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

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

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

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

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

Step 4: Separate real and imaginary parts:

$$ \frac{ 1+i }{ 1-i } = \frac{ 2i }{ 2 } = \frac{ 0 }{ 2 } + \frac{ 2 }{ 2 } i= i $$

Complex Division Calculator