Why Abelian Grapes Are Funny and Important

Why Abelian Grapes Are Funny and Important

A couple weeks ago, my son started studying complex numbers in his Advanced Algebra class.  He had caught onto the idea of adding and multiplying them, but he had lots of questions about where they came from, why imaginary numbers are called "imaginary", and what these numbers really were.  We spent some time talking about the history of complex numbers and I showed him how to graph them, making the connection between the real number line and the need for two dimensions.

This idea of two-dimensional numbers got him thinking and he wanted to know about higher-dimensional numbers.  This led us to a discussion (very limited) about quaternions.  We looked them up on Wikipedia, and the second line was about multiplication on quarternions being non-commutative.  This kind of blew his mind, since everything he had studied so far said that multiplication was commutative.  So we talked for a few minutes about defining sets of numbers as groups, rings, and fields.  I had not remembered all the properties that applied in each case, so we had a nice research session online.  I had also just been reading Edward Frenkel's Love and Math which has a nice description of an abelian group using rotations in the first couple of chapters, so I showed him that example.

He's been really excited about our math talks, and I enjoy talking to him about the subject I enjoy.*  Since we've been having these conversations, his grade in his class has gone up, even though we have not spent as much time on the actual classwork.  As I think about our math talks, I am reminded about how important context is for some students.  Simplifying rational expressions was not very interesting for him until we chatted about limits, and the quadratic formula was just a dreary procedure until we talked about Girolamo Cardano and Niccolo Tartaglia.  Frenkel uses the analogy that the math we teach in school is to the actual topic of Mathematics as painting a fence is to appreciating and talking about Art.  (You can see an interview with Edward Frenkel on The Colbert Report.)  Makes me think I need to brush up on my Math History as I start planning lessons for the fall ...

*I am slowly trying to convince my son to think about majoring in math when he gets to college in a few years, but he still thinks engineering is his thing.  He got excited about some notation and game theory ideas a few weeks ago, and when I told him the joke, "What's purple and commutes?  An abelian grape," he laughed.  So I'm winning. :)