Teaching irrational numbers?

The first thing that a teacher should know is that the very existence of irrational numbers is an astounding fact. "Anyone who is not surprised by irrational numbers does not understand them", you might say.

Here's how I would drive home how bizarre they are.

Consider a line, say a meter in length. Now, if the only unit of measurement I know is the meter, I won't be able to measure very many points on this line - just the endpoint. If I want to be able to measure out the infinite number of points in the middle, I'll need a finer unit of measurement, right?

Right.

So let's use centimeters. Now I can measure a lot more points, but there's still an infinite number of points inside each centimeter.

Yes.

That's okay, let's subdivide even further. We'll divide every centimeter into ten parts to get millimeters. We can then divide millimeters into tenths as well, and keep going further and further. In doing so, we will eventually mark off every point on this line, right? Who thinks we'll hit every single point along the line, if we just subdivide far enough into tenths?

[Show of hands]

Actually, the ones who weren't so sure are right. There's some points along the line that can't be measure out in tenths, or in tenths of tenths, or any unit like that. A third of the line is an example.

[Students may be surprised at this - as they should be! It's kind of wild that even though the subdivision mesh is tending to zero, some points will eternally slip through the contracting net. I'm unfortunately unaware of any way to show that 1/3 has an infinite decimal expression without lots of calculation.]

Okay, so one third slipped through our hands. It's not the only one - also two thirds, one seventh, four elevenths - there's actually lots of points along the line that can't be measured out in decimal units like that. So we're going to have to add in even more points. Let's add in all the thirds - one third, two thirds. All the quarters - one quarter, two quarters, three quarters. All the fifths, all the sixths, everything. We subdivide out line into every possible number of parts. Now, we're surely done, right?

[I would expect more students to say yes to this.]

Well, no. Far from it. There are certain points along this line - an infinite number of them, in fact - that cannot be measured by any subdivision of the line. It doesn't matter if you divide the line into three or twelve or 566734 parts - you will never have a fine enough unit of measurement to measure them out exactly. You'll always overshoot or undershoot just a little bit. This is why the Greeks called them incommensurable - which means "unmeasurable". We call them "irrational" numbers.

At this point I expect students will want an example, so use the square root of two. Note that you're cheating them if you just say "the square root of two is irrational", because you haven't proven that there's such a thing as the square root of two. Use the Pythagorean theorem to justify that there are lines who squared lengths should make 22, and then use the standard proof to show that such a length cannot be rational. If you just assume there's a square root of two and give the proof that it can't be rational, that's cheating. You just prove there's no fraction who's square is two - okay, kinda neat, but big whoop. There's also no integer whose square is ten, that's not cause for despair.

By the way, this leads to philosophical issues about the boundary between math and reality - another possible path would be to say "well then clearly it's impossible to draw a perfect right isosceles triangle". The thing is that math only idealizes reality, and if we're going to idealize it, we might as well idealize it in a way that's convenient.

As for the use of irrationals in algebra (as opposed to geometry) I like to think of them as symbols for "insert value as close as possible to the ideal value here". There may not be a square root of two, but I can find rationals that get as close to it as possible, and as the required perfection of your approximation grows, the range of rational numbers that will fit the bill gets slimmer and slimmer, and closes in on a particular "point" (this is one way to construct the reals - nested sequences of rationals whose lengths tend to zero). So there is a sort of a perfect value for the square root of two, and when I write that the solution to an equation is 2‾√2, I mean that you should pick a value as close to that point as you can if you want to solve the equation. If I write 1+2‾√1+2, I want a value as close to that ideal point as possible, and then you add 11 to that.

You eventually need that kind of arithmetic point of view on what irrationals are, rather than just relying on geometry to justify their existence (or at least their utility as a way of thinking about things, depending on your philosophical leanings). Otherwise, what are you going to do when you get to 31/531/5? That clearly has no geometrical meaning as a line length.

Above all, remember: if your students aren't amazed, or at least slightly fazed, you were unsuccessful. Don't let them think it's just another rule for them to plug and chug with!