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}{ 4+2i }\, $ is $ \color{blue}{ 4-2i } $.

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

$$\begin{aligned} \frac{ 1-3i }{ 4+2i } &= \frac{ 1-3i }{ 4+2i } \cdot \frac{ \color{blue}{ 4-2i } }{ \color{blue}{ 4-2i } } = \\[1 em] &= \frac{ \left( 1-3i \right) \cdot \left( 4-2i \right) }{ \left( 4+2i \right) \cdot \left( 4-2i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( 1-3i \right) \cdot \left( 4-2i \right) &= 1 \cdot 4 + 1 \cdot \left(-2 \,i \right) + \left( -3 \,i \right) \cdot \left(4 \right) + \left( -3 \,i \right) \cdot \left(-2 \,i \right) = \\[1 em] &= 4 -2 \, i -12 \, i + 6 \color{blue}{(-1)} = \\[1 em] &= -2-14i\end{aligned} $$ $$ \begin{aligned} \left( 4+2i \right) \cdot \left( 4-2i \right) &= 4 \cdot 4 + 4 \cdot \left(-2 \,i \right) + \left( 2 \,i \right) \cdot \left(4 \right) + \left( 2 \,i \right) \cdot \left(-2 \,i \right) = \\[1 em] &= 16 -8 \, i + 8 \, i -4 \color{blue}{(-1)} = \\[1 em] &= 20\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 1-3i }{ 4+2i } = \frac{ -2-14i }{ 20 } = \frac{ -2 }{ 20 } + \frac{ -14 }{ 20 } i= -\frac{ 1 }{ 10 }-\frac{ 7 }{ 10 }i $$

Complex Division Calculator