One aspect of having given so many school presentations is the feeling, the reality, that one has been saying the same thing over and over again. It is pretty much always the same few routines, employing the same tricks and the same jokes in order to deliver the same messages. It is impossible not to ponder “Haven’t they heard it all before? Aren’t they bored?”
Of course they may well be bored, although we’re sufficiently weird and gimmicky that the reaction of “Who is this nut?” is usually enough to at least keep them awake. But it seems that, in the main, students and teachers have not heard it all before. Not from us and not seemingly from anyone. Easy and excellent tricks are still largely unknown. There is still the same surprise at torus games and Klein bottles, still the weak understanding of fractions and equality (as evidenced by this), and of “patterns”. And 0.9999… seems to work largely the way it ever did.
In one sense this is hardly surprising. It is not as if the forces of reason have had much impact upon ACARA or AAMT or MAV or AMSI or AAS or MERGA or any of the prominent maths ed clowns. On the other hand it is common and natural to try to make up for a moribund curriculum by resorting to one-off aha tricks. One can appreciate the motivation but, and it is why we have strong doubts about doing the school thing, it is ultimately a losing game. In any case, although there may be some increased knowledge of such tricks, perhaps also through the increasing prevalence of maths candy on YouTube, the same routines seem to work pretty much the same way.
Long time viewers of The Burkard and Marty Show will know how often we have hammered 0.9999… = 1. More or less, always. There exist teachers and students, of course, who are aware that 0.9999… = 1, less often with a reasonable sense of why. In any case, sense or not, these teachers and students are clearly in the minority. Most, when asked if 0.9999… and 1 are equal, will try to weasel out of the question. They will reply with something like “It approaches 1″ or “0.9999… and 1 are equivalent“. When pressed for a yes or a no, when asked to state whether 0.9999… and 1 are equal, the overwhelming majority will say that they are not.
Does this matter? After hammering it about a thousand times, we definitely hope so. We’ll argue here why, for a number of reasons.
To begin, 0.9999… = 1 is true, and mathematical truth is obviously important. It is also important to state such truths in a properly precise manner. Secondly, that this intuitively implausible truth can be (sort of) proven in such a simple manner is a stellar demonstration of the power of algebra. Simply by declaring 0.9999… to equal X, by giving the thing a name, we can perform some simple and (sort of) standard arithmetic to deduce what that thing must be:
Thirdly, the demonstration provides an excellent platform for discussing what, and whom, students should believe, and why. Our demonstration is typically prefaced by asking for a show of hands: “Who thinks 0.9999… is equal to 1? … Who says no? …. Who’s chicken?” Then, and Burkard says we shouldn’t do this, and we do it anyway, we’ll specifically ask the same question of the half dozen teachers in the room; rarely do more than one or two provide the straight, correct answer. But the proper discussion follows the demonstration, which is typically part of a presentation titled, How To Cheat Like a Mathematician. So, we ask:
“Did I cheat?”
“Yes!”
“What if I promised you that I didn’t cheat?”
“We wouldn’t believe you!”
“Excellent!”
And it is excellent, the only intelligent response. If someone is willing to cheat, of course they are also willing to lie about whether or not they have cheated. Which means there is nothing we can then say to further convince the students that 0.9999… = 1. Which is fine. The point is not to convince the students that 0.9999… = 1, and certainly not by appealing to our mathematical authority. The point is to present a simple proof of a counterintuitive truth, and to have them begin to ponder, why or why not they may have been tricked, and why or why not the truth is so difficult to believe.
For the appropriate audience, this discussion can be extended to a consideration of the “sum”,
S = 1 – 1 + 1 – 1 + …
In an analogously simple manner we can “prove” that S = 1/2, which, unless the students have been infected by Numberphile’s nonsense, they’re definitely not willing to swallow. So why, then, is the one proof kosher and the other not?
