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Accedmychevron_right11thchevron_rightmathchevron_rightComplex Numberschevron_rightExercise 1.1

Solution

(i) 2−7i4+5i\frac{2 - 7i}{4 + 5i}

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}

(ii) (−2+3i)21+i\frac{(-2 + 3i)^2}{1 + i}

First expand (−2+3i)2:(−2+3i)2=(−2)2+2(−2)(3i)+(3i)2=4−12i+9i2=4−12i−9=−5−12iNow divide by (1+i) using conjugate (1−i):−5−12i1+i⋅1−i1−i=(−5−12i)(1−i)(1+i)(1−i)Expand numerator:(−5−12i)(1−i)=−5+5i−12i+12i2=−5−7i−12=−17−7iSimplify denominator:(1+i)(1−i)=12+12=2Write in a+ib form:−17−7i2=−172−72i\begin{aligned} & \boxed{\text{First expand } (-2+3i)^2:} \\ \\ & (-2+3i)^2 = (-2)^2 + 2(-2)(3i) + (3i)^2 \\ & = 4 - 12i + 9i^2 = 4 - 12i - 9 = -5 - 12i \\ \\ & \boxed{\text{Now divide by } (1+i) \text{ using conjugate } (1-i):} \\ \\ & \frac{-5-12i}{1+i} \cdot \frac{1-i}{1-i} = \frac{(-5-12i)(1-i)}{(1+i)(1-i)} \\ \\ & \boxed{\text{Expand numerator:}} \\ \\ & (-5-12i)(1-i) = -5 + 5i - 12i + 12i^2 \\ & = -5 - 7i - 12 = -17 - 7i \\ \\ & \boxed{\text{Simplify denominator:}} \\ \\ & (1+i)(1-i)=1^2+1^2=2 \\ \\ & \boxed{\text{Write in } a+ib \text{ form:}} \\ \\ & \frac{-17-7i}{2} = -\frac{17}{2} - \frac{7}{2}i \end{aligned}

(iii) i1+i\frac{i}{1 + i}

Multiply numerator and denominator by conjugate (1−i):i1+i⋅1−i1−i=i(1−i)(1+i)(1−i)Simplify numerator:i(1−i)=i−i2=i−(−1)=1+iSimplify denominator:(1+i)(1−i)=12+12=2Write in a+ib form:1+i2=12+12i\begin{aligned} & \boxed{\text{Multiply numerator and denominator by conjugate } (1-i):} \\ \\ & \frac{i}{1+i} \cdot \frac{1-i}{1-i} = \frac{i(1-i)}{(1+i)(1-i)} \\ \\ & \boxed{\text{Simplify numerator:}} \\ \\ & i(1-i)=i-i^2=i-(-1)=1+i \\ \\ & \boxed{\text{Simplify denominator:}} \\ \\ & (1+i)(1-i)=1^2+1^2=2 \\ \\ & \boxed{\text{Write in } a+ib \text{ form:}} \\ \\ & \frac{1+i}{2} = \frac{1}{2} + \frac{1}{2}i \end{aligned}

(iv) (4+3i)24−3i\frac{(4 + 3i)^2}{4 - 3i}

First expand (4+3i)2:(4+3i)2=42+2(4)(3i)+(3i)2=16+24i+9i2=16+24i−9=7+24iNow divide by (4−3i) using conjugate (4+3i):7+24i4−3i⋅4+3i4+3i=(7+24i)(4+3i)(4−3i)(4+3i)Expand numerator:(7+24i)(4+3i)=28+21i+96i+72i2=28+117i−72=−44+117iSimplify denominator:(4−3i)(4+3i)=42+32=16+9=25Write in a+ib form:−44+117i25=−4425+11725i\begin{aligned} & \boxed{\text{First expand } (4+3i)^2:} \\ \\ & (4+3i)^2 = 4^2 + 2(4)(3i) + (3i)^2 \\ & = 16 + 24i + 9i^2 = 16 + 24i - 9 = 7 + 24i \\ \\ & \boxed{\text{Now divide by } (4-3i) \text{ using conjugate } (4+3i):} \\ \\ & \frac{7+24i}{4-3i} \cdot \frac{4+3i}{4+3i} = \frac{(7+24i)(4+3i)}{(4-3i)(4+3i)} \\ \\ & \boxed{\text{Expand numerator:}} \\ \\ & (7+24i)(4+3i)=28+21i+96i+72i^2 \\ & = 28 + 117i - 72 = -44 + 117i \\ \\ & \boxed{\text{Simplify denominator:}} \\ \\ & (4-3i)(4+3i)=4^2+3^2=16+9=25 \\ \\ & \boxed{\text{Write in } a+ib \text{ form:}} \\ \\ & \frac{-44+117i}{25} = -\frac{44}{25} + \frac{117}{25}i \end{aligned}
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