# Engaging students: Adding, subtracting, multiplying, and dividing complex numbers

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Daniel Adkins. His topic, from Algebra: adding, subtracting, multiplying, and dividing complex numbers.

How has this topic appeared in pop culture?

Robot chicken aired a television episode in which students were being taught about the imaginary number. Upon the instructor’s completion of his definition of the imaginary number, one student, who understands the definition, immediately has his head explode. One student turns to him and says, “I don’t get it. No wait now I-“, and then his head also explodes.

This video can be used as a humorous introduction that only takes a few seconds. It conveys that these concepts can be difficult in a more light-hearted sense. At the same time it satirically exaggerates the difficulty, and therefore might challenge the students. All the while the video provides the definition as well.

How did people’s conception of this topic change over time?

The first point of contact with imaginary numbers is attributed to Heron of Alexandria around the year 50 A.D. He was attempting to solve the section of a pyramid. The equation he eventually deemed impossible was the sqrt(81-114). Attempts to find a solution for a negative square root wouldn’t reignite till the discovery of negative numbers, and even this would lead to the belief that it was impossible. In the early fifteenth century speculations would rise again as higher degree polynomial equations were being worked out, but for the most part negative square roots would just be avoided. In 1545 Girolamo Cardono writes a book titled Ars Magna. He solves an equation with an imaginary number, but he says, “[imaginary numbers] are as subtle as they would be useless.” About them, and most others agreed with him until 1637. Rene Descartes set a standard form for complex numbers, but he still wasn’t too fond of them. He assumed, “that if they were involved, you couldn’t solve the problem.” And individuals like Isaac Newton agreed with him.

Rafael Bombelli strongly supported the concept of complex numbers, but since he wasn’t able to supply them with a purpose, he went mostly unheard. That is until he came up with the concept of using complex numbers to find real solutions. Over the years, individuals eventually began to hear him out.

One of the major ways that helped aid with people eventually come to terms with imaginary numbers was the concept of placing them on a Cartesian graph as the Y-axis. This concept was first introduced in 1685 by John Wallis, but he was largely ignored. A century later, Caspar Wessel published a paper over the concept, but was also ignored. Euler himself labeled the square root of negative 1 as i, which did help in modernizing the concept. Throughout the 19th century, countless mathematicians aided to the growing concept of complex numbers, until Augustin Louis Cauchy and Niels Henrik Able make a general theory of complex numbers.

This is relevant to students because it shows that mathematicians once found these things impossible, then they found them unbelievable, then they found them trivial, until finally, they found a purpose. It encourages students to work hard even if there doesn’t seem to be a reason behind it just yet, and even if it seems like your head is about to blow.

How has this topic appeared in high culture?

The Mandelbrot set is a beautiful fractal set with highly complex math hidden behind it. However it is extremely complicated, and as Otto von Bismarck put it, “laws are like sausages. Better not to see them being made.”

Like most fractals, the Mandelbrot set begins with a seed to start an iteration. In this case we begin with x2 + c, where c is some real number. This creates an eccentric pattern that grows and grows.

For students, this can show how mathematics can create beautiful patterns that would interest their more artistic senses. Not only would this generate interest in complex numbers, but it might convince students to investigate recurring patterns.

Sources:

History of imaginary numbers:

http://rossroessler.tripod.com/

Mendelbrot sets:

https://plus.maths.org/content/unveiling-mandelbrot-set