Can an irrational number to an irrational power be rational? (thanks Francis Su!)

Saw this really neat lecture from Francis Su posted on Twitter today:

In the lecture he gives a really cool proof that an irrational number to an irrational power can be rational. I can’t remember ever seeing this proof before and I since it uses only really basic ideas about powers, I thought I’d share it with the boys tonight.

As an aside, by virtue of a lucky coincidence at the NCTM meeting in Boston this last spring, I had an opportunity to have lunch with Francis Su (and Fawn Nguyen, Tina Cardone, Justin Lanier, Dan Anderson, and a bunch of other folks). I wrote about that fun day here:

My surprise afternoon with the MBToS and MAA President Francis Su

Now back to the project . . . we started talking about some basic properties of powers. The opening question was what is 3^2 * 3^5? I was a little surprised that my younger son actually computed the product, but that computation led to a nice conversation to start off the project.

Next we moved on to talking about powers of powers. These numbers were large enough that we did not multiply them out. Even though this bit of the project was short, we got to talk about powers of powers in several different ways.

We have talked about powers of powers in a few different projects in the past, too. For example:

The last 4 digits of Graham’s Number

The Joy of x^x^x^x^ . . . .

Now we had the background to talk about the question (and proof) from Francis Su’s lecture. I especially like this proof because it does not show a specific example that satisfies the property of an irrational number to an irrational power being rational. Rather, it just shows that such an example exists. It is a beautiful mathematical argument.

At the end of the last movie my older son asked about what might happen if we used the square root of 3 rather than the square root of 2. I was super happy to hear this question, so we extended the project for one more video to check out a few more cases.

What a great “fun fact” to learn from Francis Su’s lecture. It was really great to share this idea with the boys – both because the idea itself is pretty amazing, and because the proof is really cool. Happy that I saw the video of his lecture this afternoon.