I like this view:
let X=0.999…
then 1+X=1.999…
also X+X=1.999…
so 1+X=X+X and thus 1+X UNLESS we want to junk the rules of arithmetic that we like to use.
Okay, let’s say the error margin concept is true.
Then I might as well say that a particle has no mass, since we can’t possibly measure it - nor has it any volume since we can’t measure that either.
I don’t think that makes it matematically correct to assume things, based on the fact that we can’t measure it.
Hence, 0.999’ will never equal 1 either. It’s convenient for mathematicians, no doubt - but that’s about it! c",)
Oh, and by the way…haileris wrote in #11:
1/3 (one third) can be represented by 0.333…
If you take each thirds and add them up (0.333… + 0.333… + 0.333…) they add up to 1.0, not 0.999…
This argument assumes falsely that 1/3 can be represented by 0.333’.
The flaw in this one is, that 0.333’ is NOT 1/3, only an approximation(!)
I don’t think it should be a question of wether 0.999… = 1, but a question of wether 0.333… = 1/3.
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The color of 42 is Kansas City. I thought everyone knew that!
Yes, 0.999… = 1. OK, fine, but will someone please tell me what the next real number is???
A Student
I found this website after the uncomfortable explanation on what the e (constant) truly meant. And the moved to some other posts, as I loved the intuitive grasp of e that now makes perfect sense. I think the central issue is communication, where I define it as understanding the true meaning of each message exchange. I see a lot of people struggling with understanding other people all the time, UNKNOWINGLY. When you don’t know the underling (or hidden) premises, both can be talking about a different topic, and thinking they are talking about the same thing.
For an amusing (at least to me) example, see the airplane in a conveyor belt myth…will it take off? See the comments to the myth buster video here:
If you read the more than 80 pages of comments, you’ll realize that those that where “not right” are those that understood the problem different. And that’s probably several thousands of them. But until they saw the video, they couldn’t stop arguing that the plane cannot take off. And rigthly so: the instructions somehow implied to most that the plane remained stationary, probably using the wrong analogy with cars. They though (myself included) that the plane couldn’t advance, as part of the “assumptions”. There are several webpages with hundreds of posts and interactions. Then there where those that had it right for the wrong reasons (they started agreeing that a stationary plane couldn’t take of…there’d be no lift), yet they developed all kind of magic because they believed it could take of anyhow. Then some others interpreted all kind of weird ideas about the setup. For a sample discussion at ~280 comments (closed to new posts for sanity), you can go to:
This is the problem description:
“A plane is standing on a runway that can move (some sort of band conveyor). The plane moves in one direction, while the conveyor moves in the opposite direction. This conveyor has a control system that tracks the plane speed and tunes the speed of the conveyor to be exactly the same (but in the opposite direction). Can the plane take off?”
What’s amazing is that when Mythbuster did the test, many people argued (as I already said) that the setup is not what they expected: that it OBVIOUSLY takes of, as it can advance in the belt.
WHat I mean with all this is that there’s one thing I like about better explained: the analogies, the relationships approach, the examples and the illustrations are really helping a lot of people get on the same page, and communicate on the same level. And also (example: this thread) to challenge our axioms as something natural, that might even lead somewhere. I like it a lot. Thanks Kalid.
About 4 years ago, I was a bit stressed and took a month off. I am a bit stressed as well, so I tried to understand the meaning of e, pi and i, in a way that would make me understand them (use them intuitively in new forms, as if those were tangible objects).
I loved the explanations and then, by curiosity, I stumbled on this post, which reminded me of a week I spend looking at prime numbers just for fun 4 years ago.
I was looking at the meaning of prime numbers beyond the obvious “divisible by 1 and itself only”, and felt that it was very much like asking if what Color is 42 (in truth, 42 is not a call…it’s actually THE answer to the meaning of life and everything…eh).
In my quest to try to make meaning, I started to think about concepts like notation limitations (is the notation lacking to get at where I want to get to). About what is the relationship of infinity to unity, how does one traverse from systems that “add up” (1,2,3,…) to systems that make groups (1/2, 1/3, 1/4, …). Is it possible? (I liked the concept of hyperreals which I wasn’t aware off). I also though about periodicals. For example, if I can say 0,999… where there are infinite digits that can take me to 1…why can’t I say 1… (infinite string of 111…, no decimals). If .9999 is close to 1, is 999… equal to infinity? And is that infinity some kind of “1” in a higher level view of things? What is that “1” plus 45? At some point I didn’t know what I was getting at, but I assumed that if there’d be a way to bridge systems that go like (1,2,3,…) and systems that did (infinite/2, inf/3, inf/4) where infinite you can see as a higher order “1” (hyperreals looked a bit like this and catched my attention), then I would have answers to some questions.
