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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?

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Here is a possibility, taken from Tristan Needham, Visual Complex Analysis (Oxford Univ. Press).
           Needham cover
The advantages of this theorem are:

  1. it is certainly not obvious,
  2. "it would require a great deal of ingenuity" to prove this without complex numbers,
  3. it is elementary planar geometry, and
  4. it is more engaging than "finding the centroid or circumcenter of a triangle."


          Fig12
          T. Needham, Fig.[12], p.16.
The proof uses rotations throughout. For example, the point $p$ is obtained by moving $a$ halfway along the $2a$ edge of the quadrilateral, and then turning $90^\circ$ counterclockwise via $i a$. So $p=a+i a = (1+i) a$. (OP: "Rotation is so much nicer with complex numbers.") Eventually the theorem is proved by showing that $A + iB = 0$, "the verification of which is a routine calculation."

Related: Visual research problems in geometry.

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(comment)

Why the need and is there really a pedagogical benefit to a non-standard presentation of complex numbers? This feels more like something that appeals to you, that thus you want to push on students. But without considering if it really benefits them or why it wasn't done before. Or even if the non-standard approach is detrimental.

Other than that, useful to think about if this is for high capability students or average students.

Oh...and the most obvious applications of complex numbers are (real life) alternating current and (math) roots to the quadratic. Oh...and those are boring and familiar to math shmarties. But for kids learning complex numbers for the first time, they are not boring.

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