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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?

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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.

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@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.

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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.

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