It worries me that as I move into leadership roles, I get fewer and fewer opportunities to teach. Hence, I was excited to give a talk at the Sonya Kovalevsky Day at Barnard College. The entire 9th grade of the Urban Assembly Institute of Math and Science for Young Women was at the event, and about a quarter were in my class. It was a challenging talk for a number of reasons. First, the students had not chosen to be there, and had no particular interest in math. Moreover, I felt that it was important to do more than give a fun math talk. I wanted the students to really learn something that would help them in their mathematics in school.
The topic I chose was “Does 0.9999… = 1?” I did this for a few reasons:
- I could pose an interesting question at the beginning and let the students think about it and come up with ideas.
- I could vary the level of the discussion based on student background. At one end I could always retreat to 1/3 = 0.3333…, multiply both sides by 3, and get 1 = 0.9999… (ignoring issues of how you multiply an infinite decimal); at the other end I could talk about infinite geometric series and convergence (although that was very unlikely). In the middle, I could talk about how you prove things in general, give the overall idea of a series, and talk about what convergence means.
- I would be able to emphasize ideas of “math makes sense”—that different pieces of math fit together and work in the same way—and I would also be able to reinforce basic mathematics skills. This is as opposed to a topic like, say, combinatorial game theory, where I could talk about logical reasoning but wouldn’t be able to tie it into what they’ve already seen or build their ability to understand numbers.
I began by introducing myself and having all of the girls tell me their name. I asked about their subway ride to Barnard and generally tried to bond a bit and be friendly. Then I put up the question “Does 0.9999… = 1?” I took some questions (“what does the bar over the 9 mean?”) and took a poll. A bunch thought no, a bunch thought yes, several said they were unsure. Great!
I next asked students to explain why they were or were not equal. I got a few ideas but nothing very deep. I knew that underneath, students were struggling with what this question even means, so I asked them straight up, what does this question mean? I got some not terribly enlightening answers.
Now it was time to actually help them understand the question. “What does = mean?” I asked (and wrote on the board). We discussed the notion of equality for a while, and gave examples of things that were equal (1 = 1, 1/4 = 0.25, 2 + 3 = 1 + 4); part of my goal was to emphasize that = means “are the same as”, not “is the outcome of an operation”. I asked them if x + 3 = 2x, and we had some debate before deciding that it is only true if x = 3 (I’m still not sure if they got this); then we had some more debate about if x + x = 2x before deciding yes. Although I had hoped to doubly address the meaning of = and the hidden quantifiers in most algebra problems, I realize in retrospect that this probably muddled the picture more than it helped.
Once I felt like they understood =, I asked what 1 means (briefly, although I wish I’d had time to talk about it more) and then I asked what 0.9999… means. I ended up getting drawn off-topic in the discussion, which is OK, but I wish I’d gotten more of a bead on “it’s 9/10 + 9/100 + 9/1000 + …”.
Instead, we ended up talking about what 0.3333… is. None of them recognized it as 1/3, which took me rather by surprise. OK, time to talk about 0.3333…
We spent some time discussing how to turn decimals into fractions in general, but quickly concluded that you couldn’t straight-up turn 0.3333… into a fraction; 333333…/100000… makes no sense. So instead I suggested looking at 0.3, 0.33, 0.333, 0.3333, and so forth. I was again a bit surprised: they seemed to have no intuition that these numbers are actually very close together. So it was time to take out a number line and draw them.
Where, I asked, is 0.3 on the number line? Only one girl knew how to put it up; she knew that 0.1, 0.2, …, 0.9 were equally spaced and so she found 0.3. But when asked about 0.33, she didn’t know. So we spent some time talking about how 0.3 is 3/10, and how to find 3/10 (divide the number line into 10ths; there’s 1/10, now we want three of them, so we go over here). Then we decided that 0.33 is 33/100, and we talked about dividing the number line into 100 pieces. Then we saw that 0.333 is 333/1000 and we talked about dividing the number line into 1000 pieces. (I actually liked this part a lot, because it required a certain amount of abstraction to visualize dividing the number line into so many pieces!)
