Motivated by Algorithmic Information Theory, we study the field Cc of the complex computable numbers. We also present some non trivial examples of computable and noncomputable real numbers and propose a definition of a partial computable function defined in C n . We develop the basic notions of convergence in the field Rc of the real computable numbers and show that every analytic function whose series representation has computable coefficients is a computable function when restricted to any c disk in its domain. Finally, we prove that Rc is a real closed field and also that Cc is an algebraic closed field containing the algebraic numbers.