## The Perils of Teacher Blogging

As a general rule, I find my blog to be a very useful teaching tool. It’s full of good resources for students to explore and they often follow up in class with interesting questions and comments. The blog helps extend our mathematical conversations *beyond* the classroom. Sometimes, it works too well.

In a recent discussion on probability, we considered the following question:

*Suppose you randomly choose a positive integer (a number such as 1, 2, 3, 4, 5, … ). What is the probability that the number you choose is divisible by five?*

The students thought about the question and discussed their ideas. I asked for their thoughts. The usual good answers came out ( as well as at least one bad one (*infinity*!?). I asked students to explain their reasoning, and an eloquent student sitting in the back volunteered to defend his answer.

“The answer is zero,” he said. “Although there are infinitely many multiples of five and infinitely many total integers, the probability is zero because a small infinity divided by a big infinity is zero.”

“A small infinity divided by a big infinity is zero?” I responded, trying to appear as perplexed as possible. “That sounds kind of crazy to me. What does that even mean?” I tried to stir up the *anti-zero* sentiment in the room.

The student persisted. “According to you, a small infinity divided by a big infinity is zero.”

“That doesn’t sound like something *I’d* say,” I said. This is what I usually say when students remember something I wish they hadn’t. I usually get away with it. Not this time.

“You didn’t say it,” replied the student. “*You wrote it on your blog*.”

In a rare moment, I had no response. What could I say? I did write it on my blog. I had nowhere to hide.

The class celebrated this clear and decisive victory.

*Click here to see more in Teaching.*

Say it in roses, say it in mink, whatever you do, don’t say it in ink.

Your advice, though poetic, seems at odds with the nature of blogging.

I understand (at least I think I understand) how you can divide a smaller infinity by a bigger infinity, and how this would result in 0. However, I was pretty sure that finding a number divisible by 5 is not 0, or rather not impossible. I went to a random number generator and set the bounds as large as possible and generated a few numbers. It did not take more several tries to get a number divisible by 5, multiple times. Did I achieve the impossible several times? Or is it because I can’t set the limits to infinity so I can’t accurately test this out? Just curious.

Thomas, the problem is that the terminology “smaller infinity” and “larger infinity” is very vague. In fact, we should probably think of “the integers” and “the multiples of 5” as having the same order of infinity; this is true in the sense of cardinality (they’re both countable sets) but more relevantly it’s true in the sense of density: a positive fraction of the first n positive integers is divisible by 5 as n gets very large. (In particular, this fraction approaches 1/5, and so the density of the multiples of 5 in the integers is 1/5.) This notion of density corresponds closely to what we mean by probability of choosing a random integer — in particular, it matches exactly what you were simulating with your experiment.

(Technical aside: it’s not really correct to talk about randomly choosing an integer, as there is no uniform probability measure on the integers, or on any countably infinite set. The notion of density is a good substitute, though.)

Thomas, here are two things to consider:

1) Part of your thought-process seems to be equating “probability = zero” with “impossible”. In some contexts this is true, but funny things happen when infinity gets involved!

2) You seem surprised, in part, because your empirical evidence contradicts the student’s argument. Are you sure the student’s answer was, in fact, correct?

Very cool stuff on density, Joel–that’s a nice way to think about it.

And you’re right, too, that the idea of “selecting an integer at random” isn’t as well-defined as people might think. James Tanton has a great video about how exploiting this vague notion of selecting things “at random” can be used to prove !

Check out the video here: http://www.youtube.com/watch?v=uI2FnUmBeeo

Cool, thanks for sharing the video — it’s a very nice presentation. I think I saw this problem once as a high school student (maybe at the CCNY summer program?) but that no one gave me a clear explanation at the time of how to resolve it (or at least, not an explanation I understood).

A few more words about density: in addition to asking what the density of a set is, we can also ask how quickly the limit converges. For example, we can show that the difference between the fraction of numbers less than that are divisible by 5 and the limiting value 1/5 of this ratio is never more than and , say. For the perfect squares, the density is 0, and the error is never more than . For the prime numbers, the density is 0, and the Prime Number Theorem tells us that the error is approximately (and how good this approximation is depends on whether or not the Riemann Hypothesis is true). Not every subset of the integers has a density, though — for example, try to compute the density of the numbers whose first (i.e., most significant) digit is 1.

I like this post very, very much.

Thanks for explaining, and for the video too. Also, I wasn’t sure the student was right, in fact I believed it was wrong from reading it first. The problem was it was it appeared to be a sound idea at first, just like how in the video it “proved” 1/2 = 1/3 when that obviously sounds wrong. I figured there was some sort of reconciliation, but couldn’t figure it out on my own.

I know this is about blogging but one last question out of curiosity. How come 1 is a “usual good answer” for the probability question? How did that student defend his or her answer? or are you just saying that was a usual answer?

Finding that reconciliation is the challenging (and fun!) part, whether you’re working by yourself, or with others, to understand something. There is a lot of fun to be had in thinking about ideas that appear sound at first but lead to strange conclusions.

As far as the original question goes, the answer “1” comes up because someone might see this as “infinity / infinity”, which

reducesto one. I consider it a “good” answer because thereissome merit to the argumentandit seems to conflict with our natural interpretations of probability (namely, probability = 1 is the same as “guaranteed”). So it’s a good place to have an interesting mathematical discussion.