I was just reading some old correspondence with a friend J who periodically writes me regarding a math question he and his son are pondering together. The exchange was pretty juicy, about how many ways can an even number be decomposed as a sum of primes. But actually, the juiciest thing we got into was this:
Is 1 a prime number?
It was kind of a fight! Since I and Wikipedia agreed on this point (it’s not prime), J acknowledged we must know something he didn’t. But regardless, he kind of wasn’t having it.
Point 1: This is awesome.
Nothing could be better mathematician training than a fight about math. Proofs are called “arguments” for a reason.
When I went to Bob and Ellen Kaplan’s math circle training in 2009, I was heading to do a practice math circle with some high schoolers and Bob asked me, “what question are you opening with?” I said, “does .9999…=1?” He smiled with knowing anticipation and said, “oooh, that one always starts a brawl.”
Well, it wasn’t quite the bloodbath Bob led me to expect, but the kids were totally divided. One kid knew the “proof” where you go
Multiplying by 10,
Subtracting,
so
and the other kids had that same sort of feeling like, “he knows something we don’t know,” but they weren’t convinced, and with only a minimal amount coaxing, they weren’t shy about it. The resulting conversation was the stuff of real growth: everybody in the room was contending with, and thereby pushing, the limits of their understanding. Even the boy who “knew the right answer” began to realize he didn’t have the whole story, as he found himself struggling to be articulate in the face of his classmates’ doubt.
Now this could have gone a completely different way. It’s common for “0.999… = 1” to be treated as a fact and the above as a proof. Similarly, since the Wikipedia entry on prime numbers says, “… a natural number greater than 1 that has no positive divisors…,” we could just leave it at that.
But in both situations, this would be to dishonor everyone’s dissatisfaction. It is so vital that we honor it. Everybody, school-aged through grown-up, is constantly walking away from math thinking “I don’t get it.” This is a useless perspective. Never let them say they don’t get it. What they should be thinking is that they don’t buy it.
And they shouldn’t! If it wasn’t already clear that I think the above “proof” that 0.999…=1 is bullsh*t, let me make it clear. I think that argument, presented as proof, is dishonest.
I mean, if you understand real analysis, I have no beef with it. But at the level where this conversation is usually happening, this is not a proof, are you kidding me?? THE LEFT SIDE IS AN INFINITE SERIES. That means to make this argument sound, you have to deal with everything that is involved with understanding infinite series! But you just kinda slipped that in the back door, and nobody said anything because they are not used to honoring their dissatisfaction. As I have pointed out in the past, if you ignore all the series convergence issues, the exact same argument proves that …999.0=-1:
Dividing by 10,
Subtracting,
so
If you smell a rat, good! My point is that that same rat is smelling up the other proof too. We need to have some respect for kids’ minds when they look funny at you when you tell them 0.999…=1. They should be looking at you funny!
Same thing with why 1 is not a prime. If a student feels like 1 should be prime, that deserves some frickin respect! Because they are behaving like a mathematician! Definitions don’t get dropped down from the sky; they take their form by mathematicians arguing about them. And they get tweaked as our understanding evolves. People were still arguing about whether 1 was prime as late as the 19th century. Today, no number theorist thinks 1 is prime; however, in the 20th century we discovered a connection between primes and valuations, which has led to the idea in algebraic number theory that in addition to the ordinary primes there is an “infinite” prime, corresponding to the ordinary absolute value just as each ordinary prime corresponds to a p-adic absolute value. Now for goodness sakes, I hope you don’t buy this! With study, I have gained some sense of the utility of the idea, but I’m not entirely sold myself.
To summarize, point 2: Change “I don’t get it” to “I don’t buy it”.
Now I think this change is a good idea for everyone learning mathematics, at any level but especially in school, and I think we should teach kids to change their thinking in this way regardless of what they’re working on. But there is something special to me about these two questions (is 0.999…=1? Is 1 prime?) that bring this idea to the foreground. They’re like custom-made to start a fight. If you raise these questions with students and you are intellectually honest with them and encourage them to be honest with you, you are guaranteed to find that many of them will not buy the “right answers.” What is special about these questions?
