Complex Numbers

Definition

A complex number is an expression of the form $z = x + i y$ where $x, y \in R$ and $i$ is the imaginary unit where $i^{2} = 1$ .

Quadratic Equations

$x^{2} = a x + b \to x = \frac{1}{2} (a \pm \sqrt{a^{2} + 4 b})$

Cubic Equations

Reduced (depressed) form of the cubic: $x^{3} = 3 p x + 2 q$

x = \sqrt[3]{q + \sqrt{q^{2} - p^{3}}} + \sqrt[3]{q - \sqrt{q^{2} - p^{3}}}

Bombelli's Example:

Consider $x^{3} = 15 x + 4$
Has $x = 4$
Here, $p = 5$ , $q = 2 \Rightarrow$ $x = \sqrt[3]{2 + \sqrt{4 - 125}} + \sqrt[3]{2 - \sqrt{4 - 125}} = \sqrt[3]{2 + 11 i} + \sqrt[3]{2 - 11 i}$

{\begin{cases} \sqrt[3]{2 + 11 i} = 2 + i a \\ \sqrt[3]{2 - 11 i} = 2 - i a \end{cases} \Rightarrow \begin{matrix} 2 + 11 i = (2 + i a)^{3} \\ 2 - 11 i = (2 - i a)^{3} \end{matrix} $ $ T h i s i s t r u e i f $ a = 1 $, $ i^{2} = - 1 $ . A r i t h m e t i c : $ $ z_{1} = x_{1} + i y_{1}, z_{2} = x_{2} + i y_{2}

then:

z_{1} + z_{2} = (x_{1} + x_{2}) + i (y_{1} + y_{2})

z_{1} \cdot z_{2} = (x_{1} + i y_{1}) (x_{2} + i y_{2}) = (x_{1} x_{2} - y_{1} y_{2}) + i (x_{1} y_{2} + x_{2} y_{1})

\frac{z_{1}}{z_{2}} = \frac{(x_{1} + i y_{1}) (x_{2} - i y_{2})}{(x_{2} + i y_{2}) (x_{2} - i y_{2})} = \frac{(x_{1} + i y_{1}) (x_{2} - i y_{2})}{(x_{2}^{2} + y_{2}^{2})}

x = R e (z)

y = I m (z)

$\bar{z} = x - i y$ complex conjugate of $z$

Modulus:
$| z | = \sqrt{x^{2} + y^{2}} = \sqrt{z \cdot \bar{z}}$

Crucial to the Triangle Inequality.