Last week we looked at the definitions of prime and composite numbers, and saw that 1 is neither. The same is true of 0. What, then, are they? That raises some deep questions that we’ll look at here.
Why not include 1?
First, here is a question from 1995:
Why is 1 Not Considered Prime? Just recently a grade six student asked me "Why is 1 not considered prime?" I tried to answer but could not, since I do not understand this either. I thought the explanation might lie in the fact that "we" don't use the true definition or we are interpreting it wrong. A prime is normally described as a number that can be expressed by only one and itself. We exclude all non-natural numbers from the set that we will be working on and then everything is fine except for when we work with 1. 1 = 1 x 1. That is, one equals 1 times itself and there is no other combination. Now to the grade six student in Faro Yukon, I said there may be a small print clause in the contract with the math gods that says you can only write it once, since 1 also equals 1x1x1x1x... This would not work for other primes such as two: 2 does not equal 1x2x2x2x... Likewise, 3 does not equal 1x3x3x3x... Patterns are very important to mathematics, I further explained, and this is a pattern I see being broken. I showed this in a slightly different way to the grade sixer but in essence the same. My question to you, Dr. Math, is what is the small print in the contract with the Math gods and how do we explain it to the grade six kids who are supposed to know it?
At one level, we could just say that his copy of the “contract” is missing a word or two. As we saw last time, our definition is “a positive number that has exactly two factors, 1 and itself”. That excludes 1.
But we can go much deeper: Why should the definition be written to exclude 1? Why not omit those extra words?
Doctor Ken answered:
Hello there! Yes, you're definitely on the right track. In fact, it's precisely because of "patterns that mathematicians don't like to break" that 1 is not defined as a prime. Perhaps you have seen the theorem (even if you haven't, I'm sure you know it intuitively) that any positive integer has a unique factorization into primes. For instance, 4896 = 2^5 * 3^2 * 17, and this is the only possible way to factor 4896. But what if we allow 1 in our list of prime factors? Well, then we'd also get 1 * 2^5 * 3^2 * 17, and 1^75 * 2^5 * 3^2 * 17, and so on. So really, the flavor of the theorem is true only if you don't allow 1 in there.
Incidentally, the full wording of this Fundamental Theorem of Arithmetic is “every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors“, because rearrangement is allowed, but not changing exponents.
So why didn't we just say something like "a prime factorization is a factorization in which there are no factors of 1" or something? Well, it turns out that if you look at some more number theory and you accept 1 as a prime number, you'd have all kinds of theorems that say things like "This is true for all prime numbers except 1" and stuff like that. So rather than always having to exclude 1 every time we use prime numbers, we just say that 1 isn't prime, end of story.
Definitions are what they are in order to be useful; they are crafted to make what we usually want to say as easy as possible. Since 1 would get in the way so often, we exclude it.
Incidentally, if you want to call 1 something, here's what it is: it's called a "unit" in the integers (as is -1). What that means is that if we completely restrict ourselves to the integers, we use the word "unit" for the numbers that have reciprocals (numbers that you can multiply by to get 1). For instance, 2 isn't a unit, because you can't multiply it by anything else (remember, 1/2 isn't in our universe right now) and get 1. This is how we think about things in Abstract Algebra, something sixth graders won't need to worry about for a long time, but I thought I'd mention it.
In 2002, an anonymous reader asked for clarification on one phrase:
Reading the explanation of why 1 isn't prime, I came across the sentence "Remember, 1/2 is not in our universe right now." What does this mean?
I explained:
This reflects the condition previously given, "if we completely restrict ourselves to the integers...". That means that we are only considering the integers, and not thinking about any other kind of number; the set of objects under consideration is called the "universe." Any object not in that universe does not exist, as far as the problem at hand is concerned. In this case, since the reciprocal of 2 is 1/2, but 1/2 is not an integer, we say that 2 _does not have_ a reciprocal, and thus is not a "unit."
Has the definition changed?