My conclusion at reading 0.9999… is that it is the same as asking if 9999… exists. And if it exists, then “1” is of some kind of higher order…and what we could think as the "universe of ways to break apart that higher order number). 0.9999… is the way to express infinity in the lower order system (in terms of units starting with 1,2,3…).
@Federico: Thanks for the comment. You’re completely right – communication is key. All those unstated assumptions and premises can mean we completely miss the point of what each other is thinking. In my head, the process goes “I have an idea. I write it down. You read the words. You recreate the idea.” If my description was fuzzy or left out key information, the idea you recreate will not be the same one I had, leading to endless confusion (great example with the conveyer belt).
I find it really helpful to talk about the intuition of things (i.e., describe the idea in your head) vs. focusing on facts/details, which are the results of that idea, which you hope the person reading will re-create properly. Thanks again for the thought provoking comment.
if would like to answer this question through simple divisibility.
consider 1/9
as we can not directly divide 1 by 9 as it is greater than 1.so we adjust a decimal point.at first division we get numerator=0.1 and remainder 1with decimal value of 1/10 which equals it to 0.1
we know that dividend=divisorquotient+remainder
applying this we get 1=90.1+0.1 equation is satisfied
now similarly at the next iteration we get quotient 0.11 and remainder 1.this time as we should divide this remainder with 100(10^2 as it is 2nd iteration) again applying same rule ie dividend=divisorquotient+remainder we get
1=0.119+0.01 equation is satisfied again.
now if we generalize this equation then what we get is
1=(0.11111…n)*9+10^(-n) that is why we can state than 1/9 is not exactly 0.1111… but it is 0.1111…n+((10^-n)/9)to be accurate
now you wont find any difficulty.
I would like to answer this question through simple division.
consider 1/9
as we can not directly divide 1 by 9 as it is greater than 1.so we adjust a decimal point.at first division we get numerator=0.1 and remainder 1with decimal value of 1/10 which equals it to 0.1
we know that dividend=divisorquotient+remainder
applying this we get 1=90.1+0.1 equation is satisfied
now similarly at the next iteration we get quotient 0.11 and remainder 1.this time as we should divide this remainder with 100(10^2 as it is 2nd iteration) again applying same rule ie dividend=divisorquotient+remainder we get
1=0.119+0.01 equation is satisfied again.
now if we generalize this equation then what we get is
1=(0.11111…n)*9+10^(-n) that is why we can state than 1/9 is not exactly 0.1111… but it is 0.1111…n+((10^-n)/9)to be accurate
now you wont find any difficulty.
From the Internet Encyclopedia of Philosophy article on Bernard Lonergan…
“Each mode of knowing has its proper criteria, although not everyone reputed to have either common sense or theoretical acumen can say what these criteria are. A major impediment in theoretical pursuits is the assumption that understanding must be something like picturing. For example, mathematicians who blur understanding with picturing will find it difficult to picture how 0.999… can be exactly 1.000…. Now most adults understand that 1/3 = 0.333…, and that when you triple both sides of this equation, you get exactly 1.000… and 0.999…. But only those who understand that an insight is not an act of picturing but rather an act of understanding will be comfortable with this explanation. Among them are the physicists who understand what Einstein and Heisenberg discovered about subatomic particles and macroastronomical events – it is not by picturing that we know how they function but rather by understanding the data.”
The first thing to realize is that we are not talking about a number, but a process, and asking whether this process is equivalent to a certain number, which it is. That this is hard to accept is merely a reflection of the fact that we abstract our abstractions into concreteness (forgetting their abstractness) and then complain that our abstractions do not act concretely. 0.999… is not a number. It is a symbol that represents a process that never ends. This is something that cannot be imagined. But it can be understood. Like all paradoxes, however, we have to be very careful (rigorous) with how we describe what we are trying to understand. That is why the first thing we have to realize is that 0.999… is not a number, but a representation of a number. An abstraction of an abstraction. It is the symbol itself which causes the confusion. 0.999… is just a shorthand way of writing “9(1/10) + 9(1/100) + 9(1/1000) …" The conjecture that follows is, all (whole? rational? real?) numbers are the result of an infinite process. The interesting thing is this relationship and whether it is general. ?
All paradoxes are a misreading of the symbols that describe the paradox.
