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

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

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

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

$$ \begin{aligned} \left( \frac{ 1 }{ 2 }-i \right) \cdot \left( 2-\sqrt{ 2 }i \right) &= \frac{ 1 }{ 2 } \cdot 2 + \frac{ 1 }{ 2 } \cdot \left(-\sqrt{ 2 } \,i \right) + \left( -1 \,i \right) \cdot \left(2 \right) + \left( -1 \,i \right) \cdot \left(-\sqrt{ 2 } \,i \right) = \\[1 em] &= 1 - \frac{\sqrt{ 2 }}{ 2 } \, i -2 \, i + \sqrt{ 2 } \color{blue}{(-1)} = \\[1 em] &= -0.4142-2.7071i\end{aligned} $$ $$ \begin{aligned} \left( 2+\sqrt{ 2 }i \right) \cdot \left( 2-\sqrt{ 2 }i \right) &= 2 \cdot 2 + 2 \cdot \left(-\sqrt{ 2 } \,i \right) + \left( \sqrt{ 2 } \,i \right) \cdot \left(2 \right) + \left( \sqrt{ 2 } \,i \right) \cdot \left(-\sqrt{ 2 } \,i \right) = \\[1 em] &= 4 -2 \sqrt{ 2 } \, i + 2 \sqrt{ 2 } \, i -2 \color{blue}{(-1)} = \\[1 em] &= 6\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ \frac{ 1 }{ 2 }-i }{ 2+\sqrt{ 2 }i } = \frac{ -0.4142-2.7071i }{ 6 } = \frac{ -0.4142 }{ 6 } + \frac{ -2.7071 }{ 6 } i= -0.069-0.4512i $$

Complex Division Calculator