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October 22

Why isn't one a prime number?

I've looked at the article, and don't understand the reason given there.

Can anyone provide a plain English explanation that will work for junior high school students? HiLo48 (talk) 06:57, 22 October 2013 (UTC)[reply]

I would say, basically, convenience. Well into the 19th century, some mathematicians considered 1 to be prime. But if you make it prime, then various things get ickier to state. For example, the fundamental theorem of arithmetic would have to allow an arbitrary power of 1, and only the rest of the prime decomposition is unique. --Trovatore (talk) 07:02, 22 October 2013 (UTC)[reply]

It is remotely possible that someone may enjoy this USEnet post that I made lo these many moons ago. --Trovatore (talk) 07:09, 22 October 2013 (UTC) [reply]

You can also consider what the idea is supposed to capture. As far as the integers are concerned, a prime is a number that has no nontrivial divisors- since 1 divides everything, being divisible by 1 is trivial, and since every number divides itself, n trivially divides n. In other words, knowing that 1 divides something doesn't tell us anything, the case 1|1 is just a special case of 1|n.Phoenixia1177 (talk) 07:17, 22 October 2013 (UTC)[reply]

I don't think that works. As you note, 1 trivially divides 1. But 1 has no other divisors (among the natural numbers). So 1 has no nontrivial divisors, and by your argument, should be prime. --Trovatore (talk) 07:20, 22 October 2013 (UTC)[reply]

...oh my God, I've been having such a stupid day today (I spent two hours making stupid mistakes in something else I was doing earlier too)! Seriously, that makes me want to do one of those "hit palm off forehead moves". Anyways, you're- obviously- absolutely right, sorry about that:-) What I was, initially thinking was that being divisible by a prime should be nontrivial, then wrote something nothing like that...Phoenixia1177 (talk) 07:30, 22 October 2013 (UTC)[reply]

This is one of those points that you keep thinking you see the right formulation, but it always reduces back to something you already considered. --Trovatore (talk) 07:59, 22 October 2013 (UTC) [reply]

I used to define a prime as a number that has exactly two (trivial) factors; or as a number that is not either a "rectangle number" or a "square number". Both of these definitions exclude 1. Dbfirs 07:37, 22 October 2013 (UTC)[reply]

Yeah, the "exactly two factors" definition is a reasonably common one, but it seems unmotivated. Or more precisely, it seems to be motivated simply by the desire to exclude 1 without saying so explicitly. --Trovatore (talk) 07:59, 22 October 2013 (UTC)[reply]

Fair comment. Perhaps "a number that is neither composite nor square" fits better, but the same criticism can be applied to the "nor square" part. The natural approach at introductory level is to define square numbers (including 1), and define composite numbers (called "rectangle numbers" if they are not squares), then define all the others to be prime. Dbfirs 08:47, 22 October 2013 (UTC)[reply]

But (in the geometric analogy), if you're considering a 1x1 square to be a square number, why not a 3x1 rectangle as a rectagular number? MChesterMC (talk) 08:55, 22 October 2013 (UTC)[reply]

The prime numbers can be defined as "row numbers" (different from rectangles), but I agree there is room for confusion. It seems to be universally accepted that 1 counts (trivially) as a square, but I agree that's a weak point in my argument. The reason for excluding 1 as a prime "because it's a unit" is not really appropriate at that level. Dbfirs 09:01, 22 October 2013 (UTC)[reply]

Gee thanks guys. Interesting discussion, but it's bedtime here now. In 11 hours time I have to explain to some students why I marked their work wrong. I don't think we've quite yet achieved the "plain English explanation that will work for junior high school students". Keep 'em coming though. HiLo48 (talk) 10:59, 22 October 2013 (UTC)[reply]

The "plain English" explanation is "1 is obviously not composite, but neither is it prime - we call it a unit, and units are neither prime nor composite". As to why this is a useful distinction to make, there is a simple explanation here. Gandalf61 (talk) 11:29, 22 October 2013 (UTC)[reply]

