I am currently writing some basic introductory texts to complex numbers for third-year high school students (Denmark). My main goal is to introduce complex numbers as a practical tool that both simplifies the overall algebraic structure of math (simplifying work with trigonometric functions and polynomials), and can work in and of itself as a practical tool for modelling certain geometric objects. Due to the nice interplay between rotation, multiplication and exponentiation, numbers in the complex plane can on occasions be a better choice to work with. Two pretty mathematical examples are:
- Finding the centroid or circumcenter of a triangle
- Working with rotated conics: Finding intersections, amount of intersections, transformations, ect.
Conics have lots of obvious applications, but circumscribed triangles is a bit too specific for me to find any good applications/modelling exercises. Rotation is so much nicer with complex numbers, so surely there must be more geometrical applications not?