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Accedmychevron_right11thchevron_rightmathchevron_rightComplex Numberschevron_rightExercise 1.1

(i) 2−7i4+5i\frac{2 - 7i}{4 + 5i}

Solution

Multiply numerator and denominator by the conjugate (4−5i):2−7i4+5i⋅4−5i4−5i=(2−7i)(4−5i)(4+5i)(4−5i)Expand numerator:(2−7i)(4−5i)=8−10i−28i+35i2=8−38i−35=−27−38iSimplify denominator:(4+5i)(4−5i)=42+52=16+25=41Write in a+ib form:−27−38i41=−2741−3841i\begin{aligned} & \boxed{\text{Multiply numerator and denominator by the conjugate } (4-5i):} \\ \\ & \frac{2-7i}{4+5i} \cdot \frac{4-5i}{4-5i} = \frac{(2-7i)(4-5i)}{(4+5i)(4-5i)} \\ \\ & \boxed{\text{Expand numerator:}} \\ \\ & (2-7i)(4-5i) = 8 - 10i - 28i + 35i^2 \\ & = 8 - 38i - 35 = -27 - 38i \\ \\ & \boxed{\text{Simplify denominator:}} \\ \\ & (4+5i)(4-5i)=4^2+5^2=16+25=41 \\ \\ & \boxed{\text{Write in } a+ib \text{ form:}} \\ \\ & \frac{-27-38i}{41} = -\frac{27}{41} - \frac{38}{41}i \end{aligned}
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(ii)
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