All of the results of the previous sections are valid for the familiar domain of the real numbers. However, when we work in any fixed-precision system, we have two difficulties:
  1. Not all numbers are even representable in the system, and
  2. Arithmetic doesn't have the properties we are used to.
For example, defining u as the smallest machine representable number which when added to 1 gives a number different from 1 when stored, we see that is computed as 0, whenever is any number between 0 and u. This effectively limits the power of the singularity of the Gauss map.