Real Numbers and the Dedekind Cut
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The irrationality of root 2
Equations are omitted for technical reasons - download the original pdf
The proof of the irrationality of prime numbers is attributed to Pythagoras, who lived during the C6th BC. It is a proof by contradiction. Proof of the irrationality of [Equation goes here - download the original to see it.] Suppose [Equation] is rational. Then [Equation] can be written as a fraction, so it can be expressed as a ratio [Equation] We may also assume that p and q do not have any common factors, because if they did these common factors could be cancelled out. Taking the equation [Equation] and squaring both sides, we obtain [Equation]. Hence [Equation] [Equation] This means that q is even. So both p and q are even. This contradicts our assumption that p and q had no common factors. Hence, by proof by contradiction, [Equation] is irrational.
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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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