But as the next question, from 2004, reveals, not everyone has always agreed with that definition:
Was 1 Ever Considered to Be a Prime Number? I learned that a prime number was one divisible by only itself and 1, but my 4th grader says that per her book a prime requires 2 different factors. I note your Greek reference for 1 not being prime, which would indicate that I'm wrong and there was no change in definition. However, Ray's New Higher Arithmetic (1880) states, "A prime number is one that can be exactly divided by no other whole number but itself and 1, as 1, 2, 3, 5, 7, 11, etc." Can you tell me when this change happened and why?
The “Greek reference” may refer to our FAQ, which refers to the Sieve of Eratosthenes (to be discussed later), which in our version starts by crossing out 1 as not being prime. (If it were called prime, then we would circle it and then cross out all its multiples – that is, every other natural number, so that only 1 would be prime!) That may not, however, be exactly how Eratosthenes saw it. In those times, 1 wasn’t even considered a number! Euclid, for example, calls 1 not a number at all, but a “unit” (not in the sense we’ve used here). So of course 1 was not a prime.
But what about this 1880 book? On page 59, it says,
Doctor Rob answered, giving much the same argument as we used before:
Thanks for writing to Ask Dr. Math, Jim! I believe the 1880 book you cited is wrong--1 has never been and will never be considered a prime. If you treated 1 as a prime, then the Fundamental Theorem of Arithmetic, which describes unique factorization of numbers into products of primes, would be false, or would have to be restated in terms of "primes different from 1." The same is true of many other theorems of number theory and commutative algebra. Rather than use this phrase, it makes more sense to define primes so as not to include 1. Also, the multiplicative inverse of 1 (reciprocal of 1) exists in the positive integers, which is true of no other positive integer. We call such numbers "units," and this property makes them different from non-units.
A couple days later, I added a different perspective:
Hi, Jim. I'm going to disagree slightly with what Dr. Rob told you: although the definition of prime never SHOULD have included 1, and DIDN'T in the late 20th century, this fact was not always recognized in the relatively distant past. This is discussed here:
http://mathworld.wolfram.com/PrimeNumber.html The number 1 is a special case which is considered neither prime nor composite (Wells 1986, p. 31). Although the number 1 used to be considered a prime (Goldbach 1742; Lehmer 1909; Lehmer 1914; Hardy and Wright 1979, p. 11; Gardner 1984, pp. 86-87; Sloane and Plouffe 1995, p. 33; Hardy 1999, p. 46), it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own. A good reason not to call 1 a prime number is that if 1 were prime, then the statement of the fundamental theorem of arithmetic would have to be modified since "in exactly one way" would be false because any n = n*1. In other words, unique factorization into a product of primes would fail if the primes included 1. A slightly less illuminating but mathematically correct reason is noted by Tietze (1965, p. 2), who states "Why is the number 1 made an exception? This is a problem that schoolboys often argue about, but since it is a question of definition, it is not arguable."
I'm assuming that the references from 1979 on, at least, say that primes were formerly defined to include 1, rather than using that definition themselves. Texts, also, may not always be careful about definitions; your "divisible by only itself and 1" may well be intended to imply that "itself and 1" are not the same number, or the question of whether 1 is a prime may not have been considered.
So the definition was refined when its unpleasant implications were fully realized.
Here is another discussion of this question that I found: http://mathforum.org/kb/message.jspa?messageID=1379707 Read especially John Conway's contributions, which point out that mathematicians recognized the need to clarify the definition when certain aspects of abstract algebra developed in the 1900's, which gave them a new perspective on the question; but that school texts, as usual, were slow to adopt the corrected definition:
In the late 1800’s and early 1900’s, mathematics was being clarified in many ways, and this greater care with definitions was part of that development.