Wow that linked explanation is so clear that even I can understand it! -- Q Chris (talk) 11:33, 22 October 2013 (UTC)[reply]

Yes, it's clear, but it is only saying that mathematicians choose to exclude 1 so that prime factorisation is unique. Dbfirs 12:13, 22 October 2013 (UTC)[reply]

I suggest that you give them a 10 by 10 grid with the numbers 1 to 100 printed (or get them to construct their own if you want them to get more involved and have plenty of time). Explain what square numbers are, and get them to colour in all the square numbers (on the main diagonal). Now explain what rectangle numbers are (non-square composite numbers if your syllabus doesn't use that term). Get them to colour in all the rectangle numbers (you can use different colours for multiples of 2, multiples of 3, multiples of 5, and multiples of 7, if you wish). The numbers uncoloured are the prime numbers. Of course, when you go on to expressing numbers as products of primes, it will be appropriate to include Gandalf61's excellent reason for not including 1 as a prime (perhaps you've already reached that stage, in which case, ignore my grid). I hope your lesson goes well! Dbfirs 12:26, 22 October 2013 (UTC)[reply]

HiLo, what kind of "why" are you asking? The "plain English" answer to why you marked their work wrong is that all the standard definitions of "prime number" deliberately exclude 1, sometimes explicitly, sometimes in a roundabout way. Assuming you told them that, they have nothing to complain about regarding the mark itself.

But if you're asking why the definitions exclude 1, the answer is, if they didn't, then prime factorizations would not be unique. I don't know if you consider that "plain English", but presumably you understand it, so reword it as plainly as you like. --Trovatore (talk) 17:26, 22 October 2013 (UTC)[reply]

If a number is prime, then all his other multiples are composite. Thus making 1 prime would imply that all other natural numbers are composite, since all natural numbers are obviously his multiples: Which is a contradiction, since it would imply that primes are composite. — 79.113.232.204 (talk) 23:25, 22 October 2013 (UTC)[reply]

I'm not sure that works. All multiples of non-primes are also composite. -- Jack of Oz ^{[pleasantries]} 23:45, 22 October 2013 (UTC)[reply]

Did I say otherwise ? — 79.113.232.204 (talk) 00:01, 23 October 2013 (UTC)[reply]

It's a trivial objection. 79 is implicitly assuming one particular definition of "composite number" that says "a composite number is any product of at least two (not necessarily distinct) primes". It's true that if you make 1 prime, and then keep that definition of "composite number", then all positive naturals are composite.

But that would be an obviously dumb thing to do. A lot of mathematicians considered 1 to be prime through the 19th century, even into the 20th, and they weren't idiots; they wouldn't have missed such an obvious point.

The truth is, it is slightly easier to come up with a natural, concise, clearly motivated definition of "prime number" that includes 1, than one that does not. But it turns out to be more convenient to exclude it, so that you don't have to keep repeating "prime number greater than 1" or some such. --Trovatore (talk) 00:08, 23 October 2013 (UTC)[reply]

(ec) No, but it seems to be a circuitous explanation, and I'm not convinced it's valid. The number 1 is the sole exception to the rule that all multiples of natural numbers are composite, but that in itself doesn't tell us whether 1 is prime or non-prime. It's essentially a definitional question, and such a thing can't be proven algebraically. -- Jack of Oz ^{[pleasantries]} 00:14, 23 October 2013 (UTC)[reply]

Eratosthenes beggs to differ, and so does his sieve. Could you imagine what would happen, were his famous process to start at 1 ? :-) — 79.113.232.204 (talk) 00:19, 23 October 2013 (UTC)[reply]

That is merely a tool for identifying primes. That it only works if you start at 2, does not prove that 1 is prime, composite or anything else. If you have to assume, before you ever start, that 1 is non-prime, how can that prove that 1 is non-prime? It can't. Yet more circular reasoning. -- Jack of Oz ^{[pleasantries]} 00:26, 23 October 2013 (UTC)[reply]

