MATH 1200: Homework 5, Proof Study and Analysis Due in class Tuesday, June 28 at 2:00 p.m. Late homework (for example, homework submitted after the start of class) will not be accepted. In this proof, we interpret k as the slope of a line through the origin, as illustrated below. y y= m kn pm kx = m n x b bc m pn x n Theorem: If k is not the square of an integer, then k is irrational. m m k= in lowest terms. Then the point on the line y = kx = x closest n n to theorigin with integer coordinates is (n, m). However, if we let p be the greatest integer less than k so that p < k < p + 1, then the point with integer coordinates (m pn, kn pm) m m2 lies on the line and is closer to the origin since (m pn) = pm = kn pm, and n n m p< < p + 1 implies 0 < m pn < n and 0 < kn pm < m. Thus we have a contradicin ton and k is irrational. Proof. Assume 1. Let be rational. Prove that the line y = x has a point both of whose coordinates are integral. m x has a point with positive integral coordinates closest to n the origin. Why does such a point exist? Why is that point (n, m)? Hint: Because m is in lowest terms, if m = st , then s = rm, t = rn for some integer r. n n 3. Why does the point (mpn, knpm) lie on the line y = k x? Provide an appropriate calculation. 2. In the proof, the line y = 1 m 4. Explain how to use p < < p + 1 to prove that 0 < mnp < n and 0 < knpm < m. n What is the relevance of these inequalities? 5. Where is the assumption that k is not the square of an integer used? 6. Verify that n( k p) = m pn and m( k p) = kn pm. How does this explain the choice of the point (m pn, kn pm). 7. The proof uses the method of contradiction. Carefully state the contradiction which is obtained. 8. Having reviewed the proof and its details, now read it again and use at most two or three sentences to summarize it briefly. You are being asked to identify the idea of the proof, not to reproduce it. You may discuss these problems with other students but may not share final written solutions. Indicate any references which you consult. 2