Additional Examples of Proof by Contradiction Paul Skoufranis October 4, 2011 The purpose of this document is to present two common proofs by contradiction that are not related to linear algebra. The ﬁrst is a very common proof where the second was developed by Euclid. Example) Prove √ 2 is irrational. Proof : Notice that either √ 2 is rational or √ 2 is irrational. Suppose to the contrary that √ 2 is ratio-nal. Then there exists integers a,b ∈ Z such that b 6 = 0, a and b have no common factors (meaning that no natural number larger than 1 divides both), and √ 2 = a b . By rearranging the equation and squaring both sides, we obtain that 2 b 2 = a 2 and thus a 2 must be an even integer. Since the square of an odd integer is odd and a 2 is even, a must be an even integer. Therefore there exists a c ∈ Z such that a = 2 c . By substituting 2 c for a in the above equation, we obtain that 2 b 2 = (2 c ) 2 = 4 c 2 so b 2 = 2 c 2 . Thus, by the same logic as before,
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