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Math 312.3 ASSIGNMENT #1 (due January 23, 9:55 AM) PROBLEM 1.1 PROBLEM 1.6 (3 pt) (3 pt) PROBLEM A (3 pt) √ Prove that 3 is an irrational number. Note. In the text-book such facts are deduced from Theorem 2.2. I√skipped this theorem and provided a simple straightforward proof of irrationality of 2. I ask you to follow this proof in Problem A. In the following problems use the fact that the sum, difference, product and ratio of rational numbers is a rational number. PROBLEM B (3 pt) √ √ √ √ √ Prove that 1 + 3, 3 − 1, 1 − 3, 2 3, 3 + 3 are irrational numbers. √ Hint: Prove by contradiction: assume that 1 + 3 = r, a rational number and come to a contradiction to the result of Problem A. Proceed the same way to prove the remaining statements. PROBLEM 2.4 (2 pt) Advise. Prove the statement in Problem 2.4 by contradiction. PROBLEM C (2 pt) Give examples 1) when the sum of two irrational numbers is a rational number. 2) when the sum of two irrational numbers is an irrational number. 3) when the product of two irrational numbers is a rational number. 4) when the product of two irrational numbers is an irrational number. Hint: use the results of Problems A and B. Note the difference: As mentioned above, the sum and the product of two rational numbers are always rational numbers. See the second page! 1 PROBLEM 3.1 (a,b) (4 pt) 2