# Motivating Complex numbers for kids

Saw this post on Twitter when I got up this morning:

Made me want to share my attempt to talk about complex numbers with my older son earlier this year.   Up until this discussion all we had really covered was basic definitions of complex numbers – hardly anything more than that $i = \sqrt{-1}$.  This series of discussion was motivated by my son asking me why the cube root of 343 had 7 as the only solution, but something like the square root of 9 had two solutions -> +3 and -3.

As with just about everything I do with my kids, this was all on the fly (except some details in the last talk).   I’m sure there are mistakes, but here’s what we did:

The first step was just talking about the question he’d posed:

The next discussion was an attempt to do an example similar to the one in the first video but not quite as complicated – the roots of $x^4 - 1 = 0$.  I probably should have started with this equation in retrospect.  The important step in this problem was looking at where two sets of roots we’ve found appear in the complex plane:

The next step was a detailed look at the geometry that we had looked at in the last video.  The focus here is the three solutions to $x^3 - 1 = 0$.  In particular, the three solutions look like they lie on our unit circle – do they?  The next thing would be to see if they do actually form an equilateral triangle.  Not sure why I skipped this second part:

The next thing we did was to circle back to the original question.  We had used a combination of factoring and the quadratic formula to find the solution to $x^3 - 1 = 0$, but we never bothered to check that all three of the numbers we found were actually solutions.  Here we crunch through that math just to get some experience calculating with complex numbers:

I wanted to end the series talking about a neat result – the fundamental theorem of algebra.  It was tricky to figure out how to cover this topic with a young kid, but I wanted to give it a shot.  Obviously I don’t get into any details of Gauss’s proof, but I’m always happy to talk about important math results.  This particular topic was actually more fun than usual since, to my son, Gauss is famous for adding up 1 + 2 + 3 + . . .  + 100 – and nothing else!! Ha ha:

I definitely enjoyed throwing this little sequence together.  Even though there was no direct follow up, every now and then I think it is fun to show my kids more advanced ideas that use some of the basic math we are covering.  Probably my favorite example along those lines – and one that turns out to be pretty closely connected to the sequence here – was a talk about regular pentagons.  A little longer than usual, but one of my favorite examples of how some basic math comes together for a neat result: