Algebraic curves (one-dimensional projective varieties) over the complex numbers are exactly Riemann surfaces. It confuses everyone at first when one is told "curves are surfaces." Almost everyone else calls \mathbb C the complex plane, but algebraic geometers call it the complex line.

One can work in any algebraically closed field, say \mathbb A, the field of algebraic numbers. But analysis only works in R or C, which are complete.

Algebraic curves (one-dimensional projective varieties) over the complex numbers are exactly Riemann surfaces. It confuses everyone at first when one is told "curves are surfaces." Almost everyone else calls $\mathbb{C}$ the complex plane, but algebraic geometers call it the complex line.

One can work in any algebraically closed field, say $\mathbb{A}$, the field of algebraic numbers. But analysis only works in $\mathbb{R}$ or $\mathbb{C}$, which are complete.