Question

Name: Show all work for each problem: include relevant steps and explain all answers. Answers should be clearly labeled. Problem 1 Prove that the v2 is irrational. Note that you must also prove the following corollary: If the square of a number is even, then the number itself is even. Use the definition below to assist you: Definition: We say that a real number r is rational if there exist integers p and such that , where is not equal to 0. A real number that is not rational is called irrational

Answer 1

suppose if possible Sz is rational 1 S2 = l q to, Piq E Z and qidl P, q) = 1 : 2.is preme : 2/2q² + 2/p² +2lp p=ak kez 71ak) ²= 2q² a 4K²= 2q² ak ² = q² 7 2/2k² + 2/q² + 2/2 :2 is preme » 21p & 219 248ed I but 8 cd (p, q) & 1 : 2/1 & 2lq is irrational let square of a number is even ta is even and ef possible a is not even e a is Odd > a= 2k+l, ke 24 - a²= (2k + 1)² = 4K²+1+4K = 4(2+2) + 1 na² is odd. 1 =2t+!, É =2k+163 but a² is even which is contradiction CS Scanned withinus a iš even. CamScanner

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