We present a weak theory BasicReal of the real numbers in the first
order specification language CASL. The aim is to provide a datatype
for practical purposes, including the central notions and results of
basic analysis. BasicReal captures for instance e and pi, as well as
the trigonometric and other standard functions. Concepts such as
continuity, differentiation, and integration are shown to be definable
and tractable in this setting; Newton's Method is presented as an
example of a numerical application. Finally, we provide a proper
connection between the specified datatype BasicReal and specifications
of the real numbers in higher order logic and various set theories.