### Quadratic Equation in One Unknown (1.8 -1.9)

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

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

• Is $$0.\dot 1\;$$a rational number?

• 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

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

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