Well this is one of the coolest things I’ve ever learned about.
So I’m in Complex Variables this semester, right? Today we talked about how to take limits of complex numbers as well as the closely related topic of infinity.
I’m assuming most people who read this regularly (or just happen to stumble upon it) know at least a little about infinity in the context of real numbers. Mainly, if we represent the reals on a number line, we have a direction that goes off towards negative infinity and a direction that goes off to positive infinity. But does this translate to complex numbers?
Well, not really. When we deal with complex numbers, we deal with the complex plane: a 2-D space with one axis representing the real part of a number and the other axis representing the imaginary part of a number. That is, one way we can think of a complex number is as a set of coordinates on the complex plane. For example, if I had the number z = 3 + i2, I could represent it with the coordinates (3,2) and plot it like this:
Since we’re now dealing with a plane, we actually have infinitely many directions that can be thought of as infinity—basically, any direction out from the origin.
So how do we define infinity in the complex plane to allow us to, among other things, take limits involving the point at infinity?
Answer: The Riemann sphere!
The Riemann sphere is a stereographic projection of the complex plane onto the unit sphere at the point (0,0,1). Piccy from Wiki:
So what does this do? Well, for each point A on the complex plane, there exists a line that intersects both the point A and the north pole of the sphere. This line hits the sphere itself at α, point unique to the position of point A in the complex plane—that’s how the plane is “mapped” to the sphere and that unique mapping point is called the “projection” of point A.
The further out you go on the complex plane—that is, the further away from the sphere you go, those projection points get closer and closer to the north pole itself. However, no point is projected directly onto the north pole. So we can think of the north pole as being the image of all the points in the complex plane that are at infinity.
Isn’t that cool? It’s a way to reduce an “infinite number of infinities” to a single point.
We didn’t have time to talk about how we’re going to use it yet, but just that idea is super cool and required a blog post.