My Favorite One-Liners: Part 54

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

The complex plane is typically used to visually represent complex numbers. (There’s also the Riemann sphere, but I won’t go into that here.) The complex plane looks just like an ordinary Cartesian plane, except the “$x-$axis” becomes the real axis and the “$y-$axis” becomes the imaginary axis. It makes sense that this visualization has two dimensions since there are two independent components of complex numbers. For real numbers, only a one-dimensional visualization is needed: the number line that (hopefully) has been hammered into my students’ brains ever since elementary school.

While I’m on the topic, it’s unfortunate that “complex numbers” are called complex, as this often has the connotation of difficult. However, that’s not why our ancestors chose the word complex was chosen. Even today, there is a second meaning of the word: a group of associated buildings in close proximity to each other is often called an “apartment complex” or an “office complex.” This is the real meaning of “complex numbers,” since the real and imaginary parts are joined to make a new number.

When I teach my students about complex number, I tell the following true story of when my daughter was just a baby, and I was extremely sleep-deprived and extremely desperate for ways to get her to sleep at night.

I tried counting monotonously, moving my finger to the right on a number line with each number:

$1, 2, 3, 4, ...$

That didn’t work, so I tried counting monotonously again, but this time moving my finger to the left on a number line with each number:

$-1, -2, -3, -4, ...$

That didn’t work either, so I tried counting monotonously once more, this time moving my finger up the imaginary axis:

$i, 2i, 3i, 4i...$

For the record, that didn’t work either. But it gave a great story to tell my students.