Last night my younger son and I were talking a little bit more about the project and he asked me why Cantor’s diagonal argument for why the set of real numbers is larger than the set of Natural numbers doesn’t work for rational numbers!! Yes!!

We explored that question today. First we did a quick review of the diagonal argument (which was the last part of yesterday’s project) and then we began talking about the rational numbers:

Next we looked at what would happen if you applied the diagonal argument to the rational numbers:

After getting our arms around the diagonal argument when applied to rational numbers, we backed up and looked at the argument why rational numbers are countable.

Unfortunately I made an easy concept hard in this part of the project. I was trying to explain the “easy for me” idea that if a set that is larger than the rationals was the same size as the natural numbers, that meant the rationals must also be the same size as the natural numbers. My explanation started off terribly and went down hill . . . .

Finally we looked at one of the strangest consequences of all of this infinity stuff. In math language – the rational numbers have measure zero.

The idea here always blows my mind and is a really fun idea about infinity to share with kids.