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2.15.1 Dedekind Cuts

A real number tex2html_wrap_inline33691 is represented by a cut tex2html_wrap_inline33693 ,   tex2html_wrap_inline33695 . Every cut has the property that for all tex2html_wrap_inline33697 :

math8872

As presented, the cut tex2html_wrap_inline33699 represents tex2html_wrap_inline32375 . Disallowing this special cut gives a representation for all non-negative real numbers. In general,

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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:

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Difficulties are encountered when generalizing this to negative real numbers. If a cut is simply redefined to be a subset of tex2html_wrap_inline32161 , 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 up previous notation contents
Next: 2.15.2 Cauchy Sequences Up: 2.15 Real Representations Previous: 2.15 Real Representations
Jeff TupperMarch 1996