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\begin{align}\frac{a+bi}{c+di} &= \frac{(a+bi)(c-di)}{(c-di)(c+di)}\\&= \frac{ac+bd+bci-adi}{c^2+d^2}\\&=\frac{ac+bd}{c^2+d^2}+\frac{bc-ad}{c^2+d^2}i\end{align}