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}{ 1-4i }\, $ is $ \color{blue}{ 1+4i } $.

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

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

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

$$ \begin{aligned} \left( 1+i \right) \cdot \left( 1+4i \right) &= 1 \cdot 1 + 1 \cdot \left(4 \,i \right) + \left( 1 \,i \right) \cdot \left(1 \right) + \left( 1 \,i \right) \cdot \left(4 \,i \right) = \\[1 em] &= 1 + 4 \, i + 1 \, i + 4 \color{blue}{(-1)} = \\[1 em] &= -3+5i\end{aligned} $$ $$ \begin{aligned} \left( 1-4i \right) \cdot \left( 1+4i \right) &= 1 \cdot 1 + 1 \cdot \left(4 \,i \right) + \left( -4 \,i \right) \cdot \left(1 \right) + \left( -4 \,i \right) \cdot \left(4 \,i \right) = \\[1 em] &= 1 + 4 \, i -4 \, i -16 \color{blue}{(-1)} = \\[1 em] &= 17\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 1+i }{ 1-4i } = \frac{ -3+5i }{ 17 } = \frac{ -3 }{ 17 } + \frac{ 5 }{ 17 } i= -\frac{ 3 }{ 17 }+\frac{ 5 }{ 17 }i $$

Complex Division Calculator