What Type of Number Are Zero and One?

Date: 11/08/2004 at 20:27:37
From: Kay
Subject: What are the numbers 1 and 0 classified as?

If 1 and 0 are not considered prime or composite numbers, then what
are they? 

This question was extra credit from school.  1 is not considered prime 
because it doesn't have only two factors, 1 and itself, and it's not 
composite because it doesn't have more than two factors.  So what are 
they considered to be? 

I did see that 1 was called a unit (or that's what I understood) from 
the following discussion:
 
    http://mathforum.org/library/drmath/view/57058.html  

But I want to know what 0 is called and I would like a little 
clarification and a confirmation for this statement.  Thanks!

Date: 11/10/2004 at 12:23:14
From: Doctor Roy
Subject: Re: What are the numbers 1 and 0 classified as?

Hi Kay,

Thanks for writing to Dr. Math.

When the Greeks first described prime numbers, they had a conception
that the only "true" numbers were whole numbers (1,2,3,4,....) and
later ratios of whole numbers (i.e. fractions).

So, when they defined prime numbers, they meant positive numbers only.

The current definition of a prime number is a positive integer greater
than 1 (i.e. the counting numbers 2, 3, 4, 5, etc.) whose only 
divisors are 1 and itself.  A composite number is a positive integer
greater than 1 which is not prime.

Zero is not positive, so it falls outside the prime/composite
categorization.  In fact, negative numbers are also neither prime nor
composite.

I think the underlying confusion lies in the idea that all numbers
must be prime or composite.  This is not true.  It is only positive
whole numbers greater than 1 which are either prime or composite.  So,
0 and 1 fall outside prime/composite categorization.  So do 1/2 and
pi and -2 and many other numbers. 

An analogy which may help: We say that some mammals are lagomorphs (if
it is a rabbit or a hare or a pika) or rodents.  So, a mouse is a
rodent and a hare is a lagomorph. 

Then what is an elephant?  If it is not a lagomorph or a rodent, what
is it?  The answer is that it is neither.  Mammals cannot simply be
split into two groups by this classification.  Specific mammals can,
however, be split into these two groups. 

1 and 0 are numbers which cannot be placed into the groups "prime" and
"composite".  There are several characterizations of numbers they fall
into, but prime and composite are not among them.

If you think about it, there are several other categorizations:

   negative/non-negative
   negative, zero, positive
   even/odd
   divisible by 6/not divisible by 6
   algebraic/transcendental

Prime/composite just happens to be a particularly interesting one that
applies only to certain numbers.

Does this help?  Please feel free to write back with any questions you
may have.

- Doctor Roy, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 11/10/2004 at 22:36:42
From: Kay
Subject: Thank you (What are the numbers 1 and 0 classified as?)

Thank you very much Doctor Math!  This reply has really made me think
more about this.  I did in fact get the same answer when I browsed the
boards after I asked.  This gave even more information.  Thank you so
much! 

- Kay