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Performing Arithmetic with Complex Numbers

Performing Arithmetic with Complex Numbers 1.13 - Solution

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We will calculate the product between aiai and b+ci.b+ci. Then, we will compare the obtained expression with -1810i.\text{-} 18-10i.
ai(b+ci)ai(b+ci)
Simplify
abi+aci2abi+aci^2
abi+ac(-1)abi+ac(\text{-} 1)
abiacabi-ac
-ac+abi\text{-} ac+abi
We want our obtained expression to be equivalent to -1810i.\text{-} 18-10i. For two complex numbers to be equal, both their real{\color{#0000FF}{\text{real}}} parts{\color{#0000FF}{\text{parts}}} and their imaginary{\color{#009600}{\text{imaginary}}} parts{\color{#009600}{\text{parts}}} must be equal. -ac+abi=-18+(-10)i{-ac=-18(I)-ab=-10(II)\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 aa and cc is 18.18. In Equation (II) we see that the product between aa and bb is -10.\text{-} 10. With this information, we can assign the numbers in the given tiles to a,a, b,b, and c.c. {ac=18ab=-10a=2, b=-5, and c=9\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. ai(b+ci)=-1810i2i(-5+9i)=-1810i\begin{gathered} ai(b+ci)=\text{-} 18-10i \\ \Downarrow \\ 2i(\text{-}5+9i)=\text{-}18-10i \end{gathered}