I was at trivia on Monday, with some friends from the department (the math department, that is). Our team consisted of six graduate students and above. As horrible as this may sound to some of you, believe me, mathematicians are AWESOME in groups.

We were winning (the questions were pretty easy, but I will admit to being mildly happy I knew a period of play in polo was called a chukker), when the following question came up: “What is the sum of the first five prime numbers?” We chuckled, more than a little. It’s 2+3+5+7+11=28.

After getting it right, and being in the minority, the teams who had gotten it wrong FREAKED OUT. “It’s 18! Everyone knows that!”

As I was wearing a shirt with a math joke on it (I often do), and as the host had noticed beforehand, she said, “where’s the mathematician I talked to before? What’s the answer?”

Calmly, I replied, “It’s 28. One is not prime. The first five primes are 2,3,5,7, and 11.”

Someone on an incorrect team said, “no, that’s not the standard definition. One is totally prime.”

Glossing over the fact that they used the word totally, I hope they got in a car accident on the way home I was irritated.

If you ignore this issue, much of mathematics works anyway. However, the Fundamental Theorem of Arithmetic, which I would say is one of the most critical facts in mathematics, needs 1 to not be prime. It says (essentially, for non-mathy readers) that the prime factorization of every number >1 is unique. The importance is clear: If this were not true, you wouldn’t be able to multiply numbers (the outcome would be unclear), so you wouldn’t be able to add, or divide, or subtract. Basically, standard mathematics couldn’t exist. But, because 6=2*3=1*2*3=1*1*1*1*1*2*3, 1 causes a failure of this (if prime) and so is not a prime number.

This is also discussed (generally, with a bit of historical context), on WikiPedia, for those further interested. It does say (and I agree) that you could allow one to be prime if you modified the statement (and the proof, which it doesn’t say) of the theorem. However, as mentioned, much of mathematics works this way; much is not all. You would need to modify many other things as well, some of which (trust the guy who’s taken algebraic number theory) get ugly.

I’m more interested in why people think one is prime. And I think I know why. It’s because it is (obviously, under either definition) not composite.

This is a problem I come across often when teaching, and it is the failure to understand logical opposites. Here are a few low-math, high-math, and non-math examples:

  • If a function is not even, that doesn’t make it odd. That means it fails the definition of an even function. Most functions are neither (a very select few are both).
  • If a set is not open, that doesn’t make it closed. There are plenty of sets which are neither (or both).
  • The opposite of “your mother” is not “your father”. It is “everyone other than your mother”.

In order for two things to be purely opposites, they need to form a partition of whatever universe they live in. That is, everything is one or the other, and nothing is both. For example, everything is either an “apple” or “not an apple”. Often we gloss over things with very small intersections, or which miss a few cases; we might say every human is “male” or “female”, and treat this (incorrectly, but with few exceptions) as a partition.

This is the problem. “Prime” and “comoposite” are not opposites. The partition of natural numbers consists of three sets: primes, composites, and {1}. There’s nothing at all wrong with that.