test3-fall90 - (vi) The product of an irrational number and...

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MAS 3300 - TEST 3 There are 7 questions. The total number of points is 55. 100% = 50. 1. (5 points) Prove 2 + 3 is irrational. 2. (5 points) Prove 2 3 is irrational. 3. (8 points) Let n and k be positive integers. Prove the following: If k n is rational then n is a perfect k -th power (i.e. n = m k for some m Z ). 4. (14 points) Determine whether the following are TRUE or FALSE. It is not necessary to give reasons. (i) The sum of a rational number and a rational number is rational. (ii) The sum of an irrational number and a rational number is rational. (iii) The sum of an irrational number and an irrational number is irrational. (iv) The product of a rational number and a rational number is rational. (v) The product of an irrational number and a rational number is irrational.
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Unformatted text preview: (vi) The product of an irrational number and an irrational number is irrational. (vii) If A = { r ∈ Q : r 2 < 3 } then lub A does not exist. 5. (5 points) Exhibit an irrational number between 7 9 and 8 9 . Give reasons. 6. (8 points) Let A = { x ∈ R : x ≥ 0 and x 2 < 3 } . Let v = lub A . Prove v 2 ≤ 3. [You may NOT assume Theorem 4.14, but you may assume FACTS 1,2,3]. 7. Suppose a and b are not both zero and a,b ∈ Q . (i) (5 points) Prove a 2-5 b 2 6 = 0. (ii) (5 points) Hence show that Axiom MIV holds for Q [ √ 5] := { a + b √ 5 : a,b ∈ Q } . 1...
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This note was uploaded on 06/03/2011 for the course MAS 3300 taught by Professor Staff during the Spring '08 term at University of Florida.

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