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A real number is represented by a cut ,
.
Every cut has the property that for all :

As presented, the cut represents
. Disallowing this
special cut gives a representation for all non-negative real numbers.
In general,
Most numbers have a
representation that cannot be written out directly since
the representation is an infinite set.
Operations on reals are inherited from the corresponding
operations on rationals. For example, a binary operation on two
real numbers, represented by cuts *X* and *Y*,
is given by:

Difficulties are encountered when generalizing this to
negative real numbers. If a cut is simply redefined to
be a subset of , then
the product of two cuts is not a cut if the
multiplicands correspond to negative numbers.
See [8, 64] for further details concerning this representation
and associated methods.

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