How do I find the complex conjugate of 12/(5i)?

Aug 12, 2015

The conjugate is $\frac{12 i}{5}$

Explanation:

To find a conjugate of a complex number we first have to convert it to a form $a + b i$. To do this here we can multiply both numerator and denominator by $i$

$z = \frac{12}{5 i} = \frac{12 i}{5 {i}^{2}} = \frac{12 i}{- 5} = - \frac{12 i}{5}$

Now to calculate the conjugate we just have to change sign of the imaginary part:

$\overline{z} = \frac{12 i}{5}$

Aug 13, 2015

In $a + b i$ form, this starts off as:

$0 + \frac{12}{5 i}$

Note that $\left[\frac{12}{5 i} = \frac{12}{5} \cdot \frac{1}{i}\right] \ne \left[\frac{12 i}{5} = \frac{12}{5} i\right]$)

$\frac{12}{5 i} \cdot \left(\frac{i}{i}\right) = \frac{12 i}{5 {i}^{2}} = \frac{12 i}{- 5}$

We currently have:

$a + b i = 0 + \frac{12 i}{- 5}$

The conjugate is $a - b i$, thus we get:

$a - b i = 0 - \frac{12 i}{- 5}$

$= \frac{12 i}{5} = \textcolor{b l u e}{\frac{12}{5} i}$