting OP is ignorant might overlook that things were less clear before 19th-century rigorous definitions.
In particular, as Indistinguishable points out, there are number systems where 1 and (1 - ɛ) are distinct, where ɛ is infinitesimal. It happens that this is not the system formalized by Dedekind and Weierstrass.
You assume what you seek to prove; i.e. that we use Dedekind’s numbers, that there is no (1 - ɛ).
I still think that best is to simply postulate The Axiom of Archimedes. If it was good enough for the greatest genius who ever lived, then it should be good enough for us. 
(ETA: Indistinguishable writes “by definition” so we arrive at the same result, the difference being that between “Do as you’re told” and “Here’s why.”)