Answer by Douba for Pushing \qed to the right within a displayed formula

For the qed symbol to appear flushed right in the last line of aligned equations, I use:

\documentclass{article}

\usepackage{amsmath}
\usepackage{amssymb}

\renewcommand{\j}{\ensuremath{\text{j}}}
\newcommand{\qed}{\tag*{$\blacksquare$}}

\begin{document}
    \begin{align*}
        \left(z_1\cdot z_2\right)^*
            &=\left(\left(a_1+\j b_1\right)\cdot \left(a_2+\j b_2\right)\right)^* \\
            &=\left(\left(a_1a_2+b_1b_2\right)+\j\left(a_1b_2+a_2b_1\right)\right)^* \\
            &=\left(a_1a_2-b_1b_2\right)-\j\left(a_1b_2+a_2b_1\right) \\
            &=a_1a_2-b_1b_2-\j a_1b_2-\j a_2b_1 \\
            &=a_1a_2-a_1\j b_2-a_2\j b_1+\left(-\j\right)^2b_1b_2 \\
            &=\left(a_1-\j b_1\right)\cdot\left(a_2-\j b_2\right) \\
            &=\left(a_1+\j b_1\right)^*\cdot\left(a_2+\j b_2\right)^* \\
            &=z_1^*\cdot z_2^*
            \qed
    \end{align*}            
\end{document}