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S1 - 92.305 Homework 1 Solutions Exercise 1.2.1(a Prove...

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92.305 Homework 1 Solutions September 18, 2009 Exercise 1.2.1. (a) Prove that 3 is irrational. Proof: Suppose that 3 is rational. Then we can write 3 = ( p/q ) 2 with p,q Z and q negationslash = 0. Without loss of generality, we can arrange this so that p,q share no common factors. We can rewrite the above equation as p 2 /q 2 = 3, or p 2 = 3 q 2 . Thus p 3 is divisible by 3, and so p itself must be divisible by 3. Writing p = 3 t with t Z , we have 9 t 2 = (3 t ) 2 = 3 q 2 , or 3 t 2 = q 2 . This implies that 3 divides q 2 and hence 3 divides q as well. We see that 3 must divide both p and q , contradicting their lack of common factors. Therefore, we cannot write 3 as the square of a rational number. squaresolid (b) Where does the proof of Theorem 1.1.1 break down if we try to use it to prove 4 is irrational? Answer: Let us try to use the method of Theorem 1.1.1 to prove that 4 is irrational. We would suppose 4 is rational, so 4 = ( p/q ) 2 with p,q Z and q negationslash = 0, and p 2 = 4 q 2 . We deduce that q 2 and hence q is even, so that we can write
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