Imaginary Numbers

A few weeks ago, when I was in London, Michael Pershan wrote an interesting piece about complex numbers on his blog:

In my jet-lagged state I came up with what seemed like a great response, but not too surprisingly, it didn’t seem so great once I wasn’t so jet-lagged!   However, Pershan’s post stayed with me for a couple of weeks.  I’m not sure why – there wasn’t anything specific that bothered me, or at least nothing that I was able to articulate well.  I still couldn’t shake it though, and because I’ve spent the week with my older son talking about logs and imaginary numbers I really wanted to talk to the boys about i.

By lucky coincidence just this morning Steven Strogatz linked  to another blog post on twitter that helped me collect my thoughts on the subject of i (even though the blog post has nothing to do with complex numbers):

The part of this post that got my attention was this line:  “Well, I’ve finally got my answer, and it only takes eleven words: Math is big ideas, approached from as many angles as possible.”

So, a few weeks of thinking things over in my head combined with a little jolt from the Math With Bad Drawing’s post led to this morning’s Family Math conversation:

(1) The first thing I wanted to talk about was the hardest – structure.  I dug up my old copies of Mike Artin’s and van der Waerden’s “Algebra” books and talked about rings.   I don’t think I did a good job here, but all I wanted to do was point out some of the things that mathematicians think are important about the number system.  A lot of this was sort of review – they know the words “associative” and “commutative” though they may not have ever seen all of this structure on the board all at the same time.  They certainly had never heard the word “ring” in this context before:

(2)  With the background structure out of the way, and in particular with the mention of 0 and 1, I wanted to start talking about some specific numbers.  We started with an “easy” number – 0.  What are the important properties of 0?  Why doesn’t 0 seem strange to anyone any more?


(3) Next up was -1.   Same sort of idea behind this talk as the talk about 0. -1 is a little hard to understand, but we seem to be pretty comfortable with it and I don’t think many people would think -1 is a number that mathematicians just made up:


(4) Next up was \sqrt{2}.  I wanted to talk through some of the ideas behind irrational numbers and hint at some of the confusion that they had caused over the years.  We talk about a simple right triangle and show how \sqrt{2} comes up pretty naturally.  We also then walk through the proof of why it is irrational:


(5) Now for two famous irrational numbers \pi and e.   We just touch on a few simple properties of these two numbers and talk about they are in some sense even more strange than \sqrt{2}.

(6) Finally we get to i.  Following Artin and van der Waerden, I introduce the complex numbers by looking at a specific quotient ring in a polynomial ring . . . . Ha!

Actually, we first quickly review a few of the interesting numbers that we’ve talked about already and then point out that we still do not know about any number that satisfies a pretty simple equation – x^2 + 1 = 0.  I call that solution “i” and then we talk about a few interesting properties ranging  from the famous e^{\pi * i} + 1 = 0 to Gauss’s Fundamental Theorem of Algebra.    I also mention a few applications that come up in physics.

So, thanks to Michael Pershan and to Ben Orlin for both causing and helping me to think through talking about i.  Definitely a fun morning.




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