# Explaining why (or whether) zero and one are prime, composite or neither to younger children

There are lots of discussions out there about whether $$1$$ is a prime number (such as this one) and even about zero (such as this question, though note zero does generate a prime ideal in $$\mathbb{Z}$$ by the standard abuse of terminology ever since Kummer).

However, I haven't seen a discussion of the related question on this site - namely:

How should one talk about the question of $$1$$ or $$0$$ being prime (or composite, or neither) with primary or middle school children?

I'm particularly interested in the question about zero.

The fundamental theorem of arithmetic uniqueness rationale is probably a little heady for them (though this book gives it a great try) or even college students at times. But I also don't really like the idea of saying "because we said so" when so much of school math feels like this to students.

Semi-sarcastic, but still possible final thoughts:

If you think you can adapt Conway's notion of factoring all integers via $$-1$$/$$1$$ as "prime powers", that would be great. Also on topic here would be discussion of non-uniqueness of factorization for this age group. But I think that those are probably asking a bit much.

• Yes, this really did come up at bedtime tonight. – kcrisman Oct 27 '19 at 1:15
• Re "because we said so": But that's exactly it... One could only try to explain why it makes sense to "say so", e.g. as in the accepted answer of the question you linked. – Jasper Oct 27 '19 at 11:57
• @Namaste You have to omit 'distinct' though, otherwise squares of primes are also included. – Paul Oct 27 '19 at 21:47
• For what it's worth, some of your questions in your more recent comment could definitely be turned into good questions on this site; I know that many even much older students struggle realizing that one can think of integers as also rational via $5 = 5/1$, asking whether that is allowed, and again for different developmental stages there could well be different strategies. – kcrisman Oct 28 '19 at 20:36
• Some Greeks, Euclid for instance, did not even consider one a number. A number meant more than one. Indeed it is generally what is meant English by “I have a number things to do today.” – user1027 Oct 28 '19 at 21:32

"Because we said so" is a bit of a conversation closer, I agree. But "Because some people agreed a long time ago to define it that way so we could have conversations where we all understood each other. Does that seem like it would be a good idea?" is both more inclusive and more correct.

I don't even think it's that hairy to talk through the FTA with anyone old enough to understand primes. Have everybody take out a sheet of paper and express, say, $$12$$ as a product of the smallest numbers possible (with repetitions being okay). You take a sheet of paper and write $$1\cdot 2\cdot 3\cdot 1\cdot 1\cdot 2$$. Then you can discuss what everyone's paper has in common and if there are any differences. Then point out that people a while ago realized that everyone's papers would have a lot more in common if they would just agree that $$1$$ isn't a prime number, and that's what lead to the convention being established.

• I like the sheet of paper idea for an activity that keeps things from giving the entire statement of FTA while maintaining the spirit. I may take exception to the first paragraph a bit, at least in terms of it feeling any better than "we said so", as it still doesn't really tell them why and feels like a conversation closer. Of course, that might just mean I'm raising too inquisitive of children :) – kcrisman Oct 28 '19 at 3:28
• Yeah, I'm a big fan of constructivism, especially in math education. Some things that are hard to teach can be relatively easy when we give students the environment in which they can "teach themselves". It's not a magic bullet for every lesson, but I like using it when it makes sense. – Matthew Daly Oct 28 '19 at 13:12
• And it is ultimately a fair question. You can tell a particularly precocious young person that mathematicians create new theories by bending "the rules" and seeing what changes. If the new theory turns out to be useful, then you're an applied mathematician. If the new theory is cool but nobody can find a use for it, then you're a pure mathematician and (as my college advisor would joke) you just need to wait 150 years for physicists and engineers to evolve to the point where they can think of an application. – Matthew Daly Oct 28 '19 at 13:33
• "Because 0 x 0 = 0 and 1 x 1 = 1 but 2 x 2 ≠ 2, 3 x 3 ≠ 3 ... If p x p ≠ p, and p can only be divided by itself or 1 and result in an integer, then p is prime. Because 0 x 0 = 0 and 1 x 1 = 1 neither 0 nor 1 can be prime. That makes 12 uniquely equal to 2 x 2 x 3. If we make the rule that you have to put the prime factors in increasing order, every integer larger than 1 either is a prime itself, or is a composite that can be written as the product of two or more primes, which will always be the same for that number." – Monty Harder Oct 28 '19 at 22:00

There was a multiplication table posted on the wall. Like this \begin{alignat}4 1 &\quad 2 &\quad 3 &\quad 4 &\quad\cdots\\ 2 &\quad 4 &\quad 6 &\quad 8 &\quad\cdots\\ 3&\quad 6 &\quad 9 &\quad 12 &\quad\cdots\\ 4&\quad 8 &\quad 12 &\quad 16 &\quad\cdots\\ \vdots&\quad \vdots &\quad \vdots &\quad \vdots &\quad\ddots\\ \end{alignat} but going up to $$10$$.

