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MIT6_042JS10_lec02_prob

# MIT6_042JS10_lec02_prob - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science February 5 Prof. Albert R. Meyer revised March 6, 2010, 795 minutes In-Class Problems Week 1, Fri. Problem 1. Generalize the proof from lecture (reproduced below) that 2 is irrational, for example, how about 3 2 ? Remember that an irrational number is a number that cannot be expressed as a ratio of two integers. Theorem. 2 is an irrational number. Proof. The proof is by contradiction: assume that 2 is rational, that is, n 2 = , (1) d where n and d are integers. Now consider the smallest such positive integer denomi- nator, d . We will prove in a moment that the numerator, n , and the denominator, d , are both even. This implies that n/ 2 d/ 2 is a fraction equal to 2 with a smaller positive integer denominator, a contradiction. Since the assumption that 2 is rational leads to this contradiction, the assumption must be false. That is, 2 is indeed irrational. This italicized comment on the

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