Fourthly, the reactions to 0.9999… = 1 tell us something interesting about the fragility of mathematical belief. Most students blithely accept that 0.3333… = 1/3 (even if any proof they have likely seen is faulty). The effect of multiplying this equation by 3, however, does not typically result in the students accepting the value of 0.9999… but, rather, has them doubting the value of 0.3333… . Similarly, and even more so, students studying geometric series 
, who will happily compute 
until the cows come home, will nonetheless tend to lose their bearings when presented with the choices a = 9/10 and r = 1/10.
Fifthly and lastly and not leastly, teachers’ reactions to 0.9999… = 1 tell us something important about their mathematical sense and about their approach to mathematical belief. It is ok, if not great, if teachers have trouble believing that 0.9999… = 1. It is ok, if not great, if they cannot be convinced by the X proof. If, however, a mathematician declares the equation to be true then, absent some compelling reason to respond otherwise, teachers must respect that and accept that. Many teachers do. Well short of all teachers do.
Teachers must have a proper respect for mathematical authority. Fundamental mathematical truths are not a matter of opinion and the truth of these truths is not open for discussion. That many teachers do not appreciate this is a serious cause for concern, and reactions to 0.9999… are a clear and strong indicator of this. This, as much as anything, is what 0.9999… means for mankind.
“Similarly, and stronger, students studying geometric series…”
What does this mean? Did you mean ‘stranger’ or misplace some commas? perhaps I’m missing something.
I meant that students who are studying geometric series have a stronger basis upon which to believe 0.9999… = 1. I’ll think about the wording and may adjust slightly tomorrow.
Did you ever try the reverse method?
Instead of starting from 0.999999… and 1.000000… and proving that they represent the same number, start from a real number as a point on a line and write it out in decimal notation.
In each step of the process, you divide an interval into ten parts and pick the part your number is in.
What if at some point the number sits exactly on the boundary between two parts?
Well, then you have a choice, and you get two representations of that number.
I tried this with students and found that for many it is more convincing than doing “calculations” with digit sequences.
I think it is because it uses an idea of what a number “is” (a point on the real line), which is independent of the digit representation. The confusing point in the “digits only” approach is that intuitively you start from defining “numbers are the things that you write out as sequences of digits”, but then you “find out” that different digit sequences represent the same number, which somehow contradicts the definition.
Thanks, hjm. Yes, the geometric inevitability definitely helps. But in a way I think it is ducking the very issue, that a number may have different representations, and one must learn to distinguish the number from its representations. It’s not easy, even in simpler contexts.
In fact, for many students it is not even easy to distinguish “the number” from its representation as digits on a calculator. So, “asking the question” already is a step into the right direction, even if the first answer turns out to be wrong. But doesn’t the point of learning that “a number may have different representations” also include representations that don’t use digit sequences at all, like points on a line?
In a broader context, it is a typical problem of school mathematics (and teachers being educated into this way of doing mathematics): Without formal definitions, and introducing terms and concepts basically by “associating them with ideas in application contexts”, you can’t go down into the rabbit hole of logical paradoxes and counter intuitive theorems.
I’m not quite sure what you’re arguing hjm. You want to add in some geometric intuition? Sure. I don’t think it really solves the problem.
There is no question that *some* formalism is good and necessary in school mathematics, but I think one can push this line only so far, and not very far. Too pure, and the majority of kids will not get it: it will simply be ritual.
Well, my point was not “adding” geometric intuition (after making a statement about digit sequences), but to use geometry in the first place to get an intuitive idea of real numbers, and then talk about digit sequences (and derive the statement).
It at least removes the kind of circular argument, starting from defining numbers as digit sequences and then derive that the definition is “wrong”, because different digit sequences “are” the same number. Which I think is the main obstacle for those who don’t get it, however formal or informal the “definition” was.
Don’t we do the same when we introduce fractions as describing, well, fractional parts of a line (or a cake)?
of the cake is the same as 
of the cake. Easy to understand and (most) students immediately get it. How would you do it without the cake? Multiply both fractions with 6, see that the result is 2 in both cases, therefore the fractions are equal. That would be the same kind of argument as the 
equation.