What numbers appear in this table somewhere? All of them. (Advanced language: all postive integers) because they are all in the first row.

What numbers appear only once in this table? Just the number $$1$$. Any other number appears at least twice, once in the first row and once in the first column (and possibly elsewhere).

What numbers appear exactly twice in this table? The numbers $${}\ge 2$$ that do not appear except in the first row and first column.
Definition these are called "prime numbers".

What numbers appears three or more times in this table? All the numbers in the table when you omit the first row and the first column.
Definition these are called "composite numbers".

• This feels very procedural to me, and doesn't explain why we bother calling numbers prime. I would not think it would help a child understand. – Sue VanHattum Oct 27 '19 at 17:47
• The OP said he didn't like the explanation "because we said so". But that's exactly what this answer does, with a slight twist "because the table says so". – IMil Oct 28 '19 at 0:38
• @IMil true but I think this one is a bit better than "I said so", because of the visualization. – kcrisman Oct 28 '19 at 3:23
• @SueVanHattum it might not help us say why we call them prime, but one advantage is it might lead to exploration of how many times a number shows up in the table etc. I agree it couldn't be an entire answer but may help visualizing. And note that extended to zero we would certainly have lots of zeros so zero isn't prime in this context. – kcrisman Oct 28 '19 at 3:25
• Another fun question: which numbers appear an odd number of times? – supercat Oct 28 '19 at 19:05

FYI: here's some pro and con: http://primefan.tripod.com/Prime1ProCon.html

One was originally considered prime. It is prime with the most convenient ("natural") definition. It got excluded from prime-ness because many other higher theorems would be complicated by leaving it as prime. Essentially "prime" -> "prime*". The definition of primeness was tweaked to exclude one.

This definition change is glossed over with the "different from itself" formalism that doesn't sound as awkward as saying "except one". But clearly the reason for the change and the effect of it was the same as if we had adjusted the definition with a suspicious sounding "except one". I personally think just saying "except one" is a little more direct and revealing. (Within the definition of prime. I'm OK with the change to prime*. But let's be real...we did it to vote one off the island. If it didn't simplify a lot of higher math statements, we would not have made the change. Certainly wouldn't have made the change if it complicated them!)

P.s. I personally think a too dogmatic "one is not a prime...how dare you think that...you are just wrong" stance is too harsh to give to the child. Just being honest and saying they tweaked the definition because it makes later math simpler is more honest and less upsetting, even though it leaves an impression of capriciousness.

• "If it didn't simplify a lot of higher math statements" - even if beyond a grade schooler, I'd love to hear the simplest thing this quote references. – JTP - Apologise to Monica Oct 27 '19 at 12:59
• @JoeTaxpayer: FTA: "There is a unique way to write any natural number above 1 as a product of primes." Not true if 1 is prime, e.g., $5 = 1 \times 5 = 1 \times 1 \times 5$, etc. Options to resolve: (1) modify/expand the statement with some exception about 1, or (2) say 1 isn't a prime number. – Daniel R. Collins Oct 27 '19 at 20:19
• @DanielR.Collins: you can easily recover uniqueness of prime factorization by demanding that factors are unique after simplifying (since $1^k = 1$ for all natural $k$). – nomen Oct 28 '19 at 21:23
• @nomen: That would fit into my 1st category of options, because that's only a statement you can make about the natural number $1$ (and seems like a burdensome way of expressing it, IMO). – Daniel R. Collins Nov 1 '19 at 4:02
• Demanding that factors are expressed in lowest terms is burdensome? – nomen Nov 1 '19 at 5:26

How should one talk about the question of 1 or 0 being prime ... with primary or middle school children?

Depending on what you did before you will have an easy or a hard task:

If the children were told:

A prime number is a natural number which cannot be divided by other numbers than by 1 and by itself.

... you will have problems explaining why 1 is not a prime number because 1 is a natural number that cannot be divided by any other number than by 1 and by itself.

However, if they were told:

A prime number is a natural number which can be divided by exactly two numbers: By 1 and by itself.

