Prime and Composite Numbers

If you’re a mathematician or a student of mathematics, you know what prime and composite numbers are.  But if you’re someone who teaches mathematics, you’ll find the standard definition of prime and composite numbers are problematic.  In the following, we’ll take apart the standard definitions, and come up with a better one.

Here’s a common definition:  A prime number is a number that can only be divided by 1 and itself; all other numbers are composite.

At this point, we have to inject some fine print:

  • Actually, any number can be divided by any (non-zero) number:  5 can be divided by 3, since 5 \div 3 = \frac{5}{3}, so we have to specify that we are talking about whole number divisors.
  • Oh, and 1 isn’t prime, even though its only whole number divisors are 1 and itself.
  • Incidentally, 1 isn’t composite either.

Thus, we get the following standard definition:  Any number larger than 1 is prime if it has no whole divisors besides 1 and itself; the number is composite otherwise.

Now there’s nothing really wrong with the standard definition…if you’re a mathematician.  But in terms of doing anything with a prime number, it’s a good example of how our choice of definitions can make mathematics easier or harder.

There are two problem with the standard definition of prime.  First, to most non-mathematicians, divisors implies division, so to find if a number is prime, we see if any number divides it.  Thus if I want to find whether 5 is prime, I’ll check 5 \div 2, 5 \div 3, 5 \div 4, and since none of them work, I’ll conclude 5 is prime.  But this means that in order to verify whether a number is prime or composite, we must divide…and division is the hardest of the elementary operations.   As a result, this definition discourages verification!

There’s a second problem:  it defines a prime number by a property it doesn’t have. Determining whether something has a property is relatively easy; determining whether something doesn’t have a property is harder, and tends to produce answers like “It just doesn’t have that property.”

What do I mean by that?  I’ve asked questions like “Prove 5 is prime” on assignments, and more often than not, I get answers like “5 has no divisors, so it’s prime.”  While this is true, it misses the point of a prove question:  A politician can claim something is true without giving any supporting evidence, but a sensible person is supposed to do better.   Mere claims are meaningless:  you’ve got to present the supporting evidence.

So here’s an alternative definition:  A number is composite if it can be expressed as a product of smaller numbers.  A number greater than 1 is prime if it cannot be expressed as a product of smaller numbers.

Note several features of this definition:

  • We don’t have to specify we’re dealing with whole number products.  While it’s true that 5 can be expressed as a product of other numbers, none of these pairs involve smaller numbers.
  • This also makes clear from the beginning that 1 is not composite, because it can’t be written as a product of smaller numbers.
  • We avoid division and focus on the easier operation of multiplication.

What about “Prove 5 is prime?”  It’s still possible to respond with a simple assertion:  “No product of smaller numbers gives 5.”  However, because the focus of the definition is on the product, and not the property, it’s more likely to cause a respondent to recognize they must provide evidence.

More importantly, students sometime have difficulties trying to prove “5 has no divisors.” I suspect it’s because they see “No divisors” and the immediate response is “But the divisors could be anything at all, so where can I start?”  In contrast, “No product of smaller numbers gives 5” automatically limits the scope of the problem:  products of smaller numbers.

I’ve started using the above definition for prime and composite numbers in all the classes I teach.  I hope you’ll start to use it as well.

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