I think it’s that the “right answers” are determined by considerations that are coming from parts of math way beyond the level where the conversation is happening. As noted above, the “full story” on 0.999…=1, in fact, the full story on the left side even having meaning, involves real analysis. We tend to slip infinite decimals sideways into the grade-school/middle-school curriculum without comment, kind of like, “oh, you know, kids, 0.3333…. is just like 0.3 or 0.33 but with more 3’s!” Students are uncomfortable with this, but we just squoosh their discomfort by ignoring it and acting perfectly comfortable ourselves, and eventually they get used to the idea and forget that they were ever uncomfortable.
Meanwhile, the full story on whether 1 is prime involves the full story on what a prime is. As above, that’s a story that even at the level of PhD study I don’t feel I fully have yet. The more I learn the more convinced I am that it would be wrong to say 1 is prime; but the learning is the point. If you tell them “a prime is a number whose only divisors are 1 and itself,” well, then, 1 is prime! Changing the definition to “exactly 2 factors” can feel like a contrivance to kick out 1 unfairly. It’s not until you get into heavier stuff (e.g. if 1 is prime, then prime factorizations aren’t unique) that it begins to feel wrong to lump 1 in with the others.
I highlight this because it means that trying to wrap up these questions with pat answers, like the phony proof above that 0.999…=1, is dishonest. Serious questions are being swept under the rug. The flip side is that really honoring students’ dissatisfaction is a way into this heavier stuff! It’s a win-win. I would love to have a big catalogue of questions like these: 3- to 6-word questions you could pose at the K-8 level but you still feel like you’re learning something about in grad school. Got any more for me?
All this puts me in mind of a beautiful 15-minute digression I witnessed about 2 years ago in the middle of Jesse Johnson’s class regarding the question is zero even or odd? It wasn’t on the lesson plan, but when it came up, Jesse gave it the full floor, and let me tell you it was gorgeous. A lot of kids wanted the answer to be that 0 is neither even nor odd; but a handful of kids, led by a particularly intrepid, diminutive boy, grew convinced that it is even. Watching him struggle to form his thoughts into an articulate point for others, and watching them contend with those thoughts, was like watching brains grow bigger visibly in real time.
Honor your dissatisfaction. Honor their dissatisfaction. Math was made for an honest fight.
p.s. Obliquely relevant: Teach the Controversy (Dan Meyer)
Research in Practice 
“Division by zero is undefined.”, “3 is a factor of 0.”
Totally! Word!
“Is infinity a number?”
“Is a line parallel to itself?”
“Is the square root of 4 plus or minus 2?”
we actually got in trouble for the third question. We started the discussion among the teachers and Admin intervened and told us that the answer is +2.
I am so mad as your administrators! I wonder if there’s some way we could help them see the light…
Any number to the zero power is one. Um, except zero. Zero to the zero power is not one.
Thanks for the great post!
And negative powers mean …
“Is pi a number?”
“Are one-third and two-sixths the same number?”
“What is the circumference of a circle?”
Ben
Curious about something here. When you discuss the idea of defining a prime as a natural number with exactly two natural number factors, you express dissatisfaction. I understand the idea that prime factorization of a natural number should be unique, but I am at a bit of a loss as to how to express the importance of that to my Algebra students. Any insights would be greatly appreciated.
Great question. First of all, honestly for myself, “exactly 2 factors” isn’t a bad definition. I experience it as a very significant difference between 1 and the primes that it only has 1 factor and they all have 2. My point is that my experience of this definition as satisfying comes from my perspective of knowing other more advanced things. To students for whom “prime” has already been defined as “only factors are 1 and itself,” which is typical, this definition can seem contrived; the reason it doesn’t for me is because I have numerous other ways of looking at it that all highlight the difference between 1 and the primes (for example, the unique factorization theorem), so the difference between 1 vs. 2 factors falls on me with the added significance that it seems to hint at all these bigger things. A student first learning about primes lacks this perspective.
To speak directly to your question, I think the best way into the importance of the unique factorization theorem is to put kids in a position where they need it in order to know something about numbers. One great example is factor counting. On some level I think this kind of experience (using the fact that prime factorizations are unique in order to gain other knowledge) is what made us all think it is important.
The ability to do this topic justice would depend on how much time you have for it. In my own Algebra I and Algebra II experience, the math related to prime factorization wasn’t really a curriculum priority. (Kind of a shame but true.)