... it will be easy to explain why 1 is not a prime number: 1 is only divisible by 1, so it is not divisible by exactly two numbers, but by only one number.

This means that the key is that the children are told a more or less correct definition of the word "prime number"; otherwise you will later have problems explaining why 1 is not a prime number.

• Best answer so far. While other answers are also correct, they are so convoluted. This is the answer that my teacher gave when I was like 8 years old in class. Based on response, most if not all of the class understood why 1 is not a prime in an instant. – n1t4chi Oct 30 '19 at 7:01
• @Namaste "A prime number is a natural number which can only be divided by two unique numbers: By 1 and by itself." – CJ Dennis Oct 30 '19 at 8:09

A good way to lead to the uniqueness of prime factorization and the convention that $$1$$ is not a prime is to build factor trees (that's common in elementary school these days in fourth grade, sometimes third grade).

       24                24                24
8    3           6      4           2    12
2   4            3   2   2  2            3    4
2  2                                      2  2


The leaf labels always turn out the same, up to order, and $$1$$ never shows up.

I did this once in a math club. Later in the day, in the classroom, Alejandro, who's in the club, volunteered the definition "A number that only $$1$$ and itself go into, except that $$1$$ is not a prime."

The teacher asked "Why isn't $$1$$ a prime?"

"Because Dr. Bolker says so."

So appeal to authority often wins over thought.

• In fact, this is one of the things one of my kids brought up while we talked about it. I suppose as long as they don't use $-2\cdot -2=4$ then we are okay, because $0$ won't show up in these - except with $0$ itself! But then the question is whether zero should be composite. – kcrisman Oct 28 '19 at 3:12
• @kcrisman Yes, it is composite by the definition that 0 is the product of 1 and itself - but it's also the product of any other number and itself as well, while primes can only be factored in one way (discounting negatives). – Darrel Hoffman Oct 28 '19 at 17:37
• I think the point is not so much that people reading this don't know that $0$ is the product of various other numbers with itself - surely they all do! Rather, the point is to think from the point of view of a younger person who may not be convinced/interested in formal definitions, or who may wonder how many "categories" there are of numbers. See e.g. Dr. Math for one person's query about exactly this distinction. – kcrisman Oct 28 '19 at 19:29

This one is really very simple. First, tell them what a prime number is:

A prime number has exactly two different factors.

(If they don't know what factors are, and they ask about primes, the correct answer is "well, first you have to know about factors...")

With that definition, it is very easy to figure out 0 and 1. Is 1 a prime? No, because it only has one factor. Is 0 prime? No, because every number is a factor of 0.

Of course, the next question is likely to be "why is that the definition?" or "what's their purpose?" or some such. The answer to that is also simple - every number bigger than one is made up of prime factors. And, for every number, there is exactly one combination of primes that makes that number.

If you build each number n using n square blocks in rectangular configurations, there are multiple configurations for each composite number. (4 is 4 by 1 or 2 by 2.) The primes are the ones that can only be built as a 1 by n rectangle.

It seems clear that 0 would be neither prime nor composite, when looked at this way.

The easiest way to understand why we don't call 1 prime is that when we factor a number like 12 down to primes, we like having just one answer. (Which is often true in mathematics.)

I was asked "Why not teach that the factorization is 1𝑛×23×32?"

The prime factorization describes how to break the number down into factors, emphasis on 'down'. 1 doesn't break it down into smaller factors, so it's not useful.

The preceding question, "Why is having one answer a good thing?", is harder to answer. My answer to it, for now, is a bit of an exploration of thoughts.

I think it feels natural to want one answer, but I'd have to have more experience with young kids to know whether it seems natural to them. Perhaps for me it comes from thinking about functions, and wanting just one thing to come out, when you put something in.

I know that square root can give even a good math student trouble, because it normally has one answer, but when we square root both sides of an equation, we put that plus or minus in front to give two answers. My student assistant had trouble with that when he was tutoring. He wanted to put the plus or minus in front when he was checking an answer.

I don't know if that helps answer this question, but I hope that it shows that things that can have more than one answer get confusing for students. (In fact, one thing many people like about math is that there is one right answer.)

• If "the primes are the ones that can only be built as a 1 by n rectangle" then 1 is prime, right? – Chris Cunningham Oct 27 '19 at 18:43
• I wish there were a way to think of this one for 0 and 1, as it is probably the best way to approach it. I think that is the trouble; how to find an argument that seems "reasonable" to concrete young thinkers and isn't too abstract? – kcrisman Oct 28 '19 at 3:14

I'm kind of restating what other answers have said, but I wanted to practice expressing it in the clearest, most concise way I could think of. (Coincidentally, this came up with my partner tonight, so I got a test-run with it, and got an entirely satisfying result).

