Complexifying our dots

It’s time to up our game a bit. Previously we have considered some cool pictures with dots and bespoked circles, looking for patterns, without really considering what sort of mathematical objects these circles might represent. In fact, they turn out to have a close connection to complex numbers.

Recall that a complex numbers are built from real numbers together with the “imaginary unit” $i$ (which is just as real as real numbers!) which has the property that $i^2 = -1$. A complex number is of the form $a + bi$ for real numbers $a$ and $b$, and we can think of complex numbers as living in a two-dimensional plane, where we usually think of real numbers as the horizontal axis, and imaginary numbers as the vertical axis.

So the dots on the circles we have been considering can be thought of as complex numbers:

If we suppose the circles to be centered at the origin and have radius 1 (which turns out to work nicely), then the above blue dots happen to correspond to the complex numbers $1/2 \pm i \sqrt{3}/2$ (as you can confirm with either some basic geometry—those dotted triangles are equilateral triangles with sides of length 1—or by remembering the sine and cosine of some special angles).

Next time, we’ll recall how multiplication of complex numbers works, and discover what is special about the dots considered as complex numbers. For now, I’ll leave you with something to play with, if you haven’t seen this before: what happens when you square $1/2 + i \sqrt{3}/2$? When you cube it? Take the fourth power? And so on?