An orthogonal system of rational functions is introduced. Some
 results on rational approximations based on various orthogonal
 projections and interpolations are established.  These results form
 the mathematical foundation of the related spectral method and
 pseudospectral method for solving differential equations on the half
 line. The error estimates of the rational spectral method and
 rational pseudospectral method for two model problems are
 established. The numerical results agree well with the theoretical
 estimates and demonstrate the effectiveness of this approach.