I only got to scroll down 1/3 of the bar (no just kidding, 2/5ths of the bar), I’ve got school tomorrow.
I quote only myself and noone else, and will post no sources. BUT.
1/3 exists only as a fraction. In decimals it immediately becomes an imaginary number. No 3 equal parts can be added to make one whole.
33+33+34=100.
333+333+334=1000.
3333+3333+3334=10000 etc.
So in practice, 1/3= 3…+3…+3…+n where n is 1-3…+3…+3…
Having a loads of decimals doesn’t make it any more difficult, it just becomes a lot more to write down. Heep hooray for n,x,y,a,b,v,t etc:)
The interesting part of this is the conception of infinity. Can you comprehend?
Hypothesis: The universe never ends.
Can you imagine? Do you understand the concept?
Some people ask "when you reach infinity, what’s beyond that?"
If there’s something beyond that, it’s not infinity.
Try to imagine infinity. If you can partially understand the idea and get butterflies or similar in your stomach, you are one step closer to understanding near-infinite numbers 
They do exist, they only near-infinitely doesn’t.
Hope you liked my 1-(3…)^3 cents
“So in practice, 1/3= 3…+3…+3…+n where n is 1-3…+3…+3…”
n is 1-3…-3…-3…
(edit)
“No 3 equal parts can be added to make one whole.”
(…) to make 1.
(edit)
Well you probably understand… sorry I am tired, long day today, long day tomorrow 
Math can be fun (considering near-infinite possibilities, noone can prove me wrong ^^ Or… can they?? But then we’d have to include hypocrisy into the algorhythm! Or… would we??) Ok I really have to go to bed now
Being a freshman in high school, I’ve never heard the word “Rigor” applied to math before. (Not at all a favorite subject of mine, despite it forming the basis of engineering/most other scientific disciplines.) Though it would certainly explain the common question of “Why can’t I just use a method that gets me the ANSWER?” Ah, the silliness of youth who expect it to be easy.
So, if for the sake of consistency we have an abstraction of an abstraction (Of an abstraction, of an abstraction…) at what point do we find our discussion to become founded in the roots of philosophy rather then math? (A very abstract field.) As I think this one has crossed that line. Assuming this is true, from a purely philosophical standpoint, an infinite never ends. (By it’s very definition, and prefix.) So if we are to say that 0.999 always fills another place with nine, then it cannot become a WHOLE one no matter how close it becomes to the number in question.
Just because one cannot reasonably observe a difference between infinite .999 and one does not mean that no difference exists between the two. Once again, from a purely philosophical point of view I see three obvious answers.
One, 0.999 does not equal one because an infinite 0.9999 will never reach a conclusion that would create a whole, complete, one.
Two, 0.999 equals one because from the human perspective no device could be invented from our current collective knowledge that would calculate to infinity, thus making the observable answer one.
Three, The problem is unsolvable because a device cannot be created that can calculate to infinity and as such it is a fools errand to make assumptions about the inherently unknown.
All three of these answers run on various assumptions about the state of our universe and knowing the conclusion to most other things I’ve written on the net, all three of those answers are almost guaranteed wrong.
Because this is math, not philosophy.
@irrelevant: Interesting comment – I’ll take a crack.
Whether 0.999… = 1 depends on what types of numbers we allow to exist.
The common case is that infinitely small numbers do not exist – i.e., a number is either measurable and comparable to other numbers, or zero. In this case, 0.999… = 1 is more of a symbolic equivalence, similar to 4/2 = 2.
0.999… is saying “What number is implied by the sequence 0.999…” and in more formal terms “What number is the limit of this sequence?”. Because we’re assuming we exist in a number system where infinitely small numbers do not exist (or rather, are 0), then 0.999… = 1 because there cannot be a difference between them.
If we do allow infinitely small numbers to exist (and hey – we allow negatives, imaginaries, irrationals, and other “strange” numbers to exist) then yes, we could represent the difference between 0.999… and 1 (as the infinitesimal “h”, say) and now we have an answer: 1 - 0.999… = h, a number which is smaller than our real numbers but still not zero. As others have mentioned, this introduces other issues, but the general insight is that the answer depends on whether you allow infinitely small numbers.
On philosophy: Math makes assumptions/axioms and explores their consequences. Certain mathematical models map better to our present understanding of the universe than others. A thousand years ago, negative numbers would have been baffling (they were only invented in the 1700s, by the way). Today, I don’t think the world would work without them. They started off as mathematical curiosities and we found ways to incorporate them into the problems/situations we face.