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0.999... = 1
however
0.99999 <> 1
The repeating decimal is identical to 1, whereas the fixed decimal is not.
let x = 0.999... (where the ellipsis denotes infinite repetition of '9')
so 10x = 9.999...
Clearly, 10x - x = 9x
10x = 9.999...
- x = - 0.999...
------ ----------
9x = 9
Dividing both sides by 9 gives x=1.
Therefore 0.999... is identically equal to 1, mathematically, theoretically, and, in fact. (Well, that's what I think anyway.)
k
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I have seen in various posts both
0,9999 and
0,9999...(meaning that we are imaginig infinite 9's)
Well, here is the question.
What do you mean by saying that a given number is given by an infinity of digits? In other words, which is the meaning of the decimal representation of a number, when it is not finite?
The number is the limit of the partial ways of writing. In our example, to say that 0,9999999........ = 1
means exactly that 1 is the limit of the succession
0,9
0,99
0,999
0,9999
........and so on
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that puzzled me a lot, indeed. But the explanation is so easy to seem obvious: roughly speaking, I would explain this way:
Suppose that you are working in the plane - the usual plane, the plane you can imagine , the plane of the blackboard prolonged to the infinity - the so called "affine plane".
Consider there two parallel lines: don't worry, they don't meet.
But,if you consider the so called "projective plane" , obtained from the above plane by adding more "points", each one representing a "direction", the two lines will have the same "point",or, in other words, they have the same direction.
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Pooh-Bah
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In reply to:
If you don't believe that 0.9999999... is equal to 1, then please tell me how far from 1 it is. That is, what is 1 - 0.9999999... = ???? If you are right, the difference must be equal to some probably quite small but finite number
That is exactly what I am saying. 1- (0.999999...) is an infinitesimally small value not quite equal to zero. When we talk about 0.99999999... we are talking about the limit of x as x approaches 1, which is represented lim(x) x --> 1. This is understood to be infinitesimally close to 1 but not quite one.
The example I offered suffered from a lack of ellipses, but it holds true in differentiating 3.000...0001 from 2.9999999.... In fact, the more zeros or nines in the decimal point, the greater the difference between the two defined outcomes.
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That is exactly what I am saying. 1- (0.999999...) is an infinitesimally small value not quite equal to zero. When we talk about 0.99999999... we are talking about the limit of x as x approaches 1, which is represented lim(x) x --> 1. This is understood to be infinitesimally close to 1 but not quite one.
I don't think so. The limit is the limit. That is, the limit is taken as exactly equal to one - well, that's the way I learned (but as Bean pointed out, we learn a lot of incorrect stuff).
lim x = 1 (identically)
x->1
For example, we don't say that integrals solved exactly are "approximately equal to blah blah."
The integral of cos(x) is sin(x) + C, not approximately sin(x) + C (and what is an integral, but a limit?).
OTOH, we don't say that differentials are identically equal to zero. It's a bit confusing. Back in HS, before actually taking calc, I read this book by a guy named Boyer called "The History of Calculus and its Conceptual Development." From what I recall (of more than 20 years ago), these concepts of differentials and limits that we take for granted today were a really big deal at one time - very controversial.
I'm a bit envious of Bean for actually having taken Real Analysis. I'm sure she's got a much better handle on the particulars because of it. (I think maybe Real Analysis is - ahem, approximately - to calculus as number theory is to arithmetic.)
k
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Carpal Tunnel
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Carpal Tunnel
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I think my problem with this concept is that saying 0.999999999…=1 seems inelegant to me.
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1- (0.999999...) is an infinitesimally small value not quite equal to zeroOK, if it's not zero, then what is it?  If you're so sure that it's a nonzero value you must be able to tell me what it is! And the leaving-out or writing-in of ellipses makes all the difference here! No ellipsis means that the number ends just where you leave off. When we talk about 0.99999999... we are talking about the limit of x as x approaches 1, which is represented lim(x) x --> 1. This is understood to be infinitesimally close to 1 but not quite one. No, we are talking about the decimal representation of a number, that is, how do we write it? We all have a concept of what the integer "one" means. We are discussing whether it is more valid to write 1.00000.... or 0.999999... to express that number. And the point here is that either way is equally acceptable, both decimals (yes! including the 1.0000000...) must be extended to infinity to correctly represent the integer we all call 1. The proof that shows that writing 0.99999... means the exact same thing as writing 1.00000... shows how to write it as an infinite series (because that's what decimal really is), that is, an infinite sum of numbers. The value of the series approaches 1 as the number of the terms in the sum becomes infinite. For any small difference from 1 that you choose to inquire about, I can always add enough terms that I am closer to 1 than the difference you've specified. (This is more or less the epsilon-delta definition of a limit: you prove that for ANY small difference you are interested in, no matter show small, you can make your function/number/decimal representation get closer to its limit than that specified difference.)
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0.999999999…=1 seems inelegant to me.So does 0.33333... = 1/3 also seem inelegant? The problem is these silly, finite sheets of paper on which we are supposed to write! 
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Carpal Tunnel
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I think the problem some of us (non-mathematicians) have with this argument is that we view 1 as a 'pure' number, being unity. we don' need no stinking infinite string of zeros after the dot. it's just one. and to say that .9999999999999999999999999999999999999999999999999999999999999999999999999999... is equal to 1 is just yentzing with perfection.
but then, math isn't an exact science.
-ron o.
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No one anywhere said it was a better or more convenient way to write it. Certainly the most convenient way to write it is just 1. But the point is that the other way is equally valid. That's all. Even a mathematician probably might not find a good USE for it. But it's important to know it's there!
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