Then I asked, “OK, where is 0.3333333333 on the number line?” The students really had no idea that it would be right by the other numbers. Some thought it would be very far out indeed! Others thought it would be close to 0; an original hypothesis had been that as you add 3s, the number gets closer and closer to 0. Eventually I got them to write it out as a fraction: 3,333,333,333/10,000,000,000. I asked them where this fraction was on the number line, and again, no idea. (Although they were amused, perhaps even impressed, at my ability to rattle off “three billion, three hundred and thirty-three million, three hundred and thirty-three thousand, three hundred and thirty-three.”) Eventually we realized that this fraction was close to the others, although I’m still not sure all of them were convinced. I finally explained that if I offered to give you that many dollars, I’d give you 34 cents and then you’d owe me some money! This seemed to help put it in perspective for them, and we briefly discussed how you might scam someone with this “trick”. Note to my future self: give the example of 5,000,000,000/10,000,000,000; it should be much easier to see that it is just 1/2 despite the large numbers.
At some point we noted that these were getting close to 1/3, although many of them didn’t have an intuition for where 1/3 is on the number line. We talked then about how to convert 1/3 into a decimal and I divided 3 into 1, at which point some of them realized that they had seen this before. That said, I realize in retrospect that I didn’t emphasize that fractions are division and so something of a learning opportunity was lost.
We’d also gotten that 1/9 = 0.1111… in this discussion, but still no insight on 0.9999…. I had them discover the relationship between 1/9 = 0.1111… and 1/3 = 0.3333…, that in both cases you can multiply by 3 to go from one to the other. Then we saw that 1/3 times 3 is 1, and that 0.3333… times 3 is 0.9999…. At this point I tried not to hint further, and despite having everything right there, the class went off on a wild tangent, thinking they could “add” something to 0.9999… to get 1. (A perfectly good theory, if they were different, and I wish I’d emphasized that more. We spent a while talking about what happens if you add 1, if you add 1/2, if you add 0.1, and 0.01, and 0.001, and so forth; I wish I’d had more time to allow them to do this themselves.) Somewhat flummoxed, I did a poll. Nearly everyone was now convinced that 0.9999… is not equal to 1!
Eventually I brought us back on track, and someone made the magical connection, and a student realized that since 1/3 times 3 is both 1 and 0.9999…, they must be equal. This part could have used more discussion, relating it back to equality, but we were really quite short on time. I ended the class by hinting at a graphical argument, and tantalizing them with the sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + …, and then we were done.
In the end, the class was engaged and had very high energy, but had such a deficit of basic facts that they really struggled. I don’t think I managed to bring everyone with me as we went forward, nor was the class nearly as student-led as I’d hoped because it was hard to get them moving in this context. I’m actually happy with the presentation itself, and I hope to continue to refine it and find ways to make it accessible to everyone.
I am reminded just how much students in schools lack, however. The insistence on moving students forward to algebra and beyond when they don’t fully have the number sense to understand how things go on the number line seems to be doing them a great disservice, and I wish we could find a way to enable teachers to address these basic challenges with them. Otherwise, they’ll be continually forced to memorize procedures and they’ll never understand why they do what they do.
Consider Learning 
++ on 5,000,000,000/10,000,000,000. For my Splash “calculus for middle school” kids who tend to be very advantaged math students, I give “what is 0.00000002/0.00000001” as a warm-up problem, and get them to tell me that if you keep inserting zeroes forever it will still be 2 (because later with introducing derivatives, they need to understand that an arbitrarily small number divided by an arbitrarily small number is not necessarily also small, and though they realize that in a snap when it’s explicitly brought up, somehow it isn’t by default in their conscious).
this also, in general, looks like it will be super fantastic next time you do it because you have a detailed reflection to look back on. I should write up thoughts like this after teaching, it would probably help me improve a lot.
That was a big part of why I wrote it!
I really don’t like the 3 times .33333… “argument.” First it’s logically circular, as in order to know that 1/3 = .33333… you need to know that 3 x .33333… = .99999… = 1. If you try to dodge this by saying 1/3 = .33333… follows from long division, you could also just get 1/1 = .99999… by doing long division. Finally any philosophical issues that come up with .9999… = 1 already come up in understanding why 1/3 is .3333… rather than infinitely many 3’s followed by a 4.