Yes, it is a tool for eliminating composite numbers. If it were to start at 1, it would eliminate all numbers > 1 as prime, and identify them as supposedly being composite. Hence, 1, though not composite, cannot be a prime either: not without making maths illogical and self-contradictory. — 79.113.232.204 (talk) 00:39, 23 October 2013 (UTC)[reply]

Sorry, that's total bullshit. Math works just fine with 1 prime. You have to adjust some other things, and in the end it's a little less convenient, which is why we don't do it. But there's no contradiction. You're inventing a contradiction by using a definition of "composite" that depends on 1 not being prime, then assuming that the statement "no number can be both prime and composite" is still true. --Trovatore (talk) 00:44, 23 October 2013 (UTC)[reply]

If it's a "definition", I did not invent it: unless, of course, you really think I'm 2,300 years old, and am posting with my IP so as to hide the fact that I'm actually a famous Greek mathematician. :-) The OP asked for a simple, down-to-earth explanation, and I gave it. Whether or not this explanation can also be used as a fertilizer, and -if so- to what species of animal it belongs, I will leave for him (or her) to decide. :-) — 79.113.232.204 (talk) 00:53, 23 October 2013 (UTC)[reply]

Referring to the Sieve of Eratosthenes is a silly argument. It's just one of many algorithms to find primes, and it can easily be formulated to match the used definition of primes. Eratosthenes' original works have not survived and I don't know his own formulation but even today where 1 is not considered prime, most presentations of the sieve explicitly say "start at 2" and not "start at the first prime". Our own article Sieve of Eratosthenes also does that. [1] from 1772 only considers odd numbers and says start at 3. PrimeHunter (talk) 01:18, 23 October 2013 (UTC)[reply]

Perhaps, but starting at 2 does not make the algorithm look inconsistent. Starting at 1, however, does. — 79.113.217.52 (talk) 09:17, 23 October 2013 (UTC)[reply]

No it doesn't. It makes the algorithm not work. But there is no inconsistency. --Trovatore (talk) 09:38, 23 October 2013 (UTC)[reply]

So there's no inconsistency about all primes > 1 being composite, or 1 being the only prime ? :-) — 79.113.217.52 (talk) 10:07, 23 October 2013 (UTC)[reply]

Why would all primes > 1 be composite? It would mean that the sieve algorithm no longer provides up with the primes, there's no reason that such a method has to work.Phoenixia1177 (talk) 11:06, 23 October 2013 (UTC)[reply]

Every multiple of a prime, save the prime itself, is a composite. If 1 were a prime, it would be the only one, and all other numbers, including the primes themselves, would be deemed among its composites. Which renders the intention behind the notion meaningless, making it a mere synonym for unit or unity or one. — 79.113.217.52 (talk) 12:36, 23 October 2013 (UTC)[reply]

Furthermore, mathematicians like unity, coherence, and generalization... (Think of how the factorial function was extended to include all complex numbers, for instance). The fact that many meaningful properties of primes cannot be extended to 1 makes its inclusion among their ranks undesirable. — 79.113.217.52 (talk) 12:46, 23 October 2013 (UTC)[reply]

If you made 1 prime, then composites would be defined differently, you're cherry picking definitions to make your argument. --as an aside: I fully understand about "unity" blah blah blah, don't talk to me like I'm stupid; and that's oversimplifying anyways.Phoenixia1177 (talk) 06:12, 24 October 2013 (UTC)[reply]

I've never talked to you that way, what are you talking about ? Anyway, I'm not saying that one cannot make definitions more complex, but usually simplicity is preferred over the alternative. (Think Ockham's razor). It's more "natural" for 1 not to be considered a prime, more logical than for it to be ranked as such. — 79.113.233.11 (talk) 11:04, 24 October 2013 (UTC)[reply]

