Today in one of my two calculus classes today, we got on the topic of .
I think it came up when we were talking about how to approximate the instantaneous rate of change in a problem: we had a function , and we wanted to estimate the instantaneous rate of change when .
So a student said let’s pick another point, such as . And we found an approximate instantaneous rate of change. Of course I asked “how could I get this answer more precise?” and someone said “add more 9s!”
So we realized we could pick a closer point, such as .
Of course, then we had some precocious youngster say: “why not get it super duper exact and plug in ?”
Ah hah. Many of them thought that was SUPER close to, but definitely not equal to, 300.
I went through the whole standard argument, which usually convinces most kids:
Let . Then .
So Which means . Which means .
I thought I had them. One student said I was breaking her worldview.
But then, THEN, they asked me an awesome question.
One said, and the others jeered: “Isn’t kinda breaking the rules of what you’ve been saying. How infinity is a concept? How this decimal goes on forever? And you said we couldn’t mix concepts with numbers. We can’t write or because we’re mixing concepts and numbers. So why can we talk about a number with a decimal that goes on FOREVER? Aren’t we mixing concepts and numbers? Isn’t this thing totally nonsensical?”
Okay, okay, they got me there. And they’re thinking deeply. And they’re getting me to think deeply.
So then I said: “okay, you have a point. So let’s see if we can mathematize this in a way that works with our understanding of things.” So I made a list:
But then I noticed we are still mixing infinity (as a concept) with these numbers because we’re adding an infinite number of them… so then below it I wrote:
Now we’ve written this decimal as what it truly is: a limit as the number of terms gets large without bound. And the limit has a definite value — the sum is getting closer and closer to 1, as close as we want to 1, infinitesimally close to 1, so we can conclude the limit of the sum is 1. So we can conclude that this sum is 300.
And then, sadly, I moved on.
The kids were interested in this conversation, and I think it could get at the heart of what we’re doing with limits (and then relating it to derivatives), and how infinity is a concept (for us) and not a number. But I don’t know what to do from here, where to go from here.
I didn’t convince everyone, and I don’t want to go too far afield with this unless someone out there can suggest a good idea. I mean, this idea of the limit, two things infinitesimally close together, is powerful . So is there a way to extend this discussion meaningfully? Philosophically? Anyone have any good activities out there, any good worksheets out there, any good readings out there, any good videos out there? I’m not even sure what sort of end goal I have. Just something that acknowledges the weirdness of repeating decimals, relating them to limits, and the concept of infinity…
For context for the class, this is the non-AP calculus class, where my kids are at very different levels of understanding.
 From my historical understanding, both Leibniz and Newton (and their followers) were still plagued with the idea that you would be kinda-ish dividing by zero when calculating the derivative, because they didn’t have the concept of limits in their formulation of calculus. This division by zero was unsettling for a number of contemporaries. And it wasn’t until Cauchy came along with his limit concept that he was able to give derivatives a solid philosophical foundation to rest upon.
If 0.9999… is the
Oops, as I was saying, by the limit process 0.9999… is the same thing as 1 at infinity, then analogously the limit of an expression as the denominator goes to zero at infinity is in fact 0. I have always thought that diff calc is really a cheating way of dividing by zero. Put another way, is going infinitely small the same thing as going infinitely large? It would appear that in order to avoid contradictions and misunderstandings calculus is nothing if not a grand mix of numbers and concepts.
I had an activity I used over the summer to illustrate the concept of repeated decimals. It will probably be far below your students, as it was designed for entering seventh graders, but it might help bring about better ideas on how to extend the discussion.
What we did was take one large sheet of poster paper and ask three students to divide it evenly between them. We said the one rule was that if they were going to cut the paper they had to cut it into ten even pieces. They began by cutting the paper into ten pieces and each taking three. With one piece left over (1/10), after some deliberation, they cut it into ten pieces and each took three. With one piece left over (1/100), after some deliberation, they cut it into ten pieces and each took three. With one piece left over (1/1000)… well, you get the picture. I see it as somewhat resembling an inductive proof, that you can go down an infinite regress of decimals getting smaller and smaller.
