Andrew's git - notes.git/blob - spec/spec-collated.tex

   1\documentclass[a4paper]{article}
   2\usepackage[a4paper,margin=2cm]{geometry}
   3\usepackage{multicol}
   4\usepackage{amsmath}
   5\usepackage{amssymb}
   6\usepackage{harpoon}
   7\usepackage{graphicx}
   8\usepackage{wrapfig}
   9\usepackage{fancyhdr}
  10\pagestyle{fancy}
  11\fancyhead[LO,LE]{Year 12 Specialist}
  12\fancyhead[CO,CE]{Andrew Lorimer}
  13\begin{document}
  14
  15\begin{multicols}{2}
  16
  17  \section{Complex numbers}
  18
  19    \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
  20
  21    \subsection*{Operations}
  22
  23      \begin{align*}
  24        z_1 \pm z_2&=(a \pm c)(b \pm d)i\\
  25        k \times z &= ka + kbi\\
  26        z_1 \cdot z_2 &= ac-bd+(ad+bc)i\\
  27        z_1 \div z_2 &= (z_1 \overline{z_2}) \div |z_2|^2
  28      \end{align*}
  29
  30    \subsection*{Conjugate}
  31
  32      \[\overline{z} = a \pm bi\]
  33
  34      \subsubsection*{Properties}
  35
  36        \begin{align*}
  37          \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
  38          \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
  39          \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\
  40          z\overline{z} &= (a+bi)(a-bi)\\
  41          &= a^2 + b^2\\
  42          &= |z|^2
  43        \end{align*}
  44
  45    \subsection*{Modulus}
  46
  47      \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
  48
  49      \subsubsection*{Properties}
  50
  51        \begin{align*}
  52          |z_1z_2|&=|z_1||z_2|\\
  53          \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
  54          |z_1+z_2|&\le|z_1|+|z_2|
  55        \end{align*}
  56
  57    \subsection*{Multiplicative inverse}
  58
  59      \begin{align*}
  60        z^{-1}&=\frac{a-bi}{a^2+b^2}\\
  61        &=\frac{\overline{z}}{|z|^2}
  62        a
  63      \end{align*}
  64
  65    \subsection*{Dividing over \(\mathbb{C}\)}
  66
  67      \begin{align*}
  68        \frac{z_1}{z_2}&=z_1z_2^{-1}\\
  69        &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
  70        &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
  71        & \qquad \text{(rationalise denominator)}
  72      \end{align*}
  73
  74\end{multicols}
  75\end{document}