We derive a description of the spatially inhomogeneous Bose-Einstein condensate which treats the system locally as a homogeneous system. This approach, similar to the ThomasFermi model for the inhomogeneous many-particle fermion system, is well-suited to describe the atomic Bose-Einstein condensates that have recently been obtained experimentally through atomic trapping and cooling. In this paper, we confine our attention to the zero temperature case, although the treatment can be generalized to finite temperatures, as we shall discuss elsewhere. Several features of this approach, which we shall call the Thomas-Fermi-Bogolubov description, are very attractive: 1. It is simpler than the Hartree-Fock-Bogolubov technique. We can obtain analytical results in the case of weakly interacting bosons for quantities such as the chemical potential, the local depletion, pairing, pressure and density of states. 2. The method provides an estimate for the error due to the inhomogeneity of the bose-condensed system. This error is a local quantity so that the validity of the description for a given trap and a given number of trapped atoms, can be tested as a function of position. We see for example that at the edge of the condensate, the Thomas-Fermi-Bogolubov theory always breaks down. 3. The Thomas-Fermi-Bogolubov description can be generalized to treat the statistical mechanics of the bose gas at finite temperatures.