imaginary number problems-－dlxpxv7的部落格

標題:

imaginary number problems?

發問:

i need some help on my math homework. 1.) if z + 1 / z = -8 + 9i , simplify z^2 + 1 / (z^2) and z^3 + 1 / (z^3) 2.) Let a and z be a real number and z = ( a-i / 1+i )^2 . a = ? thanks 更新: pls help ;_;

最佳解答:

(1a) if z + 1 / z = -8 + 9i , simplify z^2 + 1 / z^2 Square both sides: (z + 1/z)^2 = z^2 + 1/z^2 + 2 = (-8 + 9i)^2 = 64 - 81 - 144i = -17 - 144i => z^2 + 1/z^2 = -19 - 144i (2b) z^3 + 1 / (z^3) = ? = (z + 1/z)*(z^2 - 1 + 1/z^2) = (-8 + 9i)*(-19 - 144i - 1) = (8 - 9i)*(20 + 144i) = 4*(8 - 9i)*(5 + 36i) = 4*(40 + 9*36 - 45i + 8*36i) = 4*(364 + 243i) (2) Let a and z be a real number and z = [(a-i) / (1+i)]^2 . a = ? z = (a-i)^2/(1+i)^2 = (a^2 - 2ai - 1)/(1 - 2i - 1) = [(a^2-1)-2ai]/(-2i) = [(a^2-1)-2ai]i/(-2i^2) = [2a + (a^2-1)i]/2 = a + (a^2-1)i/2 = real => a^2 = 1 => a = +-1 => a = z = +-1

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