I agree with you, although for different reasons, and it was my backup plan! However, I would claim that the class was valuable for students even if the argument was not the best one, because it provides a good foil against which you can test your knowledge and understanding of fractions.
How do you get 1/1 = .999999… by long division? That is, how do you justify not simply getting 1.000000…?
Certainly you can also get 1.000000… by long division if you want to, but you could just as well get .999999999… by doing it slightly differently.
What do you mean by slightly differently? Do you mean using some other algorithm for division? Do you mean that when you divide 1 into 1 (or 9 into 9) you don’t write a “1” at the top even though you can, and instead write a 0 and see what happens? (Surely that needs more justification!) I’d be very curious about this.
It’s a little hard to explain in a blog post, but I just meant do ordinary long division but put 0.999999… at the top instead. So each step in the long division looks like 10-9 = 1, then bring down the 0 to get 10, subtract 9, etc. Keeps working just fine.
In this case it works, but you’re not following the algorithm and you need to prove that it works! This is the part I feel needs more justification. I agree that it works, because the numbers at the bottom stay bounded, but I think this example would be more confusing than helpful in a classroom unless there’s a good and easy reason why this kind of division is OK.
This is a great write-up and very useful for anyone involved in math education in any way. For cutting-edge researchers, it’s sometimes easy to forget just how much basic preparedness is missing in students today. I find it disheartening that students either don’t have any idea what fractions represent, or lack the ability to apply their ideas in unfamiliar situations such as 3,333,333,333/10,000,000,000. Key concepts and ideas such as these need constant emphasis and reinforcement or else they’ll just fall through. The teachers themselves also need to know the concepts and recognize which ones are fundamental. I suspect that the understanding deficit among students starts from the very beginning of the curriculum and grows each year until it becomes impossible to correct.
I’ve also had my share of Internet Arguments about 0.99999… = 1. I find that this is a very hard question to answer satisfactorily for the student. Congratulations to you for attempting it! The fundamental obstacle is that the building blocks in the definition of 0.99999…, such as 1/3, 1/9, and even 0.3, 0.33, 0.333, etc. are algebraic quantities: they are defined from fractions. But the final step, that of constructing the infinite decimal, is an analytic concept. The typical student who asks or considers this question is still struggling with the algebra, and is not prepared to make the leap to analysis. At best, the students might be able to take baby steps towards analysis, such as recognizing the concept of distance: 0.3 is close to 0.33 which is close to 0.333 etc. (although it seems that your class needed considerable assistance even to get this far). Still, it’s a big jump from that to the least upper bound property, which is what you need to show that 0.99999… = 1. Yes, you really do need the LUB property, since there exist number systems without this property (hyperreals, surreals) in which infinitesimals exist, and where 0.99999… can reasonably be defined to equal something different from 1. Of course, you would never be able to mention the LUB property to this kind of audience, but you cannot avoid its implicit presence in the background; although the two are not formally logically equivalent, I am not aware of any rigorous way to show that 0.99999… = 1 without invoking the LUB property (other than by fiat), and I have thought about this for a while. I find that even students with no ability whatsoever to reason logically about mathematics will still have some intuitive sense that “0.99999… = 1” is a deep equation requiring a powerful new concept for a full understanding. It can be frustrating, to say the least, when students recognize (even subconsciously) that they need something, but cannot understand what it is!
I wonder how long your talk was, and how many were in attendance? These variables would certainly affect what kind of talk I could give.
Actually, we’re saved a little bit here: I’m pretty sure that all you really need for the core issues is the Archimedean property, which is weaker than the LUB property, and much more intuitive to believe. (I’ve also wrestled extensively with how to address 0.99999… = 1 without the LUB property.)
I think the core argument is that if 0.9999… is not 1, then there must be a number between them. What is the average of 0.99999… and 1? What number could you find between them? What would you get if you subtracted one from the other if not 0? Ultimately students are forced to add extra digit after infinitely many digits, and a socratic discussion of what those extra digits might mean is usually enough to demonstrate that it’s not the number system that we usually work with, and the exercise of thinking about it with guidance can be very valuable.
