Quadratic Equation in One Unknown (1.8 -1.9)

Appreciate the development of the number systems including the system of complex numbers

  • Theory

    Numbers are classified as follows: \(\left\{ {complex\;Number\;\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{c}}{Imaginary\;Number}\\{Real\;Number\left\{ {\begin{array}{*{20}{c}}{Irrational\;Number}\\{Fractions}\\{Integers\left\{ {\begin{array}{*{20}{c}}{Natural\;Numbers}\\{Zero}\\{Negative\;Integers}\end{array}} \right.}\end{array}} \right.}\end{array}} \right.}\end{array}} \right.\)

  • Examples

    Is \(0.\dot 1\;\)a rational number?

  • Solutions

    Let \(0.\dot 1 = x\)
    \(10 = 1.\dot 1\)
    \(10{\rm{x}} - {\rm{x}} = 1.\dot 1 - 0.\dot 1\)
    \(9{\rm{x}} = 1\)
    \(\therefore 0.\dot 1 = {\rm{x}} = \frac{1}{9}\)
    Any numbers that can be expressed as \(\frac{{\rm{q}}}{{\rm{p}}}\) , where p and q are integers, is, by definition, a rational number

Perform addition, subtraction, multiplication and division of complex numbers

  • Theory

    By definition, the imaginary number \({\rm{i}}\) is defined by
    \(i = \sqrt { - 1} \)
    A complex number, z, consists of real part and imaginary part
    \({\rm{z}} = {\rm{a}} + {\rm{bi}}\)

  • Examples
    Given two complex number,
    \({{\rm{z}}_1} = 1 + 2i\)
    \({{\rm{z}}_2} = 4 - 3i\)
    Find
    a) \({{\rm{z}}_1} + {z_2}\)
    b) \({{\rm{z}}_1} - {z_2}\)
    c) \({{\rm{z}}_1}{z_2}\)
    d) \(\frac{{{{\rm{z}}_1}}}{{{z_2}}}\)
  • Solutions
    a) \({{\rm{z}}_1} + {z_2}\)
    \( = \left( {1 + 2i} \right) + \left( {4 - 3i} \right)\)
    \( = 5 - {\rm{i}}\)

    b) \({{\rm{z}}_1} - {z_2}\)
    \( = \left( {1 + 2i} \right) - \left( {4 - 3i} \right)\)
    \( = - 3 + 5{\rm{i}}\)

    c) \({{\rm{z}}_1}{z_2}\)
    \( = \left( {1 + 2i} \right)\left( {4 - 3i} \right)\)
    \( = \left( {4 - 3{\rm{i}}} \right) + 2{\rm{i}}\left( {4 - 3{\rm{i}}} \right)\)
    \( = 4 - 3{\rm{i}} + 8{\rm{i}} - 6{{\rm{i}}^2}\)
    \( = 4 + 5{\rm{i}} - 6\left( { - 1} \right)\)
    \( = 10 + 5{\rm{i}}\)

    d) \(\frac{{{{\rm{z}}_1}}}{{{z_2}}}\)
    \( = \frac{{1 + 2i}}{{4 - 3i}}\)
    \( = \frac{{1 + 2i}}{{4 - 3i}} \times \frac{{4 + 3i}}{{4 + 3i}}\)
    \( = \frac{{\left( {4 + 3i} \right) + 2i\left( {4 + 3i} \right)}}{{{4^2} - {{\left( {3i} \right)}^2}}}\)
    \( = \frac{{4 + 3i + 8i + 6{i^2}}}{{16 - 9\left( { - 1} \right)}}\)
    \( = \frac{{4 + 11i + 6\left( { - 1} \right)}}{{25}}\)
    \( = \) \(\frac{{ - 2 + 11i}}{{25}}\)
    \( = \) \( - \frac{2}{{25}} + \frac{{11}}{{25}}i\)