On teaching complex numbers

Complex numbers are usually introduced to high-school students in a rather confusing manner. Till a certain grade they are told that the square-root of a negative real number is “forbidden”, and then $i=\sqrt{-1}$ is introduced all of a sudden. It is a pity that despite being able to solve a ton of problems, most students cannot perceive complex numbers beyond mere algebraic manipulations. The true potential of complex numbers is perhaps better understood through a geometric viewpoint, followed by its applications in other topics such as geometry, trigonometry, polynomials, combinatorics, and so on.

I personally found the last approach to be more intuitive, and experimented the same on several batches of students for the past few years (2019-21). I don’t really know how successful these experiments have been. However, I did observe that the students actually enjoyed it.

References

In this YouTube video, you can find a visual introduction to the addition and multiplication of complex numbers (for younger students: don’t get intimidated by the title, you may watch at least the first 15 minutes).

The book Complex Numbers from A to … Z by Titu Andresscu served as the main reference. Other than this, following are some notes that might be helpful, especially if one wants to learn complex bashing.

Class tests

Towards the goal of making the course more application oriented, I had a wish that none of the class test papers should mention “complex number” other than in the title. Well, I am fortunate to have this wish (almost) fulfilled!

Classnotes

Following are some classnotes of the online classes I gave in 2021. These are incomplete by all means to serve as a reference, still I put them here to indicate how the courses went.

Some more (from 2020): CN-1 , CN-2, CN-3, CN-4, CN-5