As for why 0.999… = 1, I always figure it merely has to be true, since 1 = 1/3 + 1/3 + 1/3 = 0.333… + 0.333… + 0.333… = 0.999… But that only works if you already have complete belief in the formality of mathematics.
Ohhh. I might do this next year. I think things like this would work in a nonAP calc class!
Well, I usually hold off on the .9 repeating argument until we get to series and we talk about the paradox of always going 1/2 way across the room.
The funny thing there is, if you tell a kindergartner the paradox they’ll say “of course you get across the room, it’s right there” (optimists they are thinking about the distance you actually covered). Then you explain it a bit more or get kids who think harder about it and they say you can’t get there because there’s always a half left (pessimists). But THEN you hit them with the idea that you cross the half-way points faster and faster (just stride right across the room) and prove it’s possible. So, we’re back to the original way of thinking about these things.
By a similar token, 1/3 and pi are real. We have a good definition of what they are, but the decimal representations have the same infinite problem. Basically, it’s a failing of the decimal notation, not of the number or its existence.
In some way, we could connect this to the philosophy of science and the equations we have from various studies of the world. Presumably, there is some “reality” out there and we can only try to mimic it using some models with notation our brains can understand and mathematically manipulate.
Ultimately, “300” itself is just a concept, too. But (in my head) the connection between 299.9(repeating) and 300 is like the particle/wave duality of nature. One is a continuous model and one is discrete which are two things we like to categorize in separate ways, but both are equally valid in different contexts.
I love that “particle/wave” analogy. And also the idea that 300 is just a concept too. Which is true. But I teach my kids infinity isn’t a number, but it represents something without bound. You can’t tame it, quantify it. (I know, I know, we can do the whole aleph nought thing). But 300 is something concrete and definite, and tamable and it is a quantity.
Newton and Leibnitz used “infinitesimals” which were easy to manipulate but bothersome to formalists. Limit theory replaced infinitesimals, though the limit theorems were much harder to work with. In recent years, infinitesimals have been put on a sound formal foundation (though the formal proofs are messier than limit proofs), so it is possible to go back to teaching calculus using infinitesimals.
See http://www.math.wisc.edu/~keisler/calc.html for a free textbook for teaching calculus using infinitesimals.
The best book I’ve read about this subject is Rudy Rucker’s Infinity and the Mind. It’s probably too ambitious for your class though. A more accessible book might be Mario Livio’s Is God a Mathematician?
These are great questions that tragically can’t be answered with any satisfaction at a high school level. I think infinity as a concept vs. infinity as a number is the main division between the mathematical philosophies of Platonism and Intuitionism. Some questions for your class:
Is mathematics invented or discovered? (To stimulate this question, you might want to bring up pi-hat:http://everything2.com/title/trichotomy+law)
Is zero a number or a concept?
Why can’t infinity be treated as a number?
Personally, I have a hard time accepting the number/concept distinction. Aren’t all numbers concepts?
All numbers are concepts, but not all concepts are numbers.
Treating infinity as a number means giving up some very useful properties of numbers (like that (1+x)-x=1), so most mathematicians find it better to make infinity a different sort of object. There is a theory of transfinite numbers which is interesting to study, but which does not have nearly the usefulness of the real numbers and the complex numbers.
I had a similar discussion in a non-AP calculus class this year. The thing that seemed most convincing to the kids was to imagine subtracting 300 – 299.999… . The difference is 0.000…0001, but since there are an infinite number of zeroes, we never get to the one, so the difference is 0 and 300 = 299.999…
Also, we talked about that some people think that 0.999… means just a really large number of 9s, but that we were lazy and didn’t write them all down. If that’s what 0.999… means, they are right that 299.999… isn’t the same as 300. However, if 0.999… really contains an infinite number of zeroes, we have to accept the conclusion that it really IS 300.
