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

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

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

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

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

Step 4: Separate real and imaginary parts:

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

Complex Division Calculator