There are a countable number of rationals having two different expansions.
\(\displaystyle \dfrac{1}{2} = .4999999...\)
\(\displaystyle \dfrac{2}{25} = .07999999...\)
edit: I think the biggest misconception about repeating decimals is the gerund that sits in its name. The number .9999... isn't doing anything, it isn't repeating, it isn't continuing, it isn't approaching anything. It is a fixed number.
My description of the one-to-one nature of this notation does require a
canonical form for repeating decimals. I thought it was obvious -- but as you illustrate, a lack of rigor can be confusing. This is an important exercise though, because it underscores the fact that we are talking about an expression representing what you call "a
fixed number". {
Edit: I agree.} However, It is NOT a point on a number line, it
represents a point on a number line.
Edit: I agree That, "The number .9999... isn't
doing anything, it isn't
repeating, it isn't
continuing, it isn't
approaching anything." The adjective
repeating, inspires those notions. Those notions are interesting. But as you point out, they generate a lot of diversion.
Below is a response to your valid statement, "There are a countable number of rationals having two different expansions." It shows a lack of rigor (on my part) to claim that there is only one exception. Once the notation that I suggest is accepted, then rule #3) can be used to resolve all of those cases. Rule #3) is stating that 0.999.... = 1.
So, for my posts assume the following canonical form for repeating decimals (base 10) in pseudo- scientific notation to be as follows.
Example : \(\displaystyle \frac{230}{7}\text{ = 3.}\overline{285714}\) E 1
1) Digits under the vinculum represent no more than one cycle of repetition.
2) Decimal point is always located just before the first digit under the vinculum.
3) \(\displaystyle \overline{9}\) is replaced by \(\displaystyle \overline{0}\) and add 1 to the mantissa.
4) E n represents that the resulting number is multiplied by 10^n
5) Represent the additive identity as \(\displaystyle \text{ = 0.}\overline{0}\) E 0
Edit: Analogous example for repeating integer digits
{ Example : \(\displaystyle \frac{230}{7}\text{ = }\overline{571428}\text{.9}\) E 2 }
used in future posts.
Now there is only one way to express a rational number using this repeating decimal notation.
The discussion brought by the OP is that he finds it unclear what is meant when someone writes the expression 0.999... We have in this thread demonstrated that lack of clarity (as is true in message boards across the web). I submit that the root of this confusion comes from not calling 0.9999... an expression, and not considering what is the most useful (while consistent) definition.
My vote: Let 0.999... be a non-standard way of writing
\(\displaystyle \frac{1}{1}=1. \bar{0}\) or perhaps \(\displaystyle \frac{1}{1}=1. \bar{0}\) E 0
