'Extended complex plane and infinity Ahlfors, in his Complex Analysis, while beginning the discussion on spherical representation of complex numbers,  says the following: "...we can of course introduce an 'ideal' point which we call the point at infinity. The points in the plane together with the point at infinity form the extended complex plane. We agree that every straight line shall pass through the point at infinity. By contrast, no half plane shall contain the ideal point."  
What does he mean by saying that every straight line passes through "the point at infinity"? Also, with every straight passing through the ideal point, how come no half plane contains the ideal point?
 A: All points of a circle $\gamma\subset{\mathbb C}$ are ordinary complex numbers, and all of them have a distance $\leq R_\gamma$ from the origin, where $R_\gamma$ depends on the center and the radius of $\gamma$.
In the same way all points of a line $\ell\subset{\mathbb  C}$ are ordinary complex numbers, but there is no finite number $R_\ell$ such that all points of $\ell$ would have a distance $\leq R_\ell$ from the origin.
Now in certain situations it is convenient to extend the complex plane ${\mathbb C}$ by a single "artificial" point $\infty$ to the extended complex plane, or Riemann sphere, $\bar{\mathbb C}:={\mathbb C}\cup\{\infty\}$.
Things are set up in such a way that $\lim_{n\to\infty} z_n=\infty$ iff $\lim_{n\to\infty}|z_n|=\infty$ in the real sense.
A "finite" circle $\gamma$ cannot feel this extension; therefore $\gamma$ stays as it is in the extension $\bar{\mathbb C}$. On the other hand, any line $\ell\subset{\mathbb C}$ contains points $z$ of arbitrary large absolute value $|z|$. Since we want $\ell$ to be a closed set also in the extended plane we require that such an $\ell$ be extended to $\bar\ell:=\ell\cup\{\infty\}$, again denoted by $\ell$ for simplicity.
Now such an $\ell$ divides ${\mathbb C}$ into two open half planes not containing any points of $\ell$ – in the same way as a circle $\gamma$ divides ${\mathbb C}$ into an interior and an exterior. Under the extension the exterior of $\gamma$ will pick up the additional point $\infty$, but none of the two half planes created by $\ell$ will pick up this point, since $\infty\in\bar\ell$ already.
There seems to be a lot of exception handling here. In fact stereographic projection (also described by Ahlfors) allows to interpret all these ad-hoc-constructions in a unified manner: We then are only talking about ordinary circles on a $2$-sphere.
