Category: Math

One is not prime.

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.


Toilet Seat Mathematics

The complaint you always hear is that men leave the toilet seat up, while women would prefer it be left down. Well, let’s crunch a few numbers to determine the total effort required by each party in the two cases. I’m going to ignore the lid in the following calculations:

With the seat up, men need to:

  1. Nothing.
  2. Lower it before, raise it after.

Women need to:

  1. Lower it before, raise it after.
  2. Lower it before, raise it after.

So the seat up yields a number of operations of 2 for men and 4 for women.

With the seat down, men need to:

  1. Raise it before, lower it after.
  2. Nothing.

Women need to:

  1. Nothing.
  2. Nothing.

So the seat down yields a number of operations of 2 for men and 0 for women.

The conclusion here is that, to men, it makes absolutely no difference whether the seat is left up or down, while women do less work with it down. The interpretation goes both ways: It could be viewed as selfish of women to want to do less work, or it could be viewed as selfish of men, as it makes no difference to them.

A second level of analysis makes the situation clearer. Considering that everyone (unless I’m strange) finds themselves in situation… uh… 1 more often than 2, if my meaning is carried, we find that men do more work with the seat down. Let’s go with a conservatively average estimate of three 1’s and a 2 each day. We find:

  • Seat up: Men have 2 operations each day, women have 8.
  • Seat down: Men have 6 operations each day, women have 0.

We now see the emergence of the complaint. Men do much less work with the seat up, Women do none with it down. As a unit, a couple does less work with the seat down. But the true conclusion here is that couples argue about this because people are greedy.

If the couple does indeed prefer the lid down, I’ll give the totals. The assumption I’ll make is that lifting both the lid and the seat together counts as one operation. A fair assumption.

  • Seat and lid up: Men have 2 operations each day, women have 8.
  • Seat down, lid up: Men have 6 operations each day, women have 0.
  • Seat and lid down: Men have 8 operations each day, women have 8.

The only conclusion we can definitively draw is that the toilet lid belongs up.

We could continue with an expected value calculation (assigning negative values to sitting on a closed lid, peeing on the seat or even the lid, or the infamous “falling in”), but we won’t. I’ll end this with a synopsis: The toilet lid should be up (at some point, even absent), and beyond that, I might say the seat should be left down for the greater good, but the argument exists for a reason.