Dedekind CutYou are currentlybrowsing as guest. Click here to log in 

A Dedekind cut is a division of the rational numbers into two nonempty sets, A and B, such that everything in A is less than everything in B, the union is all of Q, and B does not contain its greatest lower bound.
The collection of Dedekind Cuts is then a construction of the real numbers. Defining "addition" and "negation" is relatively straightforward, although care must be exercised regarding subtraction, because if A does contain its least upper bound, that point has to be transferred to B.
"Multiplication" and "multiplicative inverse" need more care, and are quite messy in some cases.
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With this formulation some of the technicalities are avoided, but others do arise. As with many theorems and definitions in mathematics it is useful to have several, equivalent forms, and then use the form that is most convenient.
http://mathworld.wolfram.com/DedekindCut.html
http://en.wikipedia.org/wiki/Dedekind_cut
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