The razor doesn't really apply, there is no best definition, theres objects that are studied. Why do we study these things we call primes? Because of our interest. You can call the primes with the number 1 the qrimes and stidy those, it's perfectly sound. On the other hand, 2 is a bit ill behaved at times, eccluding it and just looking at odd primes is just as workable, depending on context. That the sieve method fails at 1 doesnt matter a scant bit about anything.Phoenixia1177 (talk) 12:16, 24 October 2013 (UTC)[reply]

It does matter, because the whole idea about declaring something a prime is to get rid of all its multiples, and thus simplify or reduce the numbers to a much "smaller" subset. That's the meaning behind the mathematical endeavor. But such a reasoning, when applied to 1, renders the notion of prime meaningless. — 79.113.233.11 (talk) 15:10, 24 October 2013 (UTC)[reply]

Your idea of what primes should be is a motivation, or a way of looking at them, and if you're explaining this to school children, then yes, that's what primes are. But, there are other perspectives that make including 1 quite useful. For example, going with the perspective of Tit's, or Weil and the Riemann Hyp, then it is quite useful to have a field with 1 element, and to find a way to introduce that- if you can make it work, you then have finite fields with cardinality 1 or normal primes. Like I said, there is no "right" definition, there is the "right" definition for what you are studying. When you move away from elementary number theory, sometimes 1 fits the bill, sometimes only the odd primes, etc. I'm not saying that 1 should be counted prime, but that there is no "better" answer. In other words, your idea of Occam's razor doesn't fit, nor does your idea that sieve methods "should" work (that's only relevant if you're studying sieves).Phoenixia1177 (talk) 07:00, 25 October 2013 (UTC)[reply]

if you're explaining this to school children, then yes : Well, that's precisely why the OP was asking... :-) — 79.113.235.28 (talk) 17:38, 25 October 2013 (UTC)[reply]

That was what he was asking, but that didn't seem to be what we were discussing- citing Occam's Razor is a good indicator, most school kids aren't familiar with it:-)Phoenixia1177 (talk) 03:48, 26 October 2013 (UTC)[reply]

I think that a simple (and accurate) English explanation would have to say: "Primes can be defined to include or exclude 1, and the difference is largely a matter of convenience (or even preference). Historically, both definitions have been used successfully, but the modern practice is to exclude it." To go into justification for one choice seems inappropriate at any level: there seems to be no simple clearcut argument either way. Even in more general rings, the group of units looks no different from any other set of primes that is equivalent up to multiplication by a unit. The reason for 1 being non-prime is simply that this is the definition that they are required to use. — Quondum 02:31, 23 October 2013 (UTC)[reply]

I think there's a good argument for defining primes and composites so that they BOTH exclude 1. That is, there are 3 classes of natural numbers, with 1 being the sole member of the 3rd class. -- Jack of Oz ^{[pleasantries]} 02:57, 23 October 2013 (UTC)[reply]

Yes, this is the standard convention (the third class is called the units of the semiring; it so happens that the natural numbers have only one unit). Of course, if you're a cultured gentleman, you count 0 as a natural number as well, so that makes a fourth class (not prime, not composite, not a unit). --Trovatore (talk) 09:38, 23 October 2013 (UTC)[reply]

I am, and I agree. (The 0 thing actually occurred to me a short while after I posted. True.) This is really very simple and obvious, conceptually, and I wonder why kids are still taught that all the natural numbers fall into only 2 classes, primes and composites. An appreciation of the only slightly bigger picture would put an end to these interminable questions about whether 1 is or is not a prime, and why/not. -- Jack of Oz ^{[pleasantries]} 11:09, 23 October 2013 (UTC)[reply]

