# hw3solutions - On the other hand assume a 2 b 2 = 0 If a 6...

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Math 521 - Advanced Calculus I Instructor: J. Metcalfe Due: January 27, 2010 Assignment 3 1. If r is rational ( r 6 = 0) and x is irrational, prove that r + x and rx are irrational. Suppose not. Suppose that r + x = s Q . Then x = s - r , but since Q is a ﬁeld, s - r Q which contradicts x being irrational. Similarly, if rx = w Q , then x = w/r Q which is a contradiction. 2. Prove the following using only the ﬁeld axioms, the ordering properties of R , and results that were shown to follow directly from them: If a,b R , show that a 2 + b 2 = 0 if and only if a = 0 and b = 0. If a = 0 and b = 0, then we have a 2 + b 2 = 0 2 + 0 2 = 0 + 0 = 0
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Unformatted text preview: . On the other hand, assume a 2 + b 2 = 0. If a 6 = 0 and b 6 = 0, then a 2 > 0 and b 2 > (Thm. 2.1.8). By the Ordering Properties, it follows that a 2 + b 2 > 0. Thus, we don’t have that both are non-zero. If either a 6 = 0 or b 6 = 0 but not both, then say without loss that a 6 = 0 but b = 0. Then a 2 + b 2 = a 2 + 0 2 = a 2 + 0 = a 2 > 0. This again is not possible. The only remaining case is when a = 0 and b = 0, which we showed satisﬁes the desired result. 1...
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