If you know a child in middle school, this is a great way to get them started on "cool mathematical thinking", which I called "Mathematia" when I discussed with

If you know a child in middle school, this is a great way to get them started on "cool mathematical thinking", which I called "Mathematia" when I discussed with my son when he was young (to distinguish from the horrible Math being taught in school): 1. Introduce ℤ & ℚ - this is easy. Perhaps, fingers and slices of pizza. Now s/he's ready to be as surprised as the members of the Phythagorean cult 2. Go over the classical proof for √2 given here. We now have a number that's not in ℤ or ℚ! 3. It's one thing to show a result, a very different thing to *grasp* it. Why is (2) a big deal? It smashes the simple notion Greeks had that *any* two lengths (rational numbers) are commensurable, which is a perfectly simple and obvious (and wrong) thing to believe: "Have one stick for one side of a square and another for the diagonal. You cannot cut both sticks into pieces of the same length, no matter what length you choose." *This is amazing* 4. We only discovered one such weird number. Are there others? Motivated by the above, how about checking √3. Show that it's weird, too. 5. √4 is just 2. How about √5? OMG, that's weird, too. 6. So the square root of an integer is either an integer or one of these weird numbers. It cannot be of the general ℚ form p/q. This is an interesting proof. (While thinking about that with the youngster you can think about another generalization: roots higher than second. Turns out it's true for those, too: https://math.stackexchange.com/questions/4467/how-to-prove-if-a-b-in-mathbb-n-then-a1-b-is-an-integer-or-an-irratio 7. How do we work these weird numbers? For example, can we add them up, e.g. √2 + √3? How do we do that? Is that another weird number or could it ever be an integer? Some facts about these sums are trivial to prove: https://math.stackexchange.com/questions/157245/is-the-sum-and-difference-of-two-irrationals-always-irrational 8. Using the wacky notion of adding two numbers as "mating" you can generally outline some higher algebra concepts, e.g. if a lion mates with a lion the result is always a lion. What if it mates with a tiger? (depends, liger or tigon). Can we think of adding a ℚ to one of these weird numbers the same way? Such intuitions may be misleading (remember the Greeks?) but are fun.