1.2 Geometry of Complex Numbers

$a = α + i β$ can be represented with the coordinates $(α, β)$ where the first coordinate is on the real axis and the second coordinate is on the imaginary axis. The entire plane is the complex plane.

Addition is like vector addition.

Polar Coordinates

The polar coordinate of the points $(α, β)$ are

(r, φ) {\begin{cases} α = r \cos (φ) \\ β = r \sin (φ) \end{cases}

a = α + i β ⟹ a = r (\cos (φ) + i \sin (φ))

Where $arg(a) = φ$ and $r \geq 0 = | a |$

Euler form $a = r e^{i φ}$

Multiplication:

\begin{aligned} a_{1} a_{2} & = r_{1} r_{2} [\cos φ_{1} \cos φ_{2} - \sin φ_{1} \sin φ_{2} + i (\sin φ_{1} \cos φ_{2} + \cos φ_{1} \cos ϕ_{2})] \\ = r_{1} r_{2} [\cos (φ_{1} + φ_{2}) + i \sin (φ_{1} + φ_{2})] \\ ⟹ & \arg (a_{1} a_{2}) = \arg (a_{1}) + \arg (a_{2}) \\ also \arg (\frac{a_{2}}{a_{1}}) = \arg a_{2} - \arg a_{1} \end{aligned}

Area of triangle $(a, b, 0), a, b \in C$

Area = \frac{1}{2} | a | | b | \sin (γ)

Binomial Equation

From above theorems we get:

a^{n} = r^{n} (\cos (n φ) + i \sin (n φ))

a^{- 1} = r^{- 1} (\cos φ - i \sin φ) = r^{- 1} [\cos (- φ) + i \sin (- φ)]

De Moivre's Formula

r = 1, (\cos φ + i \sin φ)^{n} = \cos n φ + i \sin n φ

Solving $z^{n} = a$ if $a \neq 0$

\begin{aligned} a = r (\cos φ + i \sin φ), z = ρ (\cos θ + i \sin θ) \\ ⟹ ρ^{n} (\cos n θ + i \sin n θ) = r (\cos φ + i \sin φ) \\ ⟹ {\begin{cases} ρ^{n} = r \\ n θ = φ + k 2 π \end{cases} \\ ⟹ z = \sqrt[n]{r} (\cos \frac{φ}{n} + i \sin \frac{φ}{n}) \end{aligned}

or $θ = \frac{φ}{n} + \frac{k \cdot 2 π}{n}$ , $k = 0, 1, \dots, n - 1$

Roots of Unity

z^{n} = 1

ω = \cos \frac{2 π}{n} + i \sin \frac{2 π}{n}

all the roots can be expressed by $1, ω_{1}, ω_{2}, \dots, ω^{n - 1}$

All $n$ th roots $\sqrt[n]{a}$ can be expressed in the form $ω^{k} \cdot \sqrt[n]{z}, k = 0, 1, \dots, n - 1$

where $z$ is the principle root above ^