No question, these algebraic arguments are nice. But in both cases, there is so much overload (mainly “formalism”) in them. You should understand
before learning how to multiply fractions, and you should (at least be able to) understand 
before you learn how to do computatons with infinite digit sequences (is this even tought in school?).
Hi, hjm. Of course you start, or close to start, with geometric intuition. You start fractions with pies or pizza or whatever, and you have to hammer the number line. But in the end, the numbers and their numerical representations are the thing, not the geometry.
If you still have to be talking about pizzas in secondary school then your kids are pretty screwed. The number line has much later uses than pizza, but in the end secondary mathematics is still dealing with decimals and n and x, not lines and dots.
Kids have to deal with decimals, including infinite decimals, all the time. Teh point of 0.9999… is it demonstrates that even the simpler infinite decimals are more complicated than they’ve been led to believe. It’s not choosing the more difficult scenario to discuss this: it is trying to deal somewhat honestly with the difficult scenario that they must deal with. In the end, the geometry here nor there.
I agree with the
and 
thing, but not with the decimals vs. lines and dots thing.
Real numbers were invented to link algebra to geometry. This is what they are almost solely used for in school after all. Everything in calculus and analytic geometry relies on the real numbers as a model for the line, plane, space, …. So, I somewhat disagree with the statement that, in the end, it is the numerical representation that makes a number, not being a point on a line. (That’s how I understood it.) It holds up on a higher level as well. Understanding Cauchy or Weierstaß without geometric intuition, but instead by imagining real numbers as infinite digit sequences, seems a bit hard to me.
That there is an “almost” one-to-one relation between real numbers and inifinite digit sequences seems to me rather an interesting fact, but not a really deep insight (ok, slighly exaggerated). And for students it might lead to the idea that numbers “are” what you see on the calculator, with all the bad consequences half of this blog is about.
But this seems to gets a bit off topic here, not directly related to the original article, so I will add another reply. Thanks to this discussion, the point I actually wanted to make just became clearer to me.
Nah. It’s all numbers now. I’m not arguing against geometric intuition and motivation, but in the end it’s all numbers.
Does
convey the same idea as 
?
It does to me.
As always thanks for writing. Please continue.
I understand the point you are making, so what follows is FWIW (and you probably already know …).
Nippers happily accept that
= 0.3333…
.
I am not sure this is the same as saying that they happily accept 0.3333… =
(Even though they do.)
They happily accept that
= 0.3333… because they create 0.3333… by dividing 1 by 3, as a result of us asking them something like, “What is the decimal form of 
?”
However, 1 ÷ 1 = 1 and so they do not have a way to think about 1 being equal to 0.9999… , even when approached from the direction of “What does 0.9999… equal?”.
In one instance, when doing the X proof stuff with a group of nippers, they were very impressed with it, as they were able to determine the fractional form for whatever decimal they wanted. They were not impressed when we did 0.9999…, all hell broke loose, “That is sick.” one said. He was really disturbed, as were others. I told them that one day I would fully explain it. Interestingly, that time is now, about 2 years on from their first meeting of the X proof stuff. We are a few lessons in. 🤞🤞
Re mathematical authority.
So true.
Lucky are those who have easy access to great mathematicians.
I am in debt to quite a few. 🙏🙏
Thanks, ST.
The kids don’t actually “create 0.3333… by dividing 1 by 3”. They create 0.3333 (no dots), with some tiny remainder. So, they should probably not be as accepting of 0.3333… as they are. That doesn’t matter, but I think the shaky grounds are somewhat exposed by X = 0.9999…
You don’t have easy access to great mathematicians, you have easy access to good enough and generous mathematicians. Which suffices. Other teachers could also easily have such access; most do not wish it.
You can actually “create”
by dividing 
. Just “miss” the first one, then you have a remainder of 
, into which the 
fits 
times, giving you a remainder of 
in the next step, …
Yeah, that’ll help.
