Step 1: Determine the conjugate of the denominator. ( to find the conjugate just change the sign of the imaginary part ).

In this example, the conjugate of $ \color{orangered}{ 80+40i }\, $ is $ \color{blue}{ 80-40i } $.

Step 2: Multiply both the numerator and denominator by the conjugate:

$$\begin{aligned} \frac{ -20+40i }{ 80+40i } &= \frac{ -20+40i }{ 80+40i } \cdot \frac{ \color{blue}{ 80-40i } }{ \color{blue}{ 80-40i } } = \\[1 em] &= \frac{ \left( -20+40i \right) \cdot \left( 80-40i \right) }{ \left( 80+40i \right) \cdot \left( 80-40i \right) } \end{aligned} $$

Step 3: Simplify numerator and denominator (use $\color{blue}{i^2 = -1}$)

$$ \begin{aligned} \left( -20+40i \right) \cdot \left( 80-40i \right) &= -20 \cdot 80 -20 \cdot \left(-40 \,i \right) + \left( 40 \,i \right) \cdot \left(80 \right) + \left( 40 \,i \right) \cdot \left(-40 \,i \right) = \\[1 em] &= -1600 + 800 \, i + 3200 \, i -1600 \color{blue}{(-1)} = \\[1 em] &= 4000i\end{aligned} $$ $$ \begin{aligned} \left( 80+40i \right) \cdot \left( 80-40i \right) &= 80 \cdot 80 + 80 \cdot \left(-40 \,i \right) + \left( 40 \,i \right) \cdot \left(80 \right) + \left( 40 \,i \right) \cdot \left(-40 \,i \right) = \\[1 em] &= 6400 -3200 \, i + 3200 \, i -1600 \color{blue}{(-1)} = \\[1 em] &= 8000\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ -20+40i }{ 80+40i } = \frac{ 4000i }{ 8000 } = \frac{ 0 }{ 8000 } + \frac{ 4000 }{ 8000 } i= \frac{ 1 }{ 2 }i $$

Complex Division Calculator