1.000… and 0.999…

 

Note: This post uses MathJax. If you see mathematical formulas with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.
 
Recently Julian Wilson sent me this letter:
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 $0.9999\ldots =1.0000\ldots$ equation is associated with several conceptual difficulties that math students have, which I will describe here.

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.

Definitions

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:

\[1+\sum_{i=1}^{\infty}\frac{0}{10^n}=1\]

and

\[0+\sum_{i=1}^{\infty}\frac{9}{10^n}=1\] 

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.

Completed Infinity

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). 

Among math ed researchers, I have learned a lot from papers by Anna Sfard and David Tall

References

 
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4 thoughts on “1.000… and 0.999…”

  1. Hi, long time reader here.
    You say that "a decimal representation of a number by definition represents the number that a certain power series converges to".  The problem is, we use decimal numbers long before actually making any such definition– and indeed, this is kind of necessary, for obvious reasons.  Power series are a cop-out when you consider that this question can (and often is) asked by junior high school students.  Therefore I have a different suggestion.
    Simon Stevin, the inventor of decimal, is the namesake of Stevin's construction (http://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Stevin.27s_construction), a construction of the reals which (IMHO) doesn't get enough play.  In the Stevin construction, the reals are obtained by taking formal decimal sequences and modding out by the equivalence relation which says that infinite tails of 9s are equivalent to the corresponding truncated number.  If we are working under this definition (which, again IMHO, is much more intuitive to ordinary people than, say, Dedekind cuts or Cauchy sequences!) then there is no need to invoke power series:  a decimal representation of a number, by definition, represents the number (equivalence class) of which it is an element.
    This has a particularly nice application to handling the "why is 0.999…=1" problem.  The answer goes something like follows:  yes, you're right, 0.999… and 1 are different decimal representations, however, if we formally compute their difference, we get 0.  We don't like that, because we don't like non-equal things to have difference zero.  Therefore, we declare 0.999… to be the same as 1, specifically to avoid the nasty situation of different things having difference zero.  And look, once we make that declaration, we can prove all these nice things like the distributive law…

    1. I didn't know about Stevin's construction; thanks.  I have taught only college students, and they get geometric series in Calc 2.  I don't know at what age you can bring in definitions as an essential part of math; I would have thought junior high school was too young.  

      1. Yeah, I wouldn't actually say the words "equivalence class" in junior high school math class 😉  But if asked why 0.999…=1, I would say, "Because in order for arithmetic to be useful, we need that numbers which are different have nonzero difference.  The difference between 1 and 0.999… turns out to be 0, so we declare 1 and 0.999… to be equal so that arithmetic doesn't break."  Depending how much leisure the student and I have at our disposal, I might also give an example to hint at well-definedness:  e.g., if we compute 2+1 we get 3, and if we computer 2+0.999… we get 2.999…, but 3 and 2.999… are declared to be equal, so it doesn't matter whether we used 1 or 0.999…

  2. Email from Julian Wilson:

    I notice that you state: "Julian's proposal provides a different interpretation of the notation, in which " 0.999… " and "1.000… " are strings of symbols generated by two different machines." Arguably, the discrete dynamical system is one machine with, from its point of view, two different inputs. Hence the distinctness of 0.999… and 1.000… . If you substituted 0.999… for 1.000… (or vice versa), claiming that these two numbers are equal, you would get different results in this context.

    I hope you don't mind my attempted clarification.

    On a different note, I agree about needing a math ed text for teachers. The key thing is that such a book should lean towards the practical rather than the theoretical, if you get my gist.

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