While I doubt it matters anymore, what I meant to say yesterday is what follows. The point of primes is not, simply, that they are irreducible, but that they are multiplicative "building blocks" of numbers. That 1 trivially divides every number does not fit well with this idea since it isn't structural, it tells you absolutely nothing about a number to know it is divisible by 1. So, while 1 fits with the common definitions of prime (without purposely excluding it), it doesn't really function like the rest of them in so far as their defining characteristic. By analogy, when considering rings with identity, some authors chose to explicitly exclude the ring with 1 = 0, the reason is that this ring doesn't really act like the rest- some don't exclude it, but, then, you're just excluding it when you state theorems. Essentially, this is the common answer of "convenience" given above, but from the perspective of what "characterizes" a thing and less "how we use it". (sorry again for yesterdays goofiness).Phoenixia1177 (talk) 06:38, 23 October 2013 (UTC)[reply]

Yes, I think you have something here. Here's another way of saying the same thing. Take the lattice of positive naturals ordered by divisibility. Now cut down to the minimum directed graph that generates the whole lattice as its transitive closure (that is, in the original lattice, 7 is below 42, but in the minimum graph you have an edge from 7 to 14, and an edge from 14 to 42, but no edge from 7 to 42).

Now, in that graph, the primes are precisely the ratios along edges. --Trovatore (talk) 09:38, 23 October 2013 (UTC)[reply]

I like that:-) The building block distinction between primes and units is also captured algebraically. The primes give factor rings that are integral domains, every composite has a factor ring that breaks down into a product of factors associated to the powers of primes maximally dividing it- the number of 0 divisors is directly related to the number of prime divisors and their exponent. The factor ring of 1 (or -1) on the other hand is the 0=1 ring, is not an integral domain, and doesn't really enter into the factoring cases (you can add it, but it in, but it does nothing). Factor rings of units are qualitatively a very different creature than those associated to primes.Phoenixia1177 (talk) 10:43, 23 October 2013 (UTC)[reply]

The number 1 would seem to be the "trivial case" in multiplication and division, just as the number 0 is the trivial case in addition and subtraction. ←Baseball Bugs ^{What's up, Doc?} carrots→ 08:32, 23 October 2013 (UTC)[reply]

Wow, what a discussion. I told the students today that it's just one of those rules that mathematicians have. They bought it. But I know I'll be asked again... HiLo48 (talk) 11:04, 23 October 2013 (UTC)[reply]

At some level, that really is the most correct answer. But it might be worth addressing at least one example of how there's an advantage to the rule, to show that it's not purely whimsical. What's wrong with the non-uniqueness of prime factorizations, as such an example? --Trovatore (talk) 18:47, 23 October 2013 (UTC)[reply]

Well, since you're ultimately dealing with "rebellious teenagers", you could just tell them that -for centuries-, the "older" mathematicians considered 1 a prime number, but then the "newer" ones "rebelled" and "rose up" against the "traditional" mathematical "establishment", and "refused" to do that anymore... :-) Or you could just tell them that calling 1 a prime got "out of fashion" about 100 years ago, and that's it "so 19th century" to still do that, and that the "trend" now is to do the exact opposite, 'cuz this is "the new big thing". :-) — 79.113.217.52 (talk) 13:18, 23 October 2013 (UTC)[reply]

If we had 'like' buttons i'd give that response a thumbs up. Perfect suggestions for a classroom full of teens, some of which are at that point where they really start to hate math because it seems incomprehensible. And this short history of '1' as prime, non-prime seems both fitting and memorable. kudos. 76.17.125.137 (talk) 01:54, 24 October 2013 (UTC)[reply]

See this. Bubba73 ^{You talkin' to me?} 02:00, 24 October 2013 (UTC)[reply]

And I don't know if this is covered above, but if 1 was considered prime, a lot of theorems about primes would have to state "except the number 1". Bubba73 ^{You talkin' to me?} 16:12, 25 October 2013 (UTC)[reply]

I was hoping that the most abstract reason (number 4) in the link would be the most persuasive. But it ranks with number 1: units are excluded only by the additional (and unnecessary) clause of exclusion. So really, the convenience of uniqueness of decomposition (plus of simplifying other theorems, as per Bubba73) provides the motivation for exclusion of units (the why), which results in consensus to add the exclusion to the definition. — Quondum 07:44, 26 October 2013 (UTC)[reply]