The Gauss map has been shown to be a good example of a chaotic discrete dynamical system, in that it exhibits in an accessible fashion all the common features of such systems. The map is simple enough that the relationship of numerical simulation of the map to the exact map can be explored effectively. We find that the numerical simulation of the map behaves significantly differently, in that the numerical simulation is not chaotic, but is still useful in that the Lyapunov exponent of the exact map can be accurately calculated from the simulation. We have in fact shown that this behaviour of numerical simulations is general. We have also exhibited a new (though impractical) method for the calculation of .