Primes, composites, and units

You may know that the prime numbers are 2, 3, 5, 7, 11, 13, 17, etc.; you may even know that a prime number is a natural numbers which:

etc. etc. These descriptions conveys the idea, but are not precise enough. For example, some people get pedantic when interpreting the word ``and'' in ``1 and itself''. Also, you are left wondering why mathematicians and books exclude 1 as a prime.

The school system did a confusing job in describing primes and composites, leaving more questions than answers.


Definition: a unit is a number which has a reciprocal (inverse, multiplicative inverse); x is a unit if and only if for some y, x*y = 1.

(Normally, you would write 1/x or x^(-1) for y.)

Now whether or not a number has a reciprocal depends on which number system you work with. If you confine yourself to the integers, 2 does not have a reciprocal, because nothing multiplies 2 to give you 1. But if you allow fractions, then 0.5 or 1/2 is the reciprocal of 2.

In number theory, we usually confine ourselves to the natural numbers, sometimes to the integers; fractions play an auxiliary role only. Thus, in the natural numbers, 1 is the only unit; in the integers, 1 and -1 are the only units.

Related to the concept of units is the concept of associates.

Definition: v and w are associates if and only if you can multiply a unit to one and get the other, i.e., for some unit u, u*v = w. Note that when this holds, you can also find some unit t so that v = t*w, namely, choose t to be 1/u since u is a unit.

So for example, 3 and -3 are associates; 3 and 3 are associates too. The concept of associates is not of much use other than stating a certain theorem later.

Primes and composites

Definition: a prime is a number which is not 0, not a unit, and not the product of two non-units; x is a prime if and only if x != 0, x is not a unit, and if x = d*e then d or e is a unit.

The idea is that a prime number cannot be further ``factorized'' into a product of two factors; if you force it, one of the factors will become a unit like 1 or -1 which is rather insignificant, and consequently the other factor is an associate, which is ``not much different'' from the original number (like, differ by a sign only), so you are not really factorizing anything.

So for example, 2, 3, 5, 7, 11, 13, 17, 19 are primes, because there is no way you can factorize them other than writing something boring like 1*2, (-1)*(-3), 5*1, etc.

The following theorem is sometimes used as an alternative definition of primes.

Theorem: x is a prime if and only if x is not 0, not a unit, and if x | s*t then x | s or x | t.

In other words, when a prime divides a product of two numbers, it must also divide at least one of them. This can easily be understood from the factorization point of view: a prime factor of the product s*t must come from s or from t, otherwise how can it be a prime factor of s*t to begin with. Although this idea as stated here is not precise enough to constitute a proof, it is the spirit of the proof. Anyway, this theorem about primes turns out to be more important than the definition.

Definition: a composite is a number which is not 0, not a unit, and not a prime.

So for example, 6, 8, 9, 10, 12, 15 are composites. A charaterizing property of composites is that they can be factorized, like 6 = 2*3, 12 = 4*3 = 2*6 = 2*2*3, etc.

The quadchotomy

The school system teaches the simple-minded dichotomy: prime vs. composite. The student is left with the question as to which one 1 belongs to. In their scheme, you have some reason to believe that 1 is a prime, and some other reason to believe that it is a composite. The culprit is that 1, -1, and in general any number bearing a reciprocal are special, and must be put into a third class: units. This is obvious when you consider that the concepts of ``prime'' and ``composite'' are built on factorization - primes cannot be further factorized and composites can be - but as soon as a number has a reciprocal it makes no sense to talk about factorizing it: no one talks about factorizing fractions because they all (except 0) have reciprocals!

For this reason you must learn the clear-cut trichotomy, which becomes a quadchotomy if you include 0 (which is special in its own way): a number is one and only one of

The school system refuses to introduce the word ``unit'', because it seems spurious: in the integers, only -1 and 1 are units; in the fractions, all except 0 are units. But the fact that some numbers have reciprocals and some do not is central to the concepts of primes and factorization; therefore an extra name is worthwhile. As I mentioned, it is not a coincidence that we do not factorize fractions. Needless to say, the school system is not run by mathematicians, so it does not know a good way to explain mathematics; worse, it cuts away a pivotal part of the story, in the name of simplification, and in doing so creates more confusion than clarity. Shakespeare's Julius Caeser certainly would become incomprehensible if you tried to omit the character Brutus in the name of accessibility.

In some less familiar systems, units are more interesting. I can give an example.