Ideally, if the students are comfortable enough with fractions and decimals, you’d establish much earlier in the class what decimals really mean. In particular, the idea that 0.99999… = 9/10 + 9/100 + 9/1000 + … is going to help tremendously with the above explanation. Then if students try to add on extra digits on the end, you can ask them what that would mean as a number. They won’t be able to come up with anything once they’ve established the meanings of decimals in general, although you can discuss how there are other number systems with infinitesimals briefly without confusing them too much, and then you’ve even seeded the idea that we can create other number systems by setting up sufficient rules at the beginning.
Another way to deal with this is to talk about geometric sequences. From the formula for finite geometric sequences, you can derive the formula for infinite geometric sequences. Then you have a formula showing you plain as day that as you add more 9s, you get arbitrarily close to 1.
One other technique I’ve seen, although I haven’t used it much myself, is to do a graphical argument. For example, you can intuitively get the idea of 1/2 + 1/4 + 1/8 + … = 1 by taking a square and dividing it in half; then taking the remaining half and dividing it in half; and so forth. It’s not hard to see that every point in the square will eventually be covered by one of the rectangles you’re drawing. You can do the same thing with 9/10 + 9/100 + 9/1000 + ….
To answer your more prosaic questions: an hour, and about a 12-15 students.
Are there self-consistent theories of infinite decimals for the hyperreals or surreals according to which .9999… is less than 1? In the hyperreals, you could cut off the sum at some randomly chosen infinite decimal place, but that’s pretty arbitrary. In the surreals, it seems like the most natural definition of an infinite decimal would be a cut defined by its finite approximations above and below, but since those are all rational, the cut will be the ordinary real number.
If we look at .9999… as the sequence of its truncations, i.e. {.9, .99, .999, …}, 1. as the constant sequence
{1., 1., …}, and our hyperreals as real sequences modulo sequences that vanish on some element of an ultrafilter that is an extension of the filter of subsets of natural numbers with finite complements we will get 1. – .999… = an infinitely small number represented by the sequence {.1, .01, .001, …}. See my article http://en.wikipedia.org/wiki/Hyperreals#An_intuitive_approach_to_the_ultrapower_construction for more details.
Oh, you want to drag in the *construction* of the hyperreals as an ultraproduct. I find that inelegant, and essentially just as arbitrary, since it amounts to picking the particular infinite natural number represented by the sequence (1,2,3,4,5,…) and cutting off the decimal expansion there.
It may be not too elegant, but at least it is understandable. Whatever way you do it, you need to postulate the existence of an ultrafilter or introduce a similar assumptions to get a hyperreal field. Anyhow, you asked a question and you got an answer, sorry you didn’t like it.
This is a beautiful description of what sounds very much like a math circle. Thank you for sharing the details.
How many girls were you working with?
I think it was in the range of 12-15, although I don’t exactly remember.
Also, the style of math circles was very much on my mind when I was planning this class! I wanted to capture the spirit of inquiry about interesting questions that seems to be a common thread among them.
On my first day teaching eighth grade, we hit a major speed bump when we got to the sequence 1, 4, 9, 16, 25… and not a single one of them knew what the pattern was.
I blame calculators.
Calculators are absolutely a factor; the beautiful thing about doing calculations by hand is that it forces you to meditate on the numbers, at least if you’re the sort to naturally think about what you’re doing. Unfortunately, I am not really convinced the situation was any better before calculators.
I was wondering the other day (when I was teaching an upper level college class on why every real number has a decimal expansion and related issues like .99999… = 1) whether students typically knew that 1/3 = .3333… before the invention of calculators. It seemed plausible to me that it’s only because the calculator turns 1/3 into .3333… that student know it. On the other hand, maybe before calculators people did enough long division by hand that they also knew this then.
Clearly, the solution is to redesign calculators so that when you divide 1/1 they give the answer .9999…
I wonder whether students would be less confused if we never taught them the concept of ‘infinite decimal’ at all. As David said, the meaning of an infinite decimal is a fundamentally analytic one, essentially involving limits, which we don’t try to teach students about in general until calculus. Of course, that’s not something we can change, but are we really explaining an important concept, or just mitigating the damage done by teaching people to trust symbols that they don’t have the conceptual background to really understand?