In light of this, allow me to stay on message: the point is not to convince students that 1 is not prime, unless you have time to really go there and get into it! As I said in the post, a pat answer for “why is 1 not prime?” such as “it only has 1 factor and the primes all have 2” is dishonoring everybody’s dissatisfaction if it is treated as settling the case. As I take you to be suggesting, “unique factorization fails if 1 is counted as prime” is not a better answer unless the kids think unique factorization matters! To make it matter to them requires the type of experiences I mentioned above – they need to see how it gives them knowledge they wouldn’t otherwise have. That is gonna take time!
Anyway, like I said, great question.
With my college students, who sometimes want to include 1 as a prime number, I say, “Hmm, let’s try that, and I make a factor tree that includes lots and lots of 1’s.”
I am assuming unique factorization, not trying to prove it. They see from my example how different 1 is from the “other” primes, and we move on.
Would you recommend a different process? (If I think like a math circle leader, I might want to give it quite a bit more time and tell them this is a good question we’ve uncovered. “What does it mean for a number to be prime?” But the last time this came up (a few days ago), we were in the middle of something else.)
I just finished researching Jerome Bruner’s theory of discovery learning and constructivism. In one of his works, he describes giving students a handful of beans, and having them construct numbers. Eight for example, can be constructed as two rows of four, or four rows of two. Nine, must be constructed in three rows of three. Certain quantities, however, will always have an incomplete table. These patterns happen to be called prime numbers. Using that as a definition, the pattern for one is one row of one. That means one is not prime.
@ Russell – sounds like a very interesting book, but actually the argument seems like it would lead you to conclude 1 *is* prime. 5 is 1 row of 5; 7 is 1 row of 7; the primes can only be organized in this one way; so 1 is just like them. Right?
Ben – touché… The table for one is unique in which the number of rows equals the number of columns. There has to be something special about that.
I would also like to clarify, Bruner never claimed his example as an argument for one not being prime, that was me. I was thinking of one as being a “complete” table.
@ Russell – don’t worry, that was clear! Regarding the specialness of 1, I don’t think it’s gonna be hard to get kids to agree that 1 is special; but if I had learned of a prime in terms of arrays, I believe my inclination would be to see 1 as prime and as special among the primes. Maybe 1 is “the most prime of all!”
And to Sue – I think that the factor tree sounds like a good limited-time solution. Another possibility is to just acknowledge the richness of the question and move on without settling it. Another possibility is to acknowledge the richness, ask them if they are curious to know what you think about it, and then do the factor tree thing only if they say yes. Either way, in an algebra context I think you’re generally right not to go into full-on math circle mode due to time constraints. (Bummer, though, right?)
“Does a point take up space?” Some variation of this question had all 5 of my colleagues’ geometry sections going at it.
Word.
Hi Ben,
This might be an interesting read.
http://arxiv.org/abs/1209.2007
The last great mathematician to consider the number 1 as a prime was none other than G.H. Hardy and this was as recently as 1933 ! It’s not surprising that this question raises discussions with younger minds.
Good day !
Awesome.
I discuss “is 1 prime” with just about every class that I have.
We (the all-knowing mathematical “we”) could decide either way. We get to define words in ways that are useful, and make sense.
I discuss, briefly, the Fundamental Theorem of Arithmetic. We decompose a number or two or three into primes, and always get the same thing (down to order).
And then I suppose that we let 1 be prime. 10 = 2 x 5. Or 1 x 2 x 5. Or 1 x 1 x 2 x 5, etc. But if we choose the other, and let 1 not be prime, then the FTA works, each number has a unique prime factorization, angels and trumpets, etc. Which way did the mathematicians choose? Does that choice make sense?
Jonathan
Are two lines that exactly coincide parallel?
In case you hadn’t thought of it: there is something to the proof that …999.0 = – 1. Namely, 1 + x + x^2 + … = 1/(1-x) when |x| < 1, and so we can interpret this sum as having value 1/(1-x) for any value of x (other than 1). If we take x to be 10, we get (formally) that …111.0 = -1/9. Multiplying both sides by 9 gives …999.0 = -1.
So the usual proof that 0.999… = 1 can be thought of as a special case of the proof that 1 + x + x^2 + … = 1/(1-x) (multiply both sides by 1-x and observe that you get 1).
