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final-fall92 - Name S.S.N MAS 3300 FINAL EXAM Fall 1992 In...

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Name: S.S.N: MAS 3300 – FINAL EXAM Fall 1992 In giving proofs make sure your reasoning is clear. The questions have equal value. Write on ONE side of your paper. 1. PROVE If a R then 0 a = 0. 2. Use mathematical induction to show that for any positive integer n 1 + 2 + · · · + n = n ( n + 1) 2 . 3. Prove 20 is irrational. You may NOT assume Theorem 4.3(c). 4. Show that if a Z n and a and n are relatively prime then a has a multiplicative inverse in Z n . 5 PROVE that no order relation can be placed on C so that C becomes an ordered field. 6. Use the Rational Root Theorem 7.14 to prove that 3 5 is irrational. 7. PROVE that there are infinitely many primes. 8. PROVE that 5 6∈ Q [ 3] := { a + b 3 : a, b Q } . 9. PROVE If z 1 , z 2 C then z 1 + z 2 = z 1 + z 2 . 10. FIND all solutions in C to x 4 + 16 = 0 . WRITE your solutions in rectangular form and simplify . EXHIBIT your solutions geometrically. 1
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11. For z = a + bi C with a, b R . Recall that a is called the real part of z and b is called the imaginary part of z . We write Re ( z ) = a
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