MIT6_042JS10_lec02_prob - Massachusetts Institute of...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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. is indeed irrational....
View Full Document

This note was uploaded on 05/27/2011 for the course CS 6.042J taught by Professor Prof.albertr.meyer during the Spring '11 term at MIT.

Page1 / 3

MIT6_042JS10_lec02_prob - Massachusetts Institute of...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online