What follows is what Conway said; the address above no longer works, so I’m glad I quoted it:
The change gradually took place over this century [the 1900's], because it simplifies the statements of almost all theorems. If you count 1 as a prime, for example, numbers don't have unique factorizations into primes, because for example 6 = 1 times 2 times 3 as well as 2 times 3. It's a bit of a nuisance that Lehmer's 1914 "List of all prime numbers below 10 million" counts 1 as a prime. I think the development of number theory for other rings played a big part, because there one finds other "units" besides 1 (for instance +-1 and +-i in the Gaussian integers), and these units clearly behave in many ways that make them different from the primes. Other examples of the kind of thing that goes wrong if you count 1 as a prime are arithmetical theorems like "If p,q,r,... are distinct primes, then the number of divisors of p^a.q^b.r^c.... is (a + 1)(b + 1)(c + 1)... ."
Gaussian integers will be mentioned again, as will units.
Mathematicians this century [the 1900's] are generally much more careful about exceptional behavior of numbers like 0 and 1 than were their predecessors: we nowadays take care to adjust our statements so that our theorems are actually true. It's easy to find lots of statements in 19th century books that are actually false with the definitions their authors used - one might well find the above one, for instance, in a work whose definitions allowed 1 to be a prime. Nowadays, we no longer regard that as satisfactory. The changeover has been very gradual, and I'll bet there are publications from the last few years in which 1 is still counted as a prime--in other words, it's not yet complete. In the 1950s and 1960s, books that chose the new definition would always be careful to point out that they were doing so, and that most authors included 1 with the primes. The real thing that gets such a change accepted is when it gets into high-school textbooks. I think that perhaps we must thank "the new math" movement, which for all its faults did get some of the terminology and conventions into the high schools that had hitherto only been used in the Universities. School textbooks don't like to muddy the waters by explaining such things as variations in usage, so would tend to give just one definition. My guess is that you'll find that schoolbooks of the 1950s defined primes so as to include 1, while those of the 1970s explicitly excluded 1.
I added:
It sounds like your textbooks, and mine, might have used the old definition!
So what do we call 0 and 1?
Are 0 and 1 prime, composite, … or something else? We’ve seen part of the answer in references to “units”. But there’s a little more to say.
Here is a 1997 question:
1 and 0: Prime or Composite? Is the number one a prime or a composite number? Why? (Please put your answer in a form that a sixth grader can understand.) What is the number zero? Prime or composite? Why?
Doctor Rob answered, necessarily expanding the question from “which is it?” to “what (else) is it?”:
One is neither a prime nor a composite number. A prime number is one with exactly two positive divisors, itself and one. One has only one positive divisor. It cannot be written as a product of two factors, neither of which is itself, so one is also not composite. It falls in a class of numbers called units. These are the numbers whose reciprocals are also whole numbers.
Note his slightly different definition of composite numbers, which I like:
- A prime is a number you can get by multiplying two numbers (not necessarily distinct) other than itself. That isn’t true of 1.
- And a unit is a number that you can multiply by some number (possibly itself) to get 1. That is true of 1 (and no other natural number).
Zero is not a prime or a composite number either. Zero has an infinite number of divisors (any nonzero whole number divides zero). It cannot be written as a product of two factors, neither of which is itself, so zero is also not composite. It falls in a class of numbers called zero-divisors. These are numbers such that, when multiplied by some nonzero number, the product is zero.
- A zero-divisor is a number that you can multiply by some number other than zero to get 0. That is true of 0 (and no other integer).
At this level, the ideas of units and zero-divisors seem silly because there is only one of each (among natural numbers). We’ll get to that in a moment!
He gives the same reason we’ve seen before:
The most important fact of multiplication of integers is called the Fundamental Theorem of Arithmetic. It says that every whole number greater than one can be written *uniquely* (except for their order) as the product of prime numbers. This is so important that we tailor our idea of what a prime number is to make it true. If 1 were a prime number, this would be false, since, for example, 7 = 1*7 = 1*1*7 = 1*1*1*7 = ..., and the uniqueness would fail.
Why name nearly empty categories?
