Section 1.1. The Complex Numbers and the Complex Plane

• Complex numbers. Definition of complex numbers ($z=x+yi$), real and complex parts of complex numbers ($x=\Re z$, $y=\Im z$). Operations: addition, conjugation, multiplication, division; their properties. Absolute value of (modulus) of a complex number ($|z|$), properties.
• Polar Representation
A polar angle of the complex number ($\arg z$ and $\Arg z$). Properties of the modulus and the polar angle.
• Complex Numbers as Vectors

$\mathbb{Z}$ -- set of integers;

$\mathbb{N}=\{1,2,3,\ldots, \}$ -- set of natural numbers;

$\mathbb{Q}$ -- set of rational numbers;

$\mathbb{R}$ -- set of real numbers (real line);

$\mathbb{C}$ -- set of complex numbers (complex plane);

Subsection 1.1.1. A Formal View of the Complex Numbers

• Formal definition

Section 1.2. Some Geometry

• The triangle inequality
• Straight lines
• Powers and roots of complex numbers
• Circles (by a circle we mean a circumference; the interior of it is called "a disk" ("the open disk" if it does not include the circumference and "the closed disk" if includes)

Section 1.3. Subsets of the Plane

• Open sets
• The boundary of a set
• Closed sets
• Connected sets
• The infinite point $\infty$ ("the point at infinity", or simply "infinity")

$\mathbb{C}^*=\mathbb{C}\cup\{\infty\}$ -- extended complex plane.