Next: 2.15.2 Cauchy Sequences Up: 2.15 Real Representations Previous: 2.15 Real Representations

## 2.15.1 Dedekind Cuts

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.

Next: 2.15.2 Cauchy Sequences Up: 2.15 Real Representations Previous: 2.15 Real Representations
 Jeff Tupper March 1996