Associated Topics || Dr. Math Home || Search Dr. Math

### Prime or Composite?

```
Date: 09/30/97 at 14:01:08
From: ~MUSKRAT~
Subject: Prime or composite?

Hello...

Wouldn't every number be composite?

The reason I think this is because the definition of PRIME is that it
only has 2 factors...itself and 1... but what about the decimals?
Those are numbers, aren't they? I think the definition of factor or
prime and composite needs to be defined better, don't you? Because
doesn't every number have another number that can go into it?

I have another question: in one of your answers you said 0 was a
number, but I dont think it is. Isn't zero considered something else?

And about factors: every number has a factor, so zero isn't a number.
I rest my case.... even 1! see...there is... .5! or .25!
```

```
Date: 09/30/97 at 16:31:10
From: Doctor Rob
Subject: Re: Prime or composite?

The definitions of prime and composite numbers are fine the way they
are. Probably you haven't seen them written out with precision and in
detail. I will make an attempt to clarify this for you below.

First of all, we shall speak only of the Natural Numbers, that is, the
counting numbers. They are all integers, or whole numbers, and they
are all positive. They begin 1, 2, 3, 4, ....

A divisor of a natural number N is a natural number D such that
N = D*Q for some unique other natural number Q. A prime number in
this set is a number with exactly two divisors. Since the number
itself and 1 are always divisors, in order to have just two divisors,
the number must be bigger than 1, and it must not have any divisors
other than 1 and itself.

The natural number 1 is very special. It is called a unit, and it is
the only natural number that has a natural number reciprocal, that is,
a natural number I such that 1*I = 1. A composite number is a natural
number which is neither a prime number nor a unit.

The big deal about prime numbers is the Fundamental Theorem of
Arithmetic. It says that every natural number can be written uniquely
as a product of powers of prime numbers. This is a very important
fact, as you might be able to tell by its name!

That disposes of most of your objections above.  1 is not a prime
because it has only one divisor, itself. Zero, negatives, and decimal
fractions are neither prime nor composite because they are not natural
numbers. They belong to a larger set, either the Integers, the
Rational Numbers, or the Real Numbers.

The next question is whether we can extend the notion of a prime
number to one of these larger sets. In the case of the Integers, this
works pretty well, but we have to be careful! Now there are two units,
1 and -1.  To every prime number P in the natural numbers there
correspond two integers that are "prime" in the integers: P and -P.
These now have exactly FOUR integer divisors: 1, -1, P, and -P divide
each of the numbers P and -P, and no other integers do. Notice,
however, that there are only two of these divisors that are natural
numbers.

Likewise to every composite number C in the natural numbers there
correspond two integers that are composite in the integers: C and -C.

Now we have to worry about zero. Zero is a special case, because it
has infinitely many divisors, since every integer except zero divides
it. Zero is relegated to a new class, neither unit, nor prime, nor
composite. The class is called the zero-divisors. Zero is the only
zero-divisor in the integers.

What happens to the Fundamental Theorem of Arithmetic in this setting?
Now it says that every non-zero integer can be written uniquely as a
unit times a product of powers of prime natural numbers (or positive
prime numbers).

When we try to extend to the Rational Numbers, we are in big trouble:
every non-zero rational number is a unit! The same happens in the real
numbers. There are no "prime" numbers and no "composite" numbers in
those sets, just units and zero-divisors (zero is the only one).

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics: