I asked a kid what his favorite number was, you won’t believe what happened next :)

A friend from graduate school and her son stopped by yesterday on their way down to NYC.  Her son and my kids have a lot of common interests, including math.  Chit chatting over dinner I asked him what his favorite number was and got quite a surprise:

\pi^{e^i} .

Now that’s a favorite number!

Although his particular favorite number is a little difficult to talk about without getting into things like cos(1) and sin(1), I though that it would be fun to show the kids a little bit about complex numbers.   My end goal was to show the kids the value of i^i rather than \pi^{e^i}, but first we had to have a little discussion about powers:

With some of the simple properties of exponents out of the way, the one last thing we needed to touch on before taking about i^i was square roots. Despite intending to be informal here, I was unfortunately a little too informal and I think that caused a little bit of confusion. It took a few extra examples to get us back on track.

Next up was the fun identity e^{\pi i} + 1 = 0. I didn’t have any interest in deriving this equation, rather I just wanted to use it as a starting point. With just a few manipulations of this equation we can come to a value for i^i. It was super fun to see all three kids react to the surprising value of i^i. At the end I mention the approximate value of \pi^{e^i} as well.

So, a nice little math talk this morning sparked by this surprise favorite number!