How do you divide imaginary numbers?

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How do you divide imaginary numbers?

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Answer 1 · 2018-04-25T02:01:18

$\frac{a + b i}{c + d i} = \frac{a c + b d}{c^{2} + d^{2}} + i \frac{b c - a d}{c^{2} + d^{2}}$ $(a+bi)/(c+di)=(ac+bd)/(c^2+d^2)+i(bc-ad)/(c^2+d^2)$

Explanation:

Suppose we wanted to determine

$\frac{a + b i}{c + d i}$ $(a+bi)/(c+di)$

We can multiply the numerator and denominator by the complex conjugate of the denominator. In this case the complex conjugate of the denominator is $c - d i$ $c-di$ .

$\frac{a + b i}{c + d i} = \frac{(a + b i) (c - d i)}{(c + d i) (c - d i)}$ $(a+bi)/(c+di)=((a+bi)(c-di))/((c+di)(c-di))$

$= \frac{a c - a d i + b c i + b d}{c^{2} - c d i + c d i + d^{2}}$ $=(ac-adi+bci+bd)/(c^2-cdi+cdi+d^2)$

$= \frac{a c + b d + (b c - a d) i}{c^{2} + d^{2}}$ $=(ac+bd+(bc-ad)i)/(c^2+d^2)$

$= \frac{a c + b d}{c^{2} + d^{2}} + i \frac{b c - a d}{c^{2} + d^{2}}$ $=(ac+bd)/(c^2+d^2)+i(bc-ad)/(c^2+d^2)$

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How do you divide imaginary numbers?

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