We’ll close with this 2013 question, which starts with a different issue before moving to primes:
Zero and One, Each Unique in Its Own Special Way Since zero isn't a positive number and it's also not a negative number, what is it? Does it have a special name?
I answered:
Hi, Gabby. Yes, its special name is "zero"! There is no need to come up with a separate name for a category that consists of only one number. So we say that every number is either positive, negative, or zero.
This led to another question:
Hello. We are Gabby's classmates. If there is only one unit (1), why is there a name for that? Rachel and Sophie
Where had they seen the term unit? What does it mean to them?
I replied, unsure of the level of their knowledge:
Hi, Rachel and Sophie. What is your understanding of the meaning of the word "unit"? One meaning is just a synonym for "one" (a single thing), and not a category containing the number one. This usage is particularly relevant in connection with fractions, where the unit tells you what the fraction is a fraction OF.
For examples, see Fractions: What Are They, and Why?.
Another meaning you might have in mind is sometimes used in connection with 1 in contrast to prime numbers and composite numbers; but the actual meaning is rather technical -- and it is used because 1 is NOT the only number of that type. For an explanation of that usage, see Why is 1 Not Considered Prime? http://mathforum.org/library/drmath/view/57058.html Prime or Composite? http://mathforum.org/library/drmath/view/57115.html What Kind of Number is One? http://mathforum.org/library/drmath/view/56062.html
Here I referred to the first answer in this post, and one we’ll see next week, and another I’ve omitted. I think their teacher had told them about one of these pages.
These tell you that the word "unit" is used for a number that has a reciprocal within a given set. When you restrict yourself to the natural numbers (as we usually do in talking about prime and composite numbers), 1 is the only unit. Extending our attention to the integers, -1 is also a unit. If we extend further to the Gaussian integers (which you may never even learn about), there are four units: 1, -1, i, and -i! We wouldn't use the word "unit" as a category if 1 were the only number EVER in the category; but these extended contexts give a reason to define a category that is relevant to primes and contains 1, even though 1 is the only unit IN THE NATURAL NUMBERS.
This is similar to the fact that we probably wouldn’t have words like “commutative” if we hadn’t started studying other kinds of “numbers” and their operations.
One of these pages also describes that in extended contexts, 0 is part of a special category, called "zero-divisors." Again, among integers there is only one of these, namely zero, and it would be silly to use the category "zero-divisors" when all we gain is a longer name. More important, this category, while somewhat relevant to prime numbers, is not relevant to Gabby's original question about positive and negative, so it wouldn't have been an appropriate answer to your original question. (If you're wondering what numbers other than 0 can be zero-divisors, the best example is in modular arithmetic, which you may have seen in the form of "clock arithmetic.")
As an example, if instead of a number line you count around a clock, then \(3\times4=12\) will take you to the same place as 0; so 3 and 4 become zero-divisors.
This may be far more than you want to know -- and may not have anything to do with the use of "unit" you asked about -- but maybe it starts to crack open the door to let you see how big math is!
Most students never get to see that math deals with “numbers” far beyond the natural or real numbers.
Then their teacher (whose email was being used) commented:
Hello, I am the teacher of the 5th graders (Gabby, Rachel and Sophie) who emailed you about zero's special name and units. They were so very excited to receive your reply. I appreciated all the information you gave and, even more so, the way that you wrote to them as though they are intelligent people capable of thinking deeply about math. They are, and your response reinforced that to them. I am very grateful. Christina Hull
I responded,
Hi, Christina. Thanks for letting me know. That's exactly what I try to do. I like "talking up to" kids, rather than talking down to them. But also, the question (especially the second one) fascinated me, and led me to put together ideas I hadn't combined before, so it was just fun to write them up.
Christina concluded:
Yes, their question and your answers led me to think about ideas I hadn't thought about in that way before, as well. And, in case you were wondering, they came up with the question while thinking about 1 fitting into a category other than prime numbers or composite numbers. Thanks again. Christina
That’s what makes it fun to be a Math Doctor!