The definition of complex numbers is not arbitrary, though. It falls out of the way real numbers work, and it's pretty much predetermined once you start investi

The definition of complex numbers is not arbitrary, though. It falls out of the way real numbers work, and it's pretty much predetermined once you start investigating "impossible" roots of polynomials. Interesting questions are 1) why do we care about polynomials and 2) how do polynomial roots motivate complex numbers.