Exact solutions are derived for the input impedances of vertical magnetic and electric short dipoles over a lossy half space. The exact solution is expressed as a simple formula and a remainder, which takes the form of a rapidly converging series. Reduction to available approximate solutions in the literature under conditions of large dipole heights, large refractive index of the half space, or both is demonstrated. The derivation of new simple formulae that tend to the exact solutions when the refractive index of the half space approaches both limits of infinity or unity is of particular interest. Numerical examples are presented and compared with published results.