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 eld. 7. Let r , s 0, , R . Let z 1 = r (cos + i sin ) , z 2 = s (cos + i sin ) . Prove that z 1 z 2 = rs (cos( + ) + i sin( + )) . 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 R . (a) Prove that cos = 1 2 ( e i + e-i ). (b) Prove that sin = 1 2 i ( e i-e-i ). (c) Expand ( e i + e-i ) 3 and simplify. (d) Hence show that cos 3 = 1 4 cos3 + 3 4 cos ....
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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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test3-summer01 - Prove that NO order relation can be placed...

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