2. Complex Conjugates

Two complex numbers that are mirror images across the real axis.

Let $\; \; \; z = a + b i$

then $\; \overline{z} = a - b i \;$ is called the complex conjugate of $z$ .

Write $\dfrac{5 + 4i}{2 - 3i}$ in the form $a + b i$ .

\begin{aligned} \dfrac{5 + 4i}{2 - 3i} &= \dfrac{(5 + 4i)(2 + 3i)}{(2 - 3i)(2 + 3i)} \\[3ex] &= \dfrac{10 + 15i + 8i + 12i^2}{4 + 6i - 6i - 9i^2} \\[3ex] &= \dfrac{-2 + 23i}{13} \\[3ex] &= -\dfrac{2}{13} + \dfrac{23}{13} i \end{aligned}