Beam asymmetries for the reactions gamma p -> p pi0 and gamma p -> n pi+ have been measured with the CEBAF Large Acceptance Spectrometer (CLAS) and a tagged, linearly polarized photon beam with energies from 1.102 to 1.862 GeV. A Fourier moment technique for extracting beam asymmetries from experimental data is described. The results reported here possess greater precision and finer energy resolution than previous measurements. Our data for both pion reactions appear to favor the SAID and Bonn-Gatchina scattering analyses over the older Mainz MAID predictions. After incorporating the present set of beam asymmetries into the world database, exploratory fits made with the SAID analysis indicate that the largest changes from previous fits are for properties of the Delta(1700) 3/2- and Delta(1905) 5/2+ states.