# Download PDF by Hans-Berndt Brinkmann, Dieter Puppe: Abelsche und exakte Kategorien Korrespondenzen By Hans-Berndt Brinkmann, Dieter Puppe

ISBN-10: 3540046151

ISBN-13: 9783540046158

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Problem 7. Let x, y, z be distinct complex numbers such that y = tx + (1 − t)z, Prove that t ∈ (0, 1). |z| − |y| |z| − |x| |y| − |x| ≥ ≥ . |z − y| |z − x| |y − x| Solution. The relation y = tx + (1 − t)z is equivalent to z − y = t(z − x). The inequality |z| − |y| |z| − |x| ≥ |z − y| |z − x| becomes |z| − |y| ≥ t (|z| − |x|), and consequently, |y| ≤ (1 − t)|z| + t|x|. This is the triangle inequality for y = (1 − t)z + tx . The second inequality can be proved similarly, by writing the equality y = tx + (1 − t)z as y − x = (1 − t)(z − x).

K = ϕk . Until now, we had n distinct roots of z0 : Z0 , Z1 , . . , Zn−1 . Consider some integer k and let r ∈ {0, 1, . . , n − 1} be the residue of k modulo n. Then k = nq + r for q ∈ Z, and ϕk = 2π t∗ 2π t∗ + (nq + r) = +r + 2qπ = ϕr + 2qπ. 2 The nth Roots of Unity 45 It is clear that Zk = Zr . Hence {Zk : k ∈ Z} = {Z0 , Z1 , . . , Zn−1 }. In other words, there are exactly n distinct nth roots of z0 , as claimed. The geometric images of the nth roots of a complex number z0 = 0 are the vertices √ of a regular n-gon inscribed in a circle with center at the origin and radius n r.

8. 9. Let z = (a, b) ∈ C. Compute z 2 , z 3 , and z 4 . Let z0 = (a, b) ∈ C. Find z ∈ C such that z 2 = z0 . Let z = (1, −1). Compute z n , where n is a positive integer. Find real numbers x and y in each of the following cases: x−3 y−3 + = i; 3+i 3−i 1 2 2 2 2 (c) (4 − 3i)x + (3 + 2i)xy = 4y − 2 x + (3xy − 2y )i. (a) (1 − 2i)x + (1 + 2i)y = 1 + i; (b) 10. Compute the following: (a) (2 − i)(−3 + 2i)(5 − 4i); (b) (2 − 4i)(5 + 2i) + (3 + 4i)(−6 − i); √ 6 √ 6 16 8 1+i −1 + i 3 1−i 7 1−i (c) + ; (d) + ; 1−i 1+i 2 2 3 + 7i 5 − 8i (e) + .