Does this strike you as a problem that leads to complex numbers? If not, then that's precisely why we should start teaching complex numbers in this way.
Complex numbers were invented for algebraic reasons (solving cubics!) and then collectively disparaged by mathematicians for a few hundred years. What did it take for complex numbers to become widely accepted? Geometry.
It is not an unreasonable demand that operations used in geometry be taken in a wider meaning than that given to them in arithmetic.
There are many situations in which it is helpful to have precise algebraic ways to describe geometric situations. This is a problem that complex multiplication were born to help with. Beginning with a "Follow That Point!" activity drives at this intersection of rotations and algebra.
That's my case for introducing students to complex numbers with a rotation activity instead of with quadratic equations.
I wrote a lesson (as part of a unit) headed towards this understanding of complex numbers. I got help from Max, Malke and Bridget. I would looove feedback and critical questions on all this. Be in touch here or (even better) on twitter.
- Activity Sheet
- Teacher's Guide (Annotated Google Doc)