Show that 1-2i3-4i-1-2i3-4i is real-

Solving given question, we get 1 2i3 4i+1+2i3+4i (1 2i)(3+4i)+(1+2i)(3 4i)(3 4i)(3+4i) 9 2i+9+2i9+16 1825 Which is real, Hence proved. ...

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Complex Numbers

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Question

Show that $\frac{1 - 2 i}{3 - 4 i} + \frac{1 + 2 i}{3 + 4 i}$ is real.

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Solution

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(\frac{1}{1 - 2 i} + \frac{3}{1 + i}) (\frac{3 + 4 i}{2 - 4 i}) =

(\frac{1}{1 - 2 i} + \frac{3}{1 + i}) (\frac{3 + 4 i}{2 - 4 i})

(\frac{1}{1 - 2 i} + \frac{3}{1 + i}) (\frac{3 + 4 i}{2 - 4 i})

(2 + 8 i) (1 - 4 i) - (3 - 2 i) (6 + 4 i)

: i = \sqrt{- 1}

z^{2} = - 3 + 4 i

z = \pm (1 + 2 i)

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