
# 2: The Complex Plane


To study geometry using Klein's Erlangen Program, we need to define a space and a group of transformations of the space. Our space will be the complex plane.

• 2.1: Basic Notions
The set of complex numbers is obtained algebraically by adjoining the number i to the set R of real numbers, where i is defined by the property that i^2=−1. We will take a geometric approach and define a complex number to be an ordered pair (x,y) of real numbers.
• 2.2: Polar Form of a Complex Number
In this section, we explore the polar form of a complex number and provide examples when r ≥ 0 and r < 0.
• 2.3: Division and Angle Measure
The division of the complex number z by w ≠ 0, denoted z/w, is the complex number u that satisfies the equation z = w ⋅ u. In practice, division of complex numbers is not a guessing game, but can be done by multiplying the top and bottom of the quotient by the conjugate of the bottom expression.
• 2.4: Complex Expressions
In this section we look at some equations and inequalities that will come up throughout the text.