It is well known that students often have trouble accepting that $0.999\ldots$ is the same number as $1.000\ldots$. However, there is at least one context in which these could be regarded as in some sense as being distinct. In a discrete dynamical system where the next iterate is formed by multiplying the current value by 10 and dropping the leading digit, and where you make a note at each iteration of the first digit after the decimal point, then 0.9999… generates a sequence of 9s, whereas 1.0000… generates a sequence of 0s. The imagery is of a stretching a circle, wrapping it ten times around itself and recording in which sector (labeled 0 to 9) you end up.From the dynamical systems perspective, being in state 9 (and remaining there after each iteration) is different from being in state 0.
The decimal representation is not the number
Another way of describing the equation is to say that "$0.999\ldots$" and "$1.000\ldots$" are distinct decimal representations of the same number, namely $1$. Julian's proposal provides a different interpretation of the notation, in which "$0.999\ldots$" and "$1.000\ldots$" are strings of symbols generated by two different machines. Of course, that is correct. But they are both correct decimal notation that correspond to the same number.
Mathematical writing will sometimes use notation to mean the abstract mathematical object it refers to, and at other times the text is referring to the notation itself. For example,
$x^2+1$ is always positive.
refers to the value of $x^2+1$, but
If you substitute $5$ for $x$ in $x^2+1$ you get $26$.
refers to the expression "$x^2+1$". Careful authors would write,
If you substitute $5$ for $x$ in "$x^2+1$" you get $26$.
This ambiguity in using mathematical notation is an example of what philosophers call the "use-mention" distinction, but they apply the phrase to many other situations as well. Mathematicians have an operational knowledge of this distinction but many of them are not consciously aware of it.
A decimal representation of a number by definition represents the number that a certain power series converges to. The two power series corresponding to 1.000… and to 0.999… both converge to 1:
They are different power series (mention) but converge (use) to the same number.
Most students new to abstract math are not aware of the importance of definition in math. As they learn more, they may still hold on to the idea that you have to discover or reason out what a math word or expression means. In purple prose, THE DEFINITION IS A DICTATOR.
This does not mean that you can understand the concept merely by reading the definition. The definition usually does not mention most of the important things about the concept.
A common remark by newbies about $0.999\ldots$ is that it gets closer and closer to $1$ but does not get there. So it can't be equal to $1$. This shows a lack of understanding of completed infinity. The point is that the notation "$0.999\ldots$" refers to a string beginning with "$0.$" and followed by an infinite sequence of $9$'s. Now "$s$ is an infinite sequence of $9$'s" means precisely that $s$ has an entry $s_n$ for every positive integer $n$, and that $s_n$ is $9$ for every positive integer $n$.
- The expression is gradually unrolling over time, and does not ever "get there".
- All the nines are already there.
Both the preceding sentences are metaphorical. They are about how you should think about "$0.999\ldots$". The first metaphor is bad, the second metaphor is good. Neither statement is a formal mathematical statement. Neither statement says anything about what the sequence really is. They are not statements about reality at all, they are about how you should think about the sequence if you are going to understand what mathematicians say about it.
Metaphors are crucial to understanding math. Too many students use the wrong metaphors, but too often no one tells them about it.
We need a math ed text for teachers
I am thinking of precalculus through typical college math major courses. The issues I have discussed in this post are occasionally written about in the math ed literature but I have had difficulty finding many articles (on the web and on JStor) about these specific ideas. Anyway, articles are not what we need. We need a modest paperback book specifically aimed at teachers, covering the kinds of cognitive difficulties math students have when faced with abstraction.
What I have written in abstractmath.org and in the Handbook are examples of what I mean, but they don't cover all the problems and they suffer from lack of focus. (Note that the material in abstractmath.org and in posts on this blog can be used freely under a Creative Commons license — click on "Permissions" in the blue banner at the top of this page).
- 0.999… (Wikipedia)
- Decimal representation of real numbers (abstractmath.org)
- Definitions (abstractmath.org)
- Metaphors and Images (abstractmath.org)
- Multiple representations (mathematics education) (Wikipedia)
- On Reform Movement and the Limits of Mathematical Discourse by Anna Sfard
- Potential versus Completed Infinity: its history and controversy by Eric Schecter
- Use–mention distinction (Wikipedia)