Well, we do have to teach them the decimal expansions for 1/2, 1/4, and 1/5. I don’t see a good way to get around mentioning the decimal expansion of 1/3.
What I was trying to say is that 1/3 doesn’t _have_ a decimal expansion in the same way that 1/2, 1/4, and 1/5 do. Maybe we are doing students a disservice by implying that it does.
Don’t students need decimal approximations? I feel like knowing that 33% is roughly 1/3 is important. And once you’ve gone there, it’s hard to avoid the question of why it’s only approximate.
Of course they need to know that 1/3 is approximately .33. I was wondering whether it would be possible to tell them that .33, .333, .3333, and so on are successively better approximations to 1/3 without telling them that it’s meaningful to write “.3333….”.
Maybe we are doing students a disservice by not explaining approximations in general on an elementary level, without dragging in the whole abstract elephant of the real numbers.
Hey Dan, I’ve got some relevant recent writing on “does 0.999… = 1?” that I think you might be interested in in light of this.
Ah yes! I’ve read that post before, and it’s really good. Now I’ve actually scrolled down to the comments, and noted them as fodder for future classes. :)
And why would you want to do that, Mike?
Oh, sorry, I got it, you don’t want to confuse them…
I remember approaching this exact same problem as a student in 7th grade, when the teacher stated with no explanation that .9 repeating = 1. Between classes the other students and I came to the conclusion that it wasn’t actually true (that it must’ve been something mathematicians decided because practically you had to eventually round /somewhere/; ah, the days when we all thought math was only for practical purposes!), because if you go 9/10 of the way to somewhere, and do it again, and again, no matter how many times you would never actually get there. You couldn’t give a distance between it and 1, but that was like how you can write x < 1, and x can be any infinitesimal amount smaller than 1 to satisfy the inequality; .9 repeating was on one side of that line and 1 on the other.
It came into a little bit better perspective when I got to infinite series a year later (where it was replaced by something unsatisfying like "it's true because inexplicably weird things happen when you add infinity things"), but I don't think I had an "aha" moment until I read Anders's answer to this post: http://www.quora.com/Is-it-possible-to-count-in-base-pi . I realized I had never even considered the possibility of 2 representations meaning the same number until I had to think about irrational bases. Don't know if that's useful to you as a teaching tool, but thought you might find it interesting.
I am an engineer who made a mid-life career change to teaching. I’m currently teaching chemistry in a public high school, and I am continually amazed by the lack of basic number sense many of my high school students exhibit. They all must have algebra I in order to take chemistry, but the majority of my students have extreme difficulties with math. Take the problem “round 299854 to the thousands place” as an example. We did this as part of a year-end review (we had covered place value and rounding early in the year). At least a third of the class had no idea where the thousands place was. What surprised me, however, was that even after a review of place value and rounding rules, some students rounded to 299 or even to 300 in addition to the usual mistake of 299000. When I talked to them about their thought process I found the same lack of general comprehension of what rounding meant and what place value meant that you found in your lesson. Unfortunately, I did not think of putting up the number line, but I am certain that these students had the same issues with the number line. Incidentally, this same issue occurred when rounding 144 to the tens place, so it was not because of the comma between 299,854.
I find myself going back to manipulatives frequently when explaining what should be a simple math concept, such as last week when we did a lab measuring the energy content of snack foods with calorimetry and I found that many of the students could not understand that to convert from 1/2 serving to 1 serving they should multiply by two. I had to get out beakers and fill two of them with 1/2 a serving of Cheetos and another with a full serving of Cheetos and show them how two half-full beakers was the same as 1 full beaker. I also took out an empty beaker and poured the two half beakers together. After that, they were able to understand that 1/2 + 1/2 = 1 whole, but a few were still unable to see that 1/2 + 1/2 = 2 x 1/2. These students seem to lack the fundamental understanding of what multiplication means, which they should have grasped in elementary school.
Re-teaching math is a significant part of my chemistry teaching job. So far in my non-quantitative observations I have to say that using real things to demonstrate math concepts works with this group of students better than anything. I suspect that many students lack a bridge from concrete math to abstract math.