Real Numbers and the Dedekind Cut
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The Dedekind cut
Equations are omitted for technical reasons - download the original pdf
We have seen that we need to extend the number system to include irrational numbers. The question of how to define a real number in general is solved by means of the Dedekind cut (1872). To illustrate this method, we take [Equation goes here - download the original to see it.] again as an example. In trying to isolate [Equation] we observe that all rational numbers may be divided into two sets; those whose squares are less than 2, and those whose squares are greater than 2. We denote the former set by L and the latter by R. The method of bisection enables us to “home in” on the numerical value of [Equation] finding successive approximations to it. We see that the set L contains the following numbers obtained by the method of bisection. [Equation] Likewise, the set R contains the numbers [Equation] The irrational number [Equation] is defined to be the cut in the set of rational numbers that follows this rule.
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Contents of
Real Numbers and the Dedekind Cut
1 Numbers as solutions to equations
2 The irrationality of root 2
3 The set of all rational numbers is dense
4 The Dedekind cut
5 The set of real numbers as an ordered field
6 The ordered field axioms
7 The sets of rational and real numbers are ordered fields.
8 Axiom of completeness
9 Bounded sets of numbers
10 Maximum and minimum
11 Bounded set
12 Least upper bound
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