Why is 1 Not Considered Prime?

Date: 20 Mar 1995 12:22:37 -0500
From: Ian Oostindie
Subject: Why 1 is 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?

Thank you very much for any consideration you make.

Date: 25 Mar 1995 16:21:45 -0500
From: Dr. Ken
Subject: Re: Why 1 is prime

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.

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.

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.

-Ken "Dr." Math

Date: 05/01/2002 at 16:41:13
From: Dominic Eldridge
Subject: 1/2 not in our universe?

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?

Date: 05/01/2002 at 17:02:14
From: Doctor Peterson
Subject: Re: 1/2 not in our universe?

Hi, Dominic.

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."

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