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
\({\rm{z}} = {\rm{a}} + {\rm{bi}}\)

ExamplesGiven 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) \(\frac{{{{\rm{z}}_1}}}{{{z_2}}}\)

Solutionsa) \({{\rm{z}}_1} + {z_2}\)
\( = \left( {1 + 2i} \right) + \left( {4  3i} \right)\)
\( = 5  {\rm{i}}\)
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) \(\frac{{{{\rm{z}}_1}}}{{{z_2}}}\)
\( = \frac{{1 + 2i}}{{4  3i}}\)
\( = \frac{{1 + 2i}}{{4  3i}} \times \frac{{4 + 3i}}{{4 + 3i}}\)
\( = \frac{{\left( {4 + 3i} \right) + 2i\left( {4 + 3i} \right)}}{{{4^2}  {{\left( {3i} \right)}^2}}}\)
\( = \frac{{4 + 3i + 8i + 6{i^2}}}{{16  9\left( {  1} \right)}}\)
\( = \frac{{4 + 11i + 6\left( {  1} \right)}}{{25}}\)
\( = \) \(\frac{{  2 + 11i}}{{25}}\)
\( = \) \(  \frac{2}{{25}} + \frac{{11}}{{25}}i\)