Ed Frenkel, the square root of 2 and i

A few weeks ago, some thoughts on twitter from Michael Person inspired this talk with my kids:

https://mikesmathpage.wordpress.com/2014/04/19/imaginary-numbers/

Last weekend I picked up the audio book version of Ed Frenkel’s “Love and Math” and Frenkel’s discussion of \sqrt{2} and i made me want to revisit this conversation about properties of numbers.

We started with \sqrt{2}.   Their reaction to hearing that we were talking about \sqrt{2} was to talk about why it was irrational, and since they nearly remembered the proof from last time, this proof made for an instructive start to the conversation today.  It is always nice to review some of the ideas behind these simple proofs with them and watch their ability to make mathematical arguments develop.

Next we moved on to talking about i.  They remembered a few basic properties about i, though my older son still thinks that it is something that math people just made up.  I’m not terribly bothered by that for now, but the ideas in Frenkel’s book are giving me some new perspective on how to present some of these more advanced concepts to the boys.  Hopefully this new perspective is going to lead to a much better approach to teaching them math.  In any case, here’s what we said about i:

The next two videos are the main point of the talk today – in what ways are \sqrt{2} and i similar?  This question is a specific example of the broad question of symmetries in math that Frenkel discusses in his book.  I felt like the book walked up a couple of stairs and then hopped into an elevator to the top floor, though.  The ideas were inspiring, but I was left (i) wanting more and (ii) wanting to fill in a few more details.  One focus of these math conversations with my kids over the next few years will be spent on (ii).  I’ll work on (i) by finishing the audio book on a drive to and from Boston this weekend!

For today, though, let’s just stick with some similarities between \sqrt{2} and i that Frenkel highlights:

So, without digging too deep into the details, it looks like the set of numbers that we get by adding \sqrt{2} to the rational numbers has some nice, simple properties.  If we add or multiply, we seem to never leave the system.  Pretty neat.  i seems to have the same property.  Frenkel make the point that is we aren’t too bothered by \sqrt{2}, we shouldn’t be that bothered by i.  This is a nice point, obviously, and a fun idea to share with kids.  I really loved that my older son made the connection between i and x from algebra.  Only one step away from polynomial rings . . .  ha ha!

So, definitely on the theoretical side, but a definitely a fun morning.  Looking forward to plucking a few more ideas  out of “Love and Math”  to share with the boys.

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