Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

We're learning about Möbius transformations and my lecturer keeps referring to $\bar{\mathbb{C}}$, i.e the extended complex plane.

How do you extend the complex plane? Isn't the complex plane all the complex numbers, so how would you have even more complex numbers if you have already included them all?

share|cite|improve this question
up vote 5 down vote accepted

The extended complex numbers is the complex numbers $\mathbb{C}$, with the complex infinity $\{\tilde{\infty}\}$ adjoined as an element of the set. There is only one infinity in this set, so if you're on the complex plane, and you set out along any line going out to infinity, all the lines meet at the same infinity.

The best way to visualize the extended complex numbers is through one of its representations, the Riemann sphere. In this representation, one can think of the north and south poles of the sphere as infinity and zero, respectively. I encourage you to look through that article (at least the opening paragraph) for more details.

share|cite|improve this answer
@Anixx That is the first time I see that notation. I dare say the standard notation is plain $\infty$, using $\tilde{\infty}$ is highly unusual. – Daniel Fischer Apr 18 '14 at 3:07
@Daniel Fischer – Anixx Apr 18 '14 at 3:09
@Anixx Interesting. That seems to be a Wolfram idiosyncrasy, however. Neither Ahlfors, Rudin, any other book I read, nor wikipedia uses that. – Daniel Fischer Apr 18 '14 at 3:13

The extended complex plane is the Alexandroff extension of $\mathbb C$. A single point, called the infinity (in symbols $\infty$), is added to $\mathbb C$; in practice $\overline{\mathbb C}=\mathbb C\cup \{\infty\}$ and obviously $\overline{\mathbb C}$ has a new topology.

If you don't know topology, you can think $\overline {\mathbb C}$ as $\mathbb C$ with "the infinity". "The infinity" collects all points having infinite distance from the origin.

share|cite|improve this answer
I think it's important to note that the extended complex numbers is also a complex manifold, not just the one-point compactification. In other words, it has manifold structure, whereas in compactification we just have some compact topological space. – Christopher A. Wong Nov 21 '12 at 18:56

For some videos of the geometry behind stereographic projection (turning the complex plane into a sphere without the north pole, and vice versa), see starting at 7:18, starting at 6:21. The entire video exhibits the geometric unification that happens with Möbius transformations when you look at them on the Riemann sphere (aka extended complex plane) instead of in the complex plane by itself.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.