How I understand that using the algebraic method is not kosher for s = 1 – 1 + 1 – 1 + … is that the infinite sum does not behave the same way as p = 0.9 + 0.09 +0.009 + … p is a convergent series and s is not.
Also, the the value of the “sum” given to s means something different what is given to p.
Thanks, Potii. I don’t understand your second sentence, both are “sums”, but of course your first sentence is spot on.
The point, of course, is that the algebra stuff ignores entirely the issues of convergence, and simply assumes that p or s or whatever makes enough sense to do the algebra. How one handles this gap, and whether to mention it all, depends upon the audience and the amount of time one has.
I mean the way the sums are attained is different and so the values given to the sums mean something different. For p, it is the sum of a converging series, and for s it is the arithmetic mean. Both are sums, yes, but I feel they should mean different things. I’m happy to be corrected.
Not sure what you mean. Dots mean the limit of the partial sums unless explicitly stated otherwise.
About your last paragraph.
Limits are used when summing a convergent infinite series. I understand that limits preserve arithmetic operations. So is that the reason the algebra works? Also, this is important because a number is a member of a group with shared characteristics (that being addition and multiplication).
Yes. The point of infinite decimals is that one can perform standard arithmetic upon them, and that is because they are the limits of convergent series.
Thanks to the other thread, I just came about a possible reason why
is so hard to understand for some. It was hinted at at several places, and perhaps this is the actual issue.
Let me tell a short anecdote. Some (very good) students were giving a talk about the Koch snowflake. It was really good and they even created it live on screen. In the end, I asked them: What IS the Koch snowflake? They were confused and claimed that they just explained it. I said no, their talk was about a (never ending) process, but not about its result. Still looking into questioning eyes, I explained further: Apparently, it is a set of points in the plane. So, there must be a way to find out if a given point belongs to it or not. That started an interessing discussion and (hopefully) lead to some deep mathematical insights for the students.
Maybe this is the obstacle here as well. Students and teachers think about (infinite) sequences of digits as processes. Coming from algebra with fractions as the previous topic, the (stereo)typical such process in mind is long division. But they have no idea what the result of this process is, because they never thought about it, because nobody told them that there is a difference between the process and the result. Clearly,
and 
(or even 
and 
) are different processes, if this is how you interprete the symbols. Before being able to understand that both have the same result, one needs to have an idea what this result actually is, and how it is different from the process.
So, what IS the result of the process
?
It is never said explicitily in the above article, nor in the Mathologer video (as far as I remember, maybe I should watch it again).
I would say it is a point on a line, but that would restart the other discussion…
Thanks, hjm. I believe that is exactly the stumbling block for students, that they cannot get past seeing 0.9999… as the process, rather than as the end result of that process. I think there is exactly the same issue with fractions, where students see 2/6 as dividing 2 by 6, rather than the result of that division.
As to what that end result is, 0.9999… is confusingly simple because the end result is simply 1. But more generally, at least for increasing sequences/sums, the end result is the least upper bound. Thsi turns into number what you want to maintain as geometry.
Yes, but maybe that’s the point where I disagree.
I see your point. But it is seems to me that you want something impossible.
Let me put it this way: I think it is impossible to understand that
means the same thing as 
, without knowing what 
means. In other words, unless you know and understand that 
(and thus also 
) stands for something else than just this particular series of digits, the question doesn’t even make sense to you, because, well, you don’t know what 
stands for.
Or, let’s say the students buy the argument as presented here. What if they then claim that
cannot be a number, because the “trick'” to convert it into something they are already familiar with fails? Would you then say that they missed something? If yes, what? Real numbers? Okay, but that’s the point. What are they?
Ugh. You’re just arguing fantasy.
No, I don’t want something impossible. I want something that is: (a) hard; (b) valuable; (c) necessary.
