Rational functions in one indeterminate - useful concept?

[WWGD]

my bad, mixing my metaphors. If I am going to speak of the projective line as the [riemann] sphere, then i should speak of a [complex curve] as a riemann surface. thanks for this clarification.

Kind of strange, to me at least, to see how Complex lines become spheres through projectivization.

 

[mathwonk]

just as the real line is one copy of the real numbers, the complex line is one copy of the complex numbers, thus homeomorphic to the real plane. projectivizing the (affine) line means adding one point at infinity, and thus the complex projective line is the one point compactification of the real plane, i.e. the usual 2-sphere.