1 4 let z 3 i w 1 3 i find polar forms for zw zw and

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4. Let z = 3 + i, w = 1 + 3 i. Find polar forms for zw , z/w and 1 /z by first putting z and w into polar form. Answer: Remember that, if ζ = x + iy is to be written in the polar form ζ = re , we know that r = | ζ | , θ = tan - 1 y x . Therefore, for the given z and w , we can determine the polar forms by computing | z | = q 3 2 + 1 2 = 3 + 1 = 4 = 2 | w | = q 1 2 + 3 2 = 1 + 3 = 4 = 2 tan - 1 1 3 = π 6 tan - 1 3 1 = π 3 . Thus, we see that z = 2 e i ( π/ 6) and w = 2 e i ( π/ 3) . Therefore, using these expressions for z and w , zw = 2 e i ( π/ 6) 2 e i ( π/ 3) = (2 · 2) e i ( π/ 6+ π/ 3) = 4 e iπ/ 2 (which is just another name for 4 i ). Similarly, z w = 2 e i ( π/ 6) 2 e i ( π/ 3) = 2 2 e i ( π/ 6 - π/ 3) = e i ( - π/ 6) . Finally, 1 z = 1 2 e i ( π/ 6) = 1 2 e - i ( π/ 6) = 1 2 e i ( - π/ 6) . 5. Find all the fifth roots of 32 and sketch them in the complex plane. Answer: Suppose α is a fifth root of 32. Write α in polar form: α = re . Then 32 = α 5 = re 5 = r 5 e i (5 θ ) . Therefore, since we can write 32 in polar form as 32 e i (0) , 32 e i (2 π ) , 32 e i (4 π ) , 32 e i (6 π ) , 32 e i (8 π ) , we see that r 5 = 32, meaning that r = 2. Also, 5 θ = 0 , 2 π, 4 π, 6 π, 8 π, so the fifth roots of 32 are 2 e i (0) = 2 , 2 e i (2 π/ 5) , 2 e i (4 π/ 5) , 2 e i (6 π/ 5) , 2 e i (8 π/ 5) .
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  • Fall '12
  • Hom
  • Calculus, Cos, Complex number, Euler's formula, polar form, zw

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