Consider only natural numbers (i.e, positive integers). It seems like the number of divisors for different numbers is interesting and important. The following terms indicate how many distinct divisors a number has:

• Unit: A number that has exactly 1 divisor. Only the number 1 satisfies this criterion.
• Prime: A number that has exactly 2 divisors. Numbers such as 2, 3, 5, 7, etc. are in this category.
• Composite: A number that has 3 or more divisors. Numbers such as 4, 6, 8, 9, etc. are in this set.

In short, we don't call $$1$$ "prime" because it has a unique number of distinct divisors; just a single one.

• Clear and concise, indeed. I'm surprised no one had used the term "unit" yet! I was sort of expecting it. – kcrisman Nov 1 '19 at 11:44

We don't need the full FTA upfront if we limit our discussion for the moment to obvious examples of what's necessary for a factorization to be unique. The FTA provides analogous sufficiency conditions they'll probably guess on their own, even if they don't know how it's proven. You can say,

$$1$$ isn't considered prime because then there wouldn't be unique prime factorizations of anything, but because $$1$$ isn't prime, it's not composite either because it has no prime factors. $$0$$ isn't prime because every integer is a factor of it, but because $$0$$ isn't prime, it's not composite either because you can't write it as a product of prime numbers.

The worst reaction a child can have to that is, "Oh, so you're saying if only we do start prime numbers at $$2$$, everything from that point will either be divisible only by itself and $$1$$, or will have a unique factorization in terms of such things?" And you can say, "yes, that can be proven, but it's a bit heavy for now; and in fact you don't need to consider those two different cases, if you'll say the prime factorization of a prime number is just that prime number". The best reaction they can have is to figure out the reply on their own.

Now, if a child does want to know how any of this is proven, you can probably make that fairly accessible by exploiting their intuition, rather than yammering on about strong induction, but that's another issue.

I would start by showing them, on paper, what they already know - that in the context of multiplication the number 1 is useless. It is the identity function. It simply reflects the original number. It is a mirror. "And just like your reflection in a mirror is not a real person, neither is 1 a real number when multiplying (not to be confused with a Real number.) Since primes are only in the context of multiplication 1 isn't relevant and should not be considered in the family of primes." I would further add that if they pursue very advanced mathematics in college there are other, more formal reasons to not let it be in the family of prime numbers.

Similarly, zero changes every number to itself. "Just like a black hole absorbs matter and energy, zero, when multiplying, destroys every other number. You have lost any information in the equation when you multiply by zero. Prime numbers are useful in solving real world problems like making your text messages unreadable to everyone except your friend. Using zero as a prime would destroy the data and not allow your friend to read your text. It doesn't work as a prime number."

(If your context does not include negative numbers, turn all the negatives below positive. This almost won't change the discussion.)

Everything divides zero, so zero can't be prime. $$0 \cdot 7 = 0$$ means $$0$$ and $$7$$ divide $$0$$. $$0 \cdot -8 = 0$$ means minus eight also divides zero. Can we see that everything divides zero, so zero is very far from being prime.

Primes are numbers that are divisible by exactly two different positive numbers. (Note that this also holds true for negative integers that are prime.) Every number is divisible by one, so that must be one of the positive divisors of a prime. Every number is divisible by its magnitude ("itself" if only talking about positives), so that must be the other positive divisor of a prime. Non-primes must have more positive divisors. If we take all the positive numbers bigger than one, take them in pairs and multiply them together, we get all the non-primes.
\begin{align*} 2 \cdot 2 &= 4, 2 \cdot 3 = 6, 2 \cdot 4 = 8, \dots \\ 3 \cdot 2 &= 6, 3 \cdot 3 = 9, 3 \cdot 4 = 12, \dots \end{align*} (This could be a good time to remind/discuss multiples of a number and to remind/discuss commutativity of multiplication to reduce redundant calculations.) \begin{align*} 4 \cdot 4 &= 16, 4 \cdot 5 = 20, 4 \cdot 6 = 24, \dots \\ 5 \cdot 5 &= 25, 5 \cdot 6 = 30, 5 \cdot 7 = 35, \dots \end{align*} Here might be a good time to point out that the smallest number we get in each of these lists is the square of the number used in every product on that row. And the products get larger as we go to the right. So is it possible that there are any composites less than $$25$$ we have missed?

