How do you divide i/(2-3i)?

Aug 24, 2016

$- \frac{3}{13} + \frac{2}{13} i$

Explanation:

The aim here is to obtain a real value on the denominator of the fraction.
This is achieved by multiplying the numerator and denominator by the $\textcolor{b l u e}{\text{complex conjugate}}$ of the denominator.

Given a complex number z=x ± yi then the complex conjugate is.

bar(z)=x∓yi#

Note that the real part remains unchanged while the $\textcolor{red}{\text{sign}}$ of the imaginary part is reversed.

The conjugate of $2 - 3 i \text{ is } 2 + 3 i$

and $\left(2 - 3 i\right) \left(2 + 3 i\right) = 4 - 9 {i}^{2} = 4 + 9 = 13 \text{ a real number}$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder }} \textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{i}^{2} = {\left(\sqrt{- 1}\right)}^{2} = - 1} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

We must multiply numerator/denominator by 2+3i

$\Rightarrow \frac{i}{2 - 3 i} \times \frac{2 + 3 i}{2 + 3 i} = \frac{i \left(2 + 3 i\right)}{\left(2 - 3 i\right) \left(2 + 3 i\right)}$

$= \frac{2 i + 3 {i}^{2}}{13} = \frac{- 3 + 2 i}{13} = - \frac{3}{13} + \frac{2}{13} i$