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\fancyhead[LO,LE]{Year 12 Specialist}
\fancyhead[CO,CE]{Andrew Lorimer}
\begin{document}

\begin{multicols}{2}

  \section{Complex numbers}

    \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]

    \subsection*{Operations}

      \begin{align*}
        z_1 \pm z_2&=(a \pm c)(b \pm d)i\\
        k \times z &= ka + kbi\\
        z_1 \cdot z_2 &= ac-bd+(ad+bc)i\\
        z_1 \div z_2 &= (z_1 \overline{z_2}) \div |z_2|^2
      \end{align*}

    \subsection*{Conjugate}

      \[\overline{z} = a \pm bi\]

      \subsubsection*{Properties}

        \begin{align*}
          \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
          \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
          \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\
          z\overline{z} &= (a+bi)(a-bi)\\
          &= a^2 + b^2\\
          &= |z|^2
        \end{align*}

    \subsection*{Modulus}

      \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]

      \subsubsection*{Properties}

        \begin{align*}
          |z_1z_2|&=|z_1||z_2|\\
          \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
          |z_1+z_2|&\le|z_1|+|z_2|
        \end{align*}

    \subsection*{Multiplicative inverse}

      \begin{align*}
        z^{-1}&=\frac{a-bi}{a^2+b^2}\\
        &=\frac{\overline{z}}{|z|^2}
        a
      \end{align*}

    \subsection*{Dividing over \(\mathbb{C}\)}

      \begin{align*}
        \frac{z_1}{z_2}&=z_1z_2^{-1}\\
        &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
        &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
        & \qquad \text{(rationalise denominator)}
      \end{align*}

\end{multicols}
\end{document}
