the Riemann sphere
/ REE-mahn /
The point at infinity feels like an awkward bolt-on when you picture the complex plane as a flat sheet — infinity is 'out there' with no real home. The Riemann sphere is the picture that fixes this beautifully: wrap the whole extended complex plane onto the surface of an ordinary sphere, so that the point at infinity becomes a perfectly ordinary point — the north pole — no more special than any other. Suddenly the lopsided plane-plus-one-extra-point becomes a smooth, symmetric, finite surface.
The wrapping is done by stereographic projection. Place a sphere touching the plane, with its south pole at the origin. To find where a plane point z lands, draw the straight line from the north pole through z; it pierces the sphere at exactly one other point, and that is the image of z. The origin maps to the south pole, the unit circle maps to the equator, points far out near infinity cluster around the north pole — and the north pole itself is precisely the point at infinity. Every point of the extended plane corresponds to exactly one point of the sphere, and vice versa.
This is not mere decoration. On the sphere, infinity is just another point, so theorems can treat it on equal footing: a function is studied 'at infinity' simply by studying it at the north pole. Lines in the plane become circles through the north pole; the family of all lines and circles in the plane becomes simply the family of all circles on the sphere; and Mobius transformations become rigid-feeling rotations and motions of the sphere. The Riemann sphere is the geometric stage on which the global theory of complex functions is most naturally told.
Under stereographic projection: the origin sits at the south pole, the unit circle |z| = 1 maps to the equator, and the entire infinite region |z| > 1 squeezes into the small cap around the north pole — with infinity itself exactly at the north pole.
The flat plane plus one point becomes a smooth sphere; infinity is just the north pole.
The Riemann sphere is the SAME object as the extended complex plane C-hat — it is a way of seeing it, not a different set. It is also the simplest example of a Riemann surface, an idea that grows enormously later.