College Algebra, Chapter 1, 1.4, Section 1.4, Problem 74

Suppose that $z = a + bi$ and $w = c + di$, then the symbol $\overline{z}$ represents the complex conjugate of $z$. Prove $\overline{zw} = \overline{z} \cdot \overline{w}$


$\begin{equation} \begin{aligned} \overline{zw} =& \overline{(a + bi)(c + di)} && \text{Model} \\ \\ =& \overline{(ac + adi + cbi + bdi^2)} && \text{Use FOIL method} \\ \\ =& \overline{(ac + adi + cbi + bd(-1))} && \text{Recall that } i^2 = -1 \\ \\ =& \overline{(ac - bd) + (ad + cb)i} && \text{Apply complex conjugate} \\ \\ =& (ac - bd) - (ad + cb)i && \\ \\ \overline{z} \cdot \overline{w} =& \overline{a + bi} \cdot \overline{c + di} && \text{Apply complex conjugate} \\ \\ =& (a - bi) \cdot (c - di) && \text{Apply FOIL method} \\ \\ =& ac - adi - cbi + (bdi^2) && \text{Recall that } i^2 = -1 \\ \\ =& ac - adi - cbi + (bd(-1)) && \text{Simplify} \\ \\ =& (ac - bd) - (ad + cb) i && \end{aligned} \end{equation}$