Real Numbers and the Dedekind Cut
DOWNLOAD
FREE
|
The set of all rational numbers is dense
Equations are omitted for technical reasons - download the original pdf
Integers may be ordered by the relation > or <. Likewise, rational numbers may be ordered. We define [Equation goes here - download the original to see it.] Theorem Between any two rational numbers there is another rational number. Proof Let [Equation] where a,b,c and d are integers. Then [Equation] Let m be any positive integer. Then [Equation] By a similar argument [Equation] Hence [Equation]. Corollary Between any two rational numbers there lies an infinite number of rational numbers. This means that the interval [Equation] contains an infinite number of rational numbers. It is said to be dense.
|
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
|
