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test3-summer01

test3-summer01 - Prove that NO order relation can be placed...

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Name: CODEWORD: MAS 3300 – Test 3 Summer A 2001 When giving proofs quote the relevant axioms and results. Proofs should be written in a proper and coherent manner. Write so that anyone in the class can follow your work. The questions have equal value. Write on ONE side of your paper. 1. Prove that Q [ 2] = { a + b 2 : a, b Q } is closed under multiplication. 2. (a) Compute the multiplication table in Z 8 : × 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 (b) Solve the equation 2 z = 4 in Z 8 ; i.e. find all z Z 8 such that 2 · z = 4. 3. Let n 2, a Z n . Prove that if a has a multiplicative inverse in Z n then a and n are relatively prime. 4. Use the Eucidean algorithm to find integers x , y such that 19 x + 81 y = 1 . Hence find the multiplicative inverse of 19 in Z 81 . 5. Let z 1 , z 2 C . Prove that z 1 z 2 = z 1 z 2 . 1

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Unformatted text preview: Prove that NO order relation can be placed on C so that C becomes an ordered ﬁeld. 7. Let r , s ≥ 0, θ , φ ∈ R . Let z 1 = r (cos θ + i sin θ ) , z 2 = s (cos φ + i sin φ ) . Prove that z 1 z 2 = rs (cos( θ + φ ) + i sin( θ + φ )) . 8. Determine all solutions z in C of the equation z 3 + 64 i = 0 . Write the solutions in rectangular form and simplify. Exhibit the solutions geomet-rically. 9. Let θ ∈ R . (a) Prove that cos θ = 1 2 ( e iθ + e-iθ ). (b) Prove that sin θ = 1 2 i ( e iθ-e-iθ ). (c) Expand ( e iθ + e-iθ ) 3 and simplify. (d) Hence show that cos 3 θ = 1 4 cos3 θ + 3 4 cos θ....
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