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}{ \frac{ 1 }{ 100 }+0.005i }\, $ is $ \color{blue}{ \frac{ 1 }{ 100 }-0.005i } $.

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

$$\begin{aligned} \frac{ 1 }{ \frac{ 1 }{ 100 }+0.005i } &= \frac{ 1 }{ \frac{ 1 }{ 100 }+0.005i } \cdot \frac{ \color{blue}{ \frac{ 1 }{ 100 }-0.005i } }{ \color{blue}{ \frac{ 1 }{ 100 }-0.005i } } = \\[1 em] &= \frac{ 1 \cdot \left( \frac{ 1 }{ 100 }-0.005i \right) }{ \left( \frac{ 1 }{ 100 }+0.005i \right) \cdot \left( \frac{ 1 }{ 100 }-0.005i \right) } \end{aligned} $$

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

$$ \begin{aligned} 1 \cdot \left( \frac{ 1 }{ 100 }-0.005i \right) &= 1 \cdot \frac{ 1 }{ 100 } + 1 \cdot \left(-0.005 \,i \right) = \\[1 em] &= \frac{ 1 }{ 100 } -0.005 \, i = \\[1 em] &= \frac{ 1 }{ 100 }-0.005i\end{aligned} $$ $$ \begin{aligned} \left( \frac{ 1 }{ 100 }+0.005i \right) \cdot \left( \frac{ 1 }{ 100 }-0.005i \right) &= \frac{ 1 }{ 100 } \cdot \frac{ 1 }{ 100 } + \frac{ 1 }{ 100 } \cdot \left(-0.005 \,i \right) + \left( 0.005 \,i \right) \cdot \left(\frac{ 1 }{ 100 } \right) + \left( 0.005 \,i \right) \cdot \left(-0.005 \,i \right) = \\[1 em] &= \frac{ 1 }{ 10000 } -0.0001 \, i + 0.0001 \, i -0 \color{blue}{(-1)} = \\[1 em] &= 0.0001\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 1 }{ \frac{ 1 }{ 100 }+0.005i } = \frac{ \frac{ 1 }{ 100 }-0.005i }{ 0.0001 } = \frac{ \frac{ 1 }{ 100 } }{ 0.0001 } + \frac{ -0.005 }{ 0.0001 } i= 79.8229-40.1322i $$

Complex number calculator