  We construct the non-standard complex (and real) numbers using the ultrapower
method in the spirit of Cauchy's construction of the real numbers. We show that
the non-standard complex numbers are a non-archimedean, algebraically closed
field, and that the non-standard real numbers are a totally ordered,
real-closed, non-archimedean field. We explore the various types of
non-standard numbers, and develop the non-standard completeness results
(Saturation Principle, Supremum Completeness of Bounded Internal Sets, etc) for
$\starr$. We give non-standard characterizations for such usual topological
objects as open, closed, bounded, and compact sets in terms of monads. We also
consider such traditional topics of real analysis as limits, continuity,
uniform continuity, convergence, uniform convergence, etc. in a non-standard
setting. In both topology and real analysis we reduce (and in some cases
eliminate) the number of quantifiers in the non-standard setting.
