In the usual approach to real number computation, one singles out a large, but finite, set 1#1 of ``machine numbers''. These are typically ``floating-point numbers''. A main problem with this approach is that if x and y are machine numbers, then x+y, xy, ...are not necessarily machine numbers. Thus, machine numbers fail to form a field, already by failing to be closed under the field operations.
A solution consists in taking machine numbers close to the mathematical value of the expressions. But is this really a solution? Ignoring overflow problems, we now have a sort of closure under the operations, but the field axioms are not satisfied, because the operations have been modified. Thus, if we use algebra and analysis to construct a numerical program, we don't necessarily get a correct or ``approximately correct'' program.