As you observe, what is missing in this formal manipulation is the considerations of convergence (e.g. that .1 is < 1, while 10 is not!). But even without taking the convergence into account, it has some meaning, as the formal power series manipulation shows.
—
Also, there is another interpretation of infinite decimals that doesn't explicitly use infinite series (although it is closely related to them in the foundations of real analysis), namely: 0.a_1a_2a_3… (here a_i are the digits) means the
following number: split the interval [0,1] into 10 subintervals labelled 0,…,9 ; then the number is in the a_1'st subinterval. Now split this interval into 10 subintervals — then the number is in the a_2'nd of these intervals. Continue … . (So we are describing our number as the intersection of these successive subintervals, each 1/10th the width of the previous one.)
From this point of view, we see that .999… is always in the last interval, however much we continue our subdivisions, and geometric reasoning (or geometric intuition, if you prefer) shows that it has to be the right hand endpoint of [0,1], i.e. the number 1.
Sure, I mean, of course, the identity 1 = (1-x)(1 + x + x^2 + …), occurring in some formal power series ring, is the engine behind both the “proofs” quoted above that 0.999… = 1 and …999.0 = -1. (And if we map x into any topological ring in which 1+x+… converges, it will become a true statement in that ring; thus …999 = -1 in the 10-adics…) I didn’t mean to be dismissive of the argument in the full scope of the mathematical universe. The manipulation (1-x)(1+x+x^2+…)=1 is a beautiful, beautiful piece of algebra.
But this is orthogonal to what the post is about, and I want to make sure my point is getting across. What I’m hoping to leave everybody with is that a student is within their mathematical rights, and actually I think should be encouraged, for being less than convinced by the formal argument that 0.999… = 1. I want us to hear kids out about why they don’t buy it. I want us to act like the full story is deep, not facile! If a teacher is in a setting where they can take the time to start playing with series or otherwise really delving into the meaning of nonterminating decimals with their middle schoolers, great, let them do it! These questions are so generative! And, I want them to respect the depth of the mathematical issues at stake. I think most teachers probably don’t feel that they have the time to get heavy into these issues, in which case what I want is that they acknowledge and encourage students’ inevitable dissatisfaction, and can tell them that there’s a deep and rich story going on here, even if they don’t have time to treat it fully.
If one is in a position to really explore this content with students, your interpretation of nonterminating decimals in terms of nested intervals is like exactly the type of new angle that it would be very thought-provoking to bring in. To stay on message, though, it would be irresponsible to act like it settles the case. Of course how it all plays out will depend on the particular students. But, for starters, since it is a different interpretation of decimals than kids typically see, you’d have to count on it raising as many if not more questions than it settles, at least initially, just because it’s new. More substantively, let me probe the geometric reasoning you’re alluding to. Perhaps it would be clear that 1 lies in the intersection. But the further claim that the intersection is a singleton would be problematic, for precisely the same reason it bothers kids that 0.999…=1 in the first place. I argue this to myself as follows: the intersection can’t contain two distinct points x,y because the lengths of the intervals will eventually become smaller than |x-y|. But if I’m a kid, and I really feel like 0.999… is just a little bit smaller than 1, then I have a belief in, or at least am willing to entertain, some kind of small nonzeroness (whatever the difference is between 1 and 0.999…) that might conceivably be less than the length of every one of those intervals. When I did the math circle I described above, it did come up at some point to ask, “what is 1 – 0.999…”? The kids, mostly high schoolers, actually re-fitted the subtraction algorithm to allow one to start from the left, in order to be able to run it on this problem! (This type of digression illustrates why really exploratory settings like math circles are so awesome.) And they saw that the answer was 0.000… But the kids who didn’t want to conclude equality felt that maybe there was a 1 “all the way at the end” (!)
My point is just, these questions are rich, and in their heart of hearts, students are not gonna buy pat answers, whether or not they say so. I want everybody to make space for this, so the standard of truth in all math classrooms hews toward folks are authentically convinced rather than they believe us ’cause we waved our hands and we’re the teacher.
Dear Ben,
Thanks for the thoughtful reply. You make very good points and it was a stimulating post to read.
Best wishes,
Matt