Another way to get at this argument is to look at the number line. I have had success with this idea by asking the following two questions:
1) If A<B, then is it true that there is some point on the number line between A and B?
2) What would a point between 299.999….. and 300 look like?
Ohh! I did that. I totally forgot I did that.
I said: “if you say these two things are different, they should be at different places on a number line. different numbers lie on different places on a number line. so if you have two different numbers, you can find a number BETWEEN them.” this worked pretty well, actually. Because no one could tell me a number between 299.9999 and 300. So I said: “then they must be the same number, if there’s nothing between them!” That hooked a few more, I think. I’m not sure though.
Is this any different from introducing the formalism of equivalence classes? I know that introducing that pushes you the wrong way through the number/concept duality by reminding them that, if we’re going to be all Cauchy about it, all numbers are concepts too. But there’s something reassuring when you’re 16-20 about knowing that math is the most internally consistent house of cards you’ve every seen.
I ask, “So what’s the difference?”
I mean really and mathematically, what is the difference between 0.9999999 … and 1?
I challenge my students to find a value between the two (this works better if they have some intuitive concept of the density of the rationals to go along with the discussion). But if you can’t find a DIFFERENCE between two numbers then there is no quantity to add to 0.9… to result in 1. Then the two quantities must be at least equivalent. And in the Real numbers this relationship is Isomorphic to equality.
This is actually one of my most favoritest concepts in math. I did a unit this year the included some of these ideas, and I’m developing a 6-week course that goes in further detail on this and issues around Zeno’s Paradox.
What’s weird is that kids are totally willing to accept that 1/3 = 0.333333… repeating forever. If we’re having infinity issues, then shouldn’t this be a problem too? Really, every infinite decimal expansion (that is, basically all of them), has this problem, as CalcDave pointed out.
At this point it would be a fun discussion to mention that there is subset (cult?) of mathematicians, the Constructivists (the other constructivists for us), who point out that most “numbers” as defined as some string of decimals cannot be described or defined in a finite number of steps, so how can we talk about them? 1/3 I can define, root(2) I can define, but give me a random number. How many digits will it have? [sorry that I’m smooshing together a bunch of problems here]
Or, as goes that favorite quote of topologists, “I could be bounded in a nutshell, and count myself a king of infinite space, were it not that I have bad dreams.”
You’re teaching a calculus class, and you have students who are enjoying forcing you to make limits formal!
Maybe I’m exaggerating, but I’d say almost every other part of the course is a place you can go with that. Enjoy it!
I am with DavidC. This is conversation is money in a calculus class. I love that this came up and that you honored their dissatisfaction.
For maximum pedagogical fertility I would entirely avoid trying to convince anyone that 299.9999…=300. You may not have time for this but I think the real payoff is in having them realize that they didn’t even know what 299.999… means, let alone what it equals. As you state in the post, 299.999… is actually the limit of an infinite series. Next natural question: what’s a limit (really)? Full answer: epsilon-delta. If that’s in your curriculum anywhere, then this is the perfect springboard.
Since you asked for related materials, I’ve totally been spending a lot of time on this exact question in my algebra-and-analysis-for-teachers class, and writing about it, including materials:
Over the Course of an Instant…
I have more where that came from that I didn’t blog about. I can send you stuff if you’re interested – just email me.
It isn’t formal by any stretch, but I usually show the following inductive process to my class and usually I’ll grab a couple students with it. I usually start with this and then (depending on the level of my class) move on into arguments that are more formal.
1/9 = .111111…
2/9 = .2222222…
3/9 = .333333…
and so on until
9/9 = .99999999…, but we also know that anything divided by itself is one, so .99999… = 1
The only context in which the infinite decimals could be treated consistently is as limits within a calculus framework. It could be common or nonstandard calculus (like the Keisler’s reference above) but a limit they need to be.
There are many infinities. While the one in “as n approaches infinity” is a concept, there are ordinal, cardinal, surreal, hyperreal infinities. However, none of these could be used in describing an infinite decimal.
For an introduction, see