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# A1 - complex numbers C(Hint Could 0 ≺ i 7 Describe...

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MATH185: Assignment 1 1. Section 3 Exercise 1 2. Section 5 Exercises 8,10,12,13,16. Use exercise 13 to show that the complex roots of any polynomial p ( x ) with real coeﬃcients come in conjugate pairs (that is, a complex number z is a root of p ( x ) if and only if z is also a root). 3. Show that the n -th roots of unity of a complex number a 6 = 0 are c, cω, cω 2 , . . . , cω n - 1 where ω = e i n and c is any n -th root of a . Is the statement still true if instead of ω we choose any other n -th root of unity? (Hint: Check n = 4). 4. Find all complex solutions to z 6 = 1 + 2 i 5. Section 7. Exercises 10,11 6. Deﬁnition. A total order in a ﬁeld K is a relation such that the following hold: For any two z, w K only one of the following holds: z w , w z or w = z . (Compatibility with addition) If z w then z + x w + x for all x K . (Compatibility with multiplication) If z w and 0 x then zx wx (a) Note that the usual ordering on the rational numbers and on the real numbers satisﬁes the above properties. (b) Show that it is impossible to deﬁne ANY total ordering on the

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Unformatted text preview: complex numbers C . (Hint: Could 0 ≺ i ?) 7. Describe geometrically the set of points z in the complex plane deﬁned by the following relations: (a) | z-1 + i | = 1 (b) | z-a | = | z-b | for given complex numbers a, b with a 6 = b . (c) 1 z = z 1 (d) Re ( z ) = 3 (e) Re ( z ) > c where c ∈ R (f) Re ( az + b ) > 0 for given complex numbers a, b . (g) | z | = Re ( z ) + 1 (h) Im ( z ) = c with c ∈ R 8. (a) Show that every convergent sequence is bounded. Give an exam-ple of a bounded sequence which is NOT convergent. (b) Give an example of a sequence of complex numbers with inﬁnitely many distinct accumulation points (c) Show that a convergent sequence has exactly one accumulation point. Give an example of a sequence which is NOT convergent with exactly one accumulation point. 2...
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A1 - complex numbers C(Hint Could 0 ≺ i 7 Describe...

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