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There are 3 complex numbers $a+bi$, $c+di$, and $e+fi$. If $b=1$, $e=-a-c$, and the sum of the numbers is $-i$, find $d+f$.

May 6, 2018

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There are 3 complex numbers $a+bi$, $c+di$, and $e+fi$. If $b=1$, $e=-a-c$,

and the sum of the numbers is $-i$, find $d+f$.

$$\begin{array}{|rcll|} \hline a+bi \\ c+di \\ e+fi \\ \hline a+c+e+(b+d+f)i &=& -i \quad & | \quad e=-a-c \\ a+c-a-c+(b+d+f)i &=& -i \\ (b+d+f)i &=& -i \quad & | \quad b = 1 \\ (1+d+f)i &=& -i \quad & | \quad :i \\ 1+d+f &=& -1 \\ \mathbf{d+f} &\mathbf{=}& \mathbf{-2} \\ \hline \end{array}$$

May 7, 2018