You simply cannot point to the number line forever. That is not what the mathematics now is. Kids need *some* number sense of real numbers which, in school, means some sense of infinite decimals. They are never going to get the full thing, but they need some working sense. They sort of kind of get it with 3.1415… , but of course they don’t really get it.
0.9999… demonstrates that they don’t really get it, that they don’t really understand the dots. 3.1415… feels easier because they don’t have to conclude much of anything. But, with 0.9999…. they do.
The point is not to give them a deeper understanding of real numbers and infinite decimals. The point is to demonstrate that it is deeper than they and their teachers have been pretending.
Okay, then I kind of missed the point.
I actually thought the article was about how to give students (and teachers) a deeper understanding of real numbers and infinite decimals, so that with this deeper understanding they can understand the now infamous eqaution.
Sorry for derailing the conversation.
I don’t think you derailed the conversation, to the extent that there were any rails in the first place. I simply think you’re wrong.
When I briefly taught a class of interested primary school students pieces of math, we went over 0.99999… = 1. My approach was not to teach them about a limiting process. I explained that the infinite sequence of digits can be simplified, like we do already with fractions, to 1. So they represent the same real number (which we had already in the class gone over as being anything but “real”).
I think if I had tried to use
as an example it would have been doomed, because the massive problem of determining the next digit of 
would have muddied the waters so that my point (which was just a very simple one) would have been lost.
I don’t know if my approach was good or bad, but I think the simple message of it is important and conspicuously absent from the curriculum.
One thing that I have found somewhat effective to nudge students, or other teachers, towards accepting 0.999…. = 1 is to ask them “if 0.999…. is not equal to 1, then what is a number in between them?”. (I’ve not found anyone who – tacitly or otherwise – doubted that for any real numbers x and y, there exists a real number z such that if x ≠ y then x < z < y). I've had some people try something like "well 0.99999 is between 0.9999 and 1, so we just 'add' an extra "9" to the end of 0.9999…." or even "just add them and divide by 2" but after you point out the errors in those responses, you can start to see them realise that there's a problem in the task you've set them. This doesn't necessarily completely eliminate their scepticism, but I've found it tends to get these people moving in the right direction.
I am currently reading Adam Gopnik’s book on mastery.
A take on your topic could be that the mastery of a mathematician is knowing when something is true. A master mathematician would know if something falls into the conjecture or proven category or could determine it.
If so your example of the 0.999… = 1 is akin to showing novices one of the master’s tricks. For some kids that will open up a world of possibilities just as learning their first card trick motivates a great magician to come.
It seems something we should expose every child to a few times over while we can.
Of course teaching the tables well and demonstrating this is entirely possible in a couple of months without much pain is worthwhile for everyone too.
Thanks, Stan. I think your basic sense of this is right. You want to give kids a view of the weird things that maths does, and the worlds it opens, but this is to help illuminate the path. It is not the path itself.
Your “mastery” summary seems a little confused. Is mastery knowing what is true, or is it being able to determine what is true? I think it is probably both.
I remember once my PhD supervisor was giving a lecture on some of his new work. At one point Brian was trying to capture a quantity, which required a string of inequalities: A ≤ B = C ≤ D ≤ E, that kind of thing, to conclude that A ≤ E. However, Brian realised at some point that he had made a mistake and that B ≠ C. So he had to look to see which of B and C was bigger. I knew enough to ponder the quantities but I had no intuition for what might happen. I remember thinking, “Uh oh, if the inequality goes the wrong way, the whole argument is screwed, and the theorem is screwed.” But Brian wasn’t concerned, because he knew that A ≤ E. For Brian it was simply a question of how to prove it, which might be fussier or less fussy, but was never really in doubt. In the end, he quickly showed B ≤ C.
Yes I meant both being able to prove things are true and knowing. I had meant the knowing in the sense of awareness of what is established knowledge whether it is a particular integral or Fermat’s last theorem. But you add another sense of intuition which I agree is also part of it.
Yes I agree the path is not the illuminating weirdness. It is more like the practice at tables so that playing with numbers is available to the mind at any time.