Let's list our composites up to $$10$$: \begin{align*} 4 = 2 \cdot 2 &\text{, so 2 also divides 4.} \\ 6 = 2 \cdot 3 &\text{, so 2 also divides 6.} \\ 8 = 2 \cdot 4 &\text{, so 2 and 4 also divide 8.} \\ 9 = 3 \cdot 3 &\text{, so 3 also divides 9.} \end{align*} This means the ones we did not produce in the table above, $$2$$, $$3$$, $$5$$, and $$7$$ must be prime -- they are only divisible by $$1$$ and themselves.

We can test this by checking each one for divisibility by smaller numbers. For two, there is nothing to check since there are no smaller positive numbers between one and two, so two is prime. For three, we see that two does not divide three, so three is prime. For five, we check two, three, and four, and discover five is prime. (This is a good time to notice that if four divides five, then two divides five, so we really only need to test for divisibility by primes.) We easily check that seven has no divisors among two, three, four, five, and six. (This could be a good time to discuss that we only need to test divisors whose square is smaller than seven, otherwise the cofactor is smaller and we have already checked the smaller potential divisors.)

When I was at school, after being taught about integer division, I was told:

"A natural number is prime if it has exactly 4 integer divisors"

Then 2 is prime, as it can be divided by -2, -1, 1, and 2.

Then one can be divided by -1 and 1, and those are only 2 divisors. Hence not prime.

Zero can be divided by anything but itself, yielding zero, hence not exactly 4 options.

\begin{align} & \begin{array}{cccccccccccccccccccc} & & & & & & & & & 840 \\[12pt] = {} & & & & & 28 & & & & \times & & & & & 30 \\[12pt] = {} & & & & 4 & \times & 7 & & & \times & & & & 5 & \times & 6 \\[12pt] = {} & & & 2 & \times & 2 & \times & 7 & & \times & & & 5 & \times & 2 & \times & 3 \end{array} \\[10pt] & = 1\times2\times1\times1\times2\times7\times1\times5\times2\times1\times1\times3\times 1\times 1 \\[10pt] & = 1\times1\times1\times2\times \cdots \end{align} Once you start taking out $$1\text{s,}$$ you can keep doing that without adding any more information about the factorization of the number you started with. So the number $$1$$ plays a different role from the role of a number that you factor and the role of a number that you end up with when you're done factoring.

• 84 should be 840. – Peter Taylor Oct 29 '19 at 8:09

The problem here is defining primes in their own right rather than defining them in terms of factorization.

Start with a number like 30. Writing 30 = 2 x 15 tells us something new; writing 30 = 1 × 30 doesn't. Writing 30 = 2 x 3 x 5 tells us something new again, whereas writing 30 = -2 x -15 doesn't.

Once we get to 30 = 2 x 3 x 5, we can't break down any of the components any further in a way that tells us anything new.

Now define primes as those numbers we can't break down any further in a way that tells us anything new.

• Hmm, of course the point of what we define first (factoring or primes or something else) is well taken. But how does that apply to 1 and 0 in your paradigm? They certainly can't be broken down any further in this sense to tell us something new. – kcrisman Nov 1 '19 at 1:44
• Well, do you want a formalization or do you want something that's easy to understand? Start with a large positive number and factor it until you can't factor it any more. The terms you're left with are primes. That's the motivation for defining primes in the first place. Investigate the phenomenon, then make names for its salient features. – Daniel McLaury Nov 1 '19 at 2:03
• In doing this you'll discover why zero and one don't fit into the picture, and then it's obvious why you don't want to consider them primes. – Daniel McLaury Nov 1 '19 at 2:08
• No, what I mean is that if you start with zero and one and use this same argument, then the break down into (from a kid's point of view) ... 0 and 1, respectively. Which might lead a kid to say they should also be prime. Do you follow the reasoning? Indeed, a similar issue with negative numbers would lead one to say -1 is prime as well (see my reference to John Horton Conway in the OP). – kcrisman Nov 1 '19 at 11:42
• Yeah, you have to get a full understanding of the situation before you try to say what a prime is. And you choose the definition of prime because it's exactly the thing you want to describe this behavior. – Daniel McLaury Nov 1 '19 at 12:50

Primes are called primes because all other integers above 1 are (multiplicatively) "built out of them."

You can't build anything else (multiplicatively) out of 1s. No matter how many 1s you multiply together.