A great complex number question from Art of Problem Solving

My son picked a great problem from Art of Problem Solving’s Introduction to Algebra to talk through today:

Interesting problem from my son's algebra book – find all complex number a + bi such that the conjugate is equal to (a + bi)^2

— Mike Lawler (@mikeandallie) May 25, 2016

A lot of people on Twitter thought the problem was fun and many people commented on the difference between a geometric and an algebraic approach.

A geometric approach is beyond my son’s understanding right now, but the algebraic approach has so many great lessons going for it – many more than you think if you aren’t generally around kids working through math problems:

 

After we were done I talked through some of the geometry for just a little bit – I don’t think introducing the geometry solely through this problem is such a great idea, but I wanted to try anyway. Here’s how it went:

 

Finally, as an extra shout out to AoPS – here’s one of my favorite math videos of all time – Richard Rusczyk illustrating how powerful a geometric approach to complex numbers can be with a stunning solution to an old math contest problem: