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# Performing Arithmetic with Complex Numbers

## Performing Arithmetic with Complex Numbers 1.13 - Solution

We will calculate the product between $ai$ and $b+ci.$ Then, we will compare the obtained expression with $\text{-} 18-10i.$
$ai(b+ci)$
Simplify
$abi+aci^2$
$abi+ac(\text{-} 1)$
$abi-ac$
$\text{-} ac+abi$
We want our obtained expression to be equivalent to $\text{-} 18-10i.$ For two complex numbers to be equal, both their ${\color{#0000FF}{\text{real}}}$ ${\color{#0000FF}{\text{parts}}}$ and their ${\color{#009600}{\text{imaginary}}}$ ${\color{#009600}{\text{parts}}}$ must be equal. $\begin{gathered} {\color{#0000FF}{\text{-} ac}}+{\color{#009600}{ab}}i={\color{#0000FF}{\text{-} 18}}+({\color{#009600}{\text{-} 10}})i \\ \Updownarrow \\ \begin{cases}{\color{#0000FF}{\text{-} ac}}={\color{#0000FF}{\text{-} 18}} & \, \text {(I)}\\ \phantom{\text{-} }{\color{#009600}{ab}}={\color{#009600}{\text{-} 10}} & \text {(II)}\end{cases} \end{gathered}$ From Equation (I) we can deduce that the product between $a$ and $c$ is $18.$ In Equation (II) we see that the product between $a$ and $b$ is $\text{-} 10.$ With this information, we can assign the numbers in the given tiles to $a,$ $b,$ and $c.$ $\begin{gathered} \begin{cases}ac=18 \\ ab=\text{-} 10 \end{cases} \quad \Rightarrow \quad a=2,\ b=\text{-}5, \ \text{and} \ c=9 \end{gathered}$ Now that we have the values, we can write the statement. $\begin{gathered} ai(b+ci)=\text{-} 18-10i \\ \Downarrow \\ 2i(\text{-}5+9